AnalysisofaNewHiddenAttractorCoupledChaoticSystemand … · 2020. 10. 1. · KxF Kxz K Kxz KxF /yapunovAonents K KxF Kx[ Kx9 Kx3 z zxF zx[ zx9 zx3 F f (b) Figure 3:(a)ebifurcationdiagramof(f)forsystem(2).(b)elyapunovexponentdiagramof(f)

Research ArticleAnalysis of a New Hidden Attractor Coupled Chaotic System andApplication of Its Weak Signal Detection

Wenhui Luo 1 Qingli Ou1 Fei Yu2 Li Cui1 and Jie Jin13

1Hunan University of Science and Technology Xiangtan 411201 Hunan China2School of Computer and Communication Engineering Changsha University of Science and TechnologyChangsha 410114 China3College of Information Science and Engineering Jishou University Jishou 416000 China

Correspondence should be addressed to Wenhui Luo wenhuiluomailhnusteducn

Received 1 October 2020 Revised 4 November 2020 Accepted 10 December 2020 Published 23 December 2020

Academic Editor Viet-ampanh Pham

Copyright copy 2020 Wenhui Luo et al ampis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to improve the complexity of the chaotic system and the accuracy of the weak signal detection this paper propose a newhidden attractor coupled chaotic system and a corresponding weak signal detection system which can be used to obtain the phasediagram of the proposed system using the fourth order of the Runge-Kutta methodampe dynamic behavior of the chaotic system isanalyzed through the bifurcation diagram Lyapunov exponent and power spectrumampe Lyapunov exponent is used to depict thebasins of attraction for the system After research it is discovered that symmetry exists in the system Comparative analysis hasdemonstrated that the system has higher detection accuracy and excellent antinoise performance Finally the circuit simulationand FPGA realization of the system indicated that the numerical simulation results are consistent with the FPGA implementationresults proving the theoretical analysis to be correct and the accuracy of the detection results

1 Introduction

After Lorenz introduced the Lorenz system [1] in 1963the chaos theory has become a hot spot of nonlinear fieldresearch With the deepening of theoretical researchresearchers have conducted a lot of analysis into theattractor and basins of attraction of the chaotic system In[2] after an investigation into the basins of attraction ofcoexistence attractors in the coupled Duffing system theriddled basins of attraction of the Chen system arestudied and the riddled basins of attraction have a pos-itive repulsive set of Lebesgue measures via mathematicalproofs It is found in [3] that the number of coexistingattractors is different when the initial values are differentA numerical method for estimating the shortening cal-culation time of the relative area of the basins of attractionis proposed in [4] and its application scope is verified In[5] the mixed basins of attraction formed when multipleattractors coexist in a chaotic system are analyzed and theresearch shows that the basins of attraction have a fractal

regular structure In addition the hidden attractors [6]and hidden bifurcations in the system have also attractedextensive attention of researchers For example thephenomenon of hidden attractor and hidden bifurcationin multiscroll Chua chaotic system is studied in [7] ampedynamic characteristics of multistable chaotic systemswith hidden attractors are studied in [8] A four-winghidden attractor chaos is studied in [9] In [10] a three-dimensional hidden attractor chaotic system with Lya-punov dimension of 29075 is studied and a Bluetoothdevice for autonomous wireless mobile robot based onchaotic motion controller is proposed At the same timeas a special state of chaotic system multistability has beenwidely studied ampe chaotic system with multiple attrac-tors coexisting and multistability are analyzed in [11 12]Reference [13] analyzed the multistability of a chaoticsystem with two circles of equilibrium points andimplemented the system using FPGA technology Withthe research of electronic components the realization ofmemristors provides a new method for the construction of

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 8849283 15 pageshttpsdoiorg10115520208849283

chaotic systems Nowadays chaotic systems based onmemristors [14ndash16] and chaotic neural networks [17ndash19]have received great attention from researchers

With the in-depth theoretical research of chaotic sys-tems their engineering application has become more ma-ture In recent years the application of chaotic systems hasspread throughout many fields including electronic circuit[20ndash24] image processing [25ndash30] secure communication[31ndash34] complex networks [35ndash38] and random signalgeneration [39ndash41] Traditional stochastic resonance [42]wavelet transform [43] and empirical mode decomposition[44] are widely used in early fault signal detection ampestochastic resonance method enhances the signal utilizingnoise destroying the original information of the signalImmune to certain noises and sensitive to specific signalsthe chaotic system is featured by simple detection methodsand retention of the original signal information In 1992Birx and Pipenberg first proposed detecting weak signals byusing chaotic oscillators [45] and since then the chaoticsystems have been represented by the Duffing system and thevan der Pol system [46 47] began to be widely applied inweak signal detection ampis research based on the timedomain and frequency domain detection methods for weaksignals [48 49] and the antinoise of chaotic system [50]enables chaotic system detection methods to be more ma-ture ampe criteria for judging the state of chaotic systems willdirectly affect the accuracy of detection thus scholars havealso conducted in-depth studies on the calculation of thethreshold of chaotic systems In [51] Melnikov method isused to identify chaotic features In [52] the threshold of thesystem is determined by solving the Lyapunov exponent Atpresent most of the researches on weak signals based onchaos theory use a single chaotic system for detection ampesystem proposed in [53] can only detect signals with a size of001 and the SNR can only reach minus4585 dB ampe system in[54] can detect signals with a size of 1 times 10minus 9 but a noisesignal that can be immune to it can only reach the order of1 times 10minus 9 ampe system in [55] can detect a signal with a widerfrequency range but the detection range of the initial phaseangle of the signal is too large and thus is unfavorable fordetermining the signal parameters At present most of theresearches on weak signals based on chaos theory use a singlechaotic system for detection Compared with the singlechaotic system the coupled system has excellent stabilityhigh antinoise performance and low detection thresholdamperefore the coupled chaotic system has high theoreticalresearch value and practical application value

To improve the chaotic system complexity and the ac-curacy of signal detection this paper proposes a new hiddenattractor coupled chaotic system amprough an analysis of thesystemrsquos phase diagram the Lyapunov index and so forththe complex dynamic behavior of the system is explained indetail Moreover the basins of attraction for the system aredrawn using the Lyapunov index to study the characteristicsof the systemrsquos attractor ampeoretical analysis indicates thatthe system is sensitive to initial values and immune to noiseand the coupled chaotic system is applied to weak signaldetection Finally through OrCAD circuit simulation andFPGA implementation the correctness of the theoretical

analysis and the accuracy of signal detection are verifiedamprough data analysis and experimental data the systemproposed in this paper can detect signals sized 1 times 10minus 6 andthe signal-to-noise ratio reaches minus73892 dB Moreover theimmune signal reached the order of 35 times 10minus 3 and the errorrange of the initial phase angle isplusmn 009 amperefore thedetection performance of the coupled system proposed inthis paper is better than the single chaotic system proposedin [45 46 53ndash55]

2 Coupled Chaotic System Model andDynamic Analysis

21 Coupled Chaotic System Model amprough [45ndash47] asingle chaotic system with different nonlinear terms is an-alyzed and the limitations of signal detection performancedetection are studied at the same time For example [53] candetect low signal amplitude and [54] can be immune ampenoise is small and [55] detects that the initial phase angleerror of the signal is too large Based on the above reasonsthis paper proposes a new system

eurox + c1 _x minus x + c2x5

+ k(x minus y) f cos(ωt + θ)

euroy + c3 1 minus y3

1113872 1113873 _y + y + y3

+ k(y minus x) f cos(ωt + θ)

⎧⎨

⎩

(1)

where k is the linear coupling coefficient of system (1)ampe larger the coefficient the higher the degree of cou-pling c1 c2 and c3 respectively represent the coeffi-cients of the nonlinear terms f cos(ωt + θ) is the drivingforce of the system and f ω and θ represent the am-plitude frequency and initial phase angle of the drivingforce respectively

22 Analysis of the Equilibrium Point and LyapunovExponent of a Coupled Chaotic System We obtained system(2) by coupling system (1) and taking the coefficientsc1 05 c2 02 c3 08 and k 04 with the state spacemodel expression of system (2) shown in the followingequation

_x z

_y q

_z minusc1z + x minus c2x5

minus k(x minus y) + f cos(ωt + θ)

_q minusc3 1 minus y2

1113872 1113873q minus y minus y3

minus k(y minus x) + f cos(ωt + θ)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

To simulate the coupled chaotic system (2) take thefrequency ω 1 and the initial phase angle θ 0 ampesystemrsquos initial value is (minus1 minus1 0 0) When the magnitude fof the driving force takes 03 046 08 and 12 thehomoclinic orbit the period-doubling bifurcation chaoticstate and periodic state will appear in system (2) ampedriving signal of system (2) is defined as H and the signal asa whole system (2) is analyzed Let the left side of theequation be equal to 0 then

2 Mathematical Problems in Engineering

z 0

q 0

minus05z + x minus 02x5

minus 04(x minus y) + H 0

minus08 1 minus y2

1113872 1113873q minus y minus y3

minus 04(y minus x) + H 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(3)

When H 1 equation (3) has a real number solutionx1 16299 y1 08060 z1 q1 0 and fourteen complexnumber solutions With the changes of H equation (3) willhave an infinite number of equilibrium points so system (2)is a hidden attractor coupled chaotic system ampe Jacobianmatrix of system (2) at the equilibrium pointM(x1 y1 z1 q1) is

J

0 0 1 0

0 0 0 1

1 minus x41 minus 04 04 minus05 0

04 minus14 minus 3y21 0 minus08 1 minus y

211113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

Let |J minus λI| 0 its characteristic values λ1 minus02486+25373i λ1 minus02486 minus 25373i λ3 minus01415 + 16983iand λ4 minus01415 minus 16983i can be obtained According tothe Routh-Hurwitz criterion the real parts of the eigenvaluesare all less than 0 so system (2) is stable at the equilibriumpoint (x1 y1 z1 q1) When H 1 the waveform diagram ofeach state variable phase changing over time and phasediagram within system (2) are shown in Figure 1

Derivation of equation (2) can achieve

nablaV z _x

zx+

z _y

zy+

z _z

zz+

z _q

zq

minus13 + 08y2

(5)

When y isin [minus12748 12748] nablaVlt 0 and system (2) is adissipative system

Let βωt and take ω 1 and θ 0 Convert system (2)into the form shown in the following equation

_x z

_y q

_z minus05z + x minus 02x5

minus 04(x minus y) + f cos β

_q minus08 1 minus y2

1113872 1113873q minus y minus y3

minus 04(y minus x) + f cos β

_β 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

and the previous equation can be expressed by vectormethod as

_X(t) w(X(t) η) (7)

ampe state space variable X(t) [x(t) y(t) z(t)

q(t) β(t)]T w [w1 w2 w3 w4 w5]T η represents the

parameter set and [middot middot middot]T represents the transposed matrixampe deviation δX equation from orbit X (t) can be expressedas δ _X(t) Lij(X(t) η)δX i j 1 2 5 Lij (zwizxj)

and the Jacobian matrix form is

J

0 0 1 0 0

0 0 0 1 0

1 minus x4

minus 04 04 minus05 0 minusf sin β

04 16qy minus 3y2

minus 14 0 minus08 1 minus y2

1113872 1113873 minusf sin β

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

When f 08 the Lyapunov exponents areLE1 0184134 LE2 0 LE3 minus0305903 LE4 minus0312796and LE5 minus0697360 respectivelyampe value of LE1 is greaterthan 0 and system (2) is in a chaotic state ampe sum ofLyapunov exponents is less than 0 and system (2) is in adissipative system which is consistent with the previousresults In a chaotic state the phase diagram of system (2)and the cross section diagram of Poincare are shown inFigure 2

3 Bifurcation of the Driving Force Parametersand Lyapunov Exponent Spectrum Analysis

As the various parameters of the driving force have a verysignificant impact on the dynamics of the system focus onthe analysis of the parameters f ω and θ was implementedand the initial values of system (2) take (minus1 minus1 0 0) on auniform basis When ω 1 θ 0 When the amplitude f

changes within the interval [0 2] the bifurcation diagramand Lyapunov exponent spectrum of system (2) are shownin the following diagrams 3(a) and 3(b) As shown inFigure 3(a) system (2) enters chaos from the period-dou-bling bifurcation the chaotic state interval is approximately[041 088] and the system is in a periodic state in thesubsequent interval According to the comparison betweenFigures 3(a) and 3(b) the intervals of the chaotic and pe-riodic states of the system are consistent

When f 08 θ 0 and when the frequency ω changesin the interval [0 3] the bifurcation diagram and Lyapunovexponent spectrum of system (2) are shown in Figures 4(a)and 4(b) As shown in Figure 4(a) the system enters achaotic state from a period-doubling bifurcation As seenfrom Figures 4(a) and 4(b) the change of frequency ω has amore significant impact on the system than the amplitude fampe bifurcation diagram of ω in the interval in [0 08] and[1 2] is shown in Figures 4(c) and 4(d) As seen from the

Mathematical Problems in Engineering 3

ndash15 ndash1 ndash05 0 05 1 15 2ndash2x

f = 08 ω = 1 θ = 0

ndash15

ndash1

ndash05

0

05

1

15

z

(a)

ndash15

ndash1

ndash05

0

05

1

15z


(b)

Figure 2 (a) X-y phase diagram of system (2) (b) Poincare section view of the x-z plane

Bifurcation diagram of the system (2)

ndash3

ndash2

ndash1

0

1

2

3

x

02 04 06 08 1812 14 16 20 1w = 1 θ = 0

(a)

ndash08ndash07ndash06ndash05ndash04ndash03ndash02ndash01

00102

Lyap

unov

expo

nent

s

02 04 06 08 1 12 14 16 18 20f

(b)

Figure 3 (a) ampe bifurcation diagram of (f ) for system (2) (b) ampe lyapunov exponent diagram of (f )

xy

zq

ndash15ndash1

ndash050

051

152

25

50 100 150 200 250 300 350 4000t

(a)

215

x y1050ndash05ndash1

x-zy-q

(x z) = (16299 0)

(y q) = (08060 0)

01020304050

t

2

z q

10

ndash1ndash2

(b)

Figure 1 (a) When H 1 system (2) state variables are x (y) z q and the waveform diagram (b) When (H) 1 system (2) xminus y and zminus qphase diagram


local bifurcation diagram a chaotic state occurs in system (2)within a small range of about ω 05 and a periodic stateappears within a small range of about ω 13

When ω 1 f becomes 08 and 1 respectively Accordingto the above analysis when ω 1 and f 08 system (2) is ina chaotic state When ω 1 and f 1 system (2) is in aperiodic state In the two different kinetic states of system(2) the initial phase angle θ is made to change at an interval[0 2π] and the bifurcation diagram and Lyapunov exponentdiagram are shown in Figure 5 As seen from the analysis inFigure 5 when system (2) is in a chaotic state with a changeto the initial phase angle θ the dynamic state of system (2)does not change ampis is also the case when system (2) is in aperiodic stateamperefore the change of the initial phase angleθ of the driving force will not cause a change in the dynamicstate of system (2)

4 Basins of Attraction

According to the set limit of system dynamics the phase spaceformed by all possible initial points is divided into severaldisjoint subsets of the same dimension and these subsets arecalled the basins of attraction When a system with complexdynamics has multiple attractors the boundary of the basinsof attraction may have fractal characteristics Within a certaintime range we compare the transverse maximum Lyapunovindex of system (2) to the Lyapunov index of a particular trackand further obtain the image of the basins of attraction for x(0)minus y (0) and x (0)minus z) (0) for system (2) ampe initial valuesare set to [x (0) y (0) 0 0] x(0) isin [minus5 5] y(0) isin [minus5 5] and[x (0) minus1 z (0) 0] x(0) isin [minus5 5] z(0) isin [minus5 5] respec-tively As shown in Figure 6 the basins of attraction formedby the attractor in system (2) have a complex basins structureincluding both riddled basins of attraction and mixed basinsof attraction In Figure 7 the basins of attraction are featuredby an overall symmetry and global mixing In Figures 6 and 7the red area represents the basins of attraction of the attractorat infinity that is the point set where the trajectory of thesystem diverges ampe yellow area represents the basins ofattraction of the chaotic attractor that is the overall system isstable andmultiple attractors coexistampe blue area representsthe transition state

As far as current research is concerned [56 57] theconditions for the existence of basins of attraction in anattractor are as follows

(i) ampere is a smooth and invariant subspace con-taining chaotic attractors

(ii) ampere is another asymptotic final state outside theinvariant subspace (not necessarily chaotic)

(iii) ampe horizontal Lyapunov exponent of the invariantsubspace is negative

(iv) ampe positive finite time change is associated with thelateral stability of the unstable periodic orbit of theattractor

According to research in [58] the area of the basins ofattraction will change with the initial value ampe research in[5] shows that when a sieve shape appears in the basins of

attraction for the Chen system the existence of a repulsiveset of positive Lebesgue measures is obtained throughmathematical justification As seen in Figure 6 the change ofthe basins of attraction is affected by the initial value and thearea of |y|gt 2 will show riddled basins of attraction indi-cating that the attractor contains the repulsive set of thepositive Lebesgue measures in the neighborhood of this areaampe overall symmetry of the basins of attraction alsomanifests the synchronization of the system Further re-search shows that it is impossible to predict the asymptoticattractor of the complex basins of attraction ampe basins ofattraction are shown in Figure 6 when y is in the intervals[minus15 5] and [2 5] the initial value of the attractor will beunpredictableampere are six attractors in a coupled system in[6] wherein the basins of attraction of two attractors are notconnected and basins of attraction of the remaining twosynchronous attractors and two asynchronous attractorsconstitute a basin of attraction with a symmetrical mixingdistribution When 35gt |y|gt 2 there will be a yellow areanot connected to the left and right and there is also a verynarrow blue area In this paper whether the basins of at-traction of different connected states represent the basins ofattraction of different attractors the relationship betweenthe characteristics of the sieve shape of the connected areaand the state changes of coexisting attractors should befurther studied ampe above analysis proves that there aremathematical conditions for the existence of hiddenattractors and riddled basins of attraction in system (2) Wealso show that the horizontal Lyapunov exponents of hiddenattractors are locally unstable

As seen from Figure 7 the basins of attraction are asymmetrical figure and based on fractal theory are self-similar According to Milnorrsquos generalization of the classicdefinition of attractors assuming that D is a smoothcompact manifold and bounded η represents the Lebesguemeasure ampe compact invariant set A on D satisfies thefollowing conditions

(1) Basins of attraction σ(A) x x isin dω(x) sub A with a positive measure ((A))gt 0

(2) ampere is no set Aprime making AprimeA η(σ(Aprime)) minus η(σ(A)) 0

A is called an attractor ampat is when A has an attractionset of positive Lebesgue measures the attractor will have anattractive neighborhood Based on studies of attractors if allxσ(A) and δ gt 0 it was discovered that when attractor A hasriddled basins of attraction Φ (A) it satisfies

η σδ(x)cap σ(A)( 1113857η σδ(x)cap σ(A)c

( 1113857gt 0 (9)

where σδ(x) is the δ neighborhood of x As seen from thiswhen the basin of attraction formed by the attractor has asieve-shaped characteristic any neighborhood of theattractor contains the attraction set of the positive Lebesguemeasures and the repulsive set of positive Lebesguemeasures In the riddled basins of attraction there are pointsthat are attracted by the attractor as well as points thatdiverge


Our research from [6] shows that when the systemcoexistence attractor exhibits synchronous changes theattraction basinsrsquo structure is symmetrical ampe researchfrom [59] on the basins of attraction of coexistence attractorsshows that when the system has multiple attractors thebasins of attraction formed by attractors will develop amixing conditionampe basins of attraction shown in Figure 7also manifest with the characteristics of mixing and sym-metry simultaneously ampe study of the two subsystems inthis system in this paper shows that the attractors of the twosubsystems maintain synchronization between chaos pe-riods Based on the above references and the research in thispaper the structure of the basins of attraction in the coupledsystem can reflect a synchronization of the system to acertain extent Furthermore the structure of the attractor

and the structure of the basins of attraction can also bemapped to each other under certain circumstances that isthe symmetry of the basins of attraction can reflect thechanging features of the system and vice versa

5 Application of Coupled System in WeakSignal Detection

According to the analysis in Figure 3 when the driving forcefrequency ω is a constant value the state of system (2) willchange from a chaotic state to a periodic state as the am-plitude f changes Weak signals can be detected by using thechange to system (2) state amperefore this paper proposes adetection system corresponding to system (2) as shown inthe following equation

_x z

_y q


minus 04(x minus y) + f0 cos ω0t + θ0( 1113857 +[b cos(ωt + θ) + S(t)]


1113872 1113873q minus y minus y3

minus 04(y minus x) + f0 cos ω0t + θ0( 1113857 +[b cos(ωt + θ) + S(t)]

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)

In system (10) bcos (ωt+θ) represents a weak signal andS (t) represents a noise signal f0 cos(ω0t + θ0) represents

the driving force of the system First the driving forceamplitude f0 is adjusted so that the system is at a critical


ndash3

ndash2

ndash1

0

1

2

3

x

15105 2 25 30ω

(a)

05 1 15 2 25 30ω

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

03 04 0701 05 0602 080ω

(c)


ndash3

ndash2

ndash1

0

1

2

3

x

1412 16 181 2ω

(d)

Figure 4 (a) ampe ω bifurcation diagram of system (2) (b) ampe Lyapunov exponent diagram of ω (c) ampe partially enlarged diagram ofbifurcation of ω in the interval [0 08] (d) ampe partially enlarged diagram of bifurcation of ω in the interval [1 2]



ndash25ndash2

ndash15ndash1

ndash050

051

152

25x

1 53 4 60 2θ

(a)

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

1 53 4 60 2θ

(c)

ndash025

ndash02

ndash015

ndash01

ndash005

0

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(d)

Figure 5 (a) ampe bifurcation diagram of θ when f 08 and ω 1 (b) ampe Lyapunov exponent diagram of θ when f 08 and ω 1 (c) ampebifurcation diagram of θ when f 1 and ω 1 (d) ampe Lyapunov exponent diagram of θ when f 1 and ω 1

ndash4 ndash3 ndash2 ndash1 0 1 2 3 4 5ndash5x (0)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

y (0)

Figure 6 Diagram of x (0)minus y (0) basins of attraction for system (2)


point of state transition and at this time f0 is called thesystem threshold When the driving force amplitudef0 08747909 system (10) is in a critical state

51 Research on the Influence of Weak Signal Parameters onthe System When the frequency ω0 1 of system (10) istaken the initial phase angle θ0 0 When amplitudef0 08747909 Lyapunov exponent values LE1 0026845LE2 0 LE3 minus0256585 LE4 minus0256435 andLE5 minus0508001 At this time the phase diagram and thepower spectrum of system (10) are shown in Figure 8Research shows that the power spectrum of the periodicsignals is a discrete spectrum ampe power spectrum ofnonperiodic signals is a continuous spectrum ampe chaoticsignal is a nonperiodic signal and the power spectrum is acontinuous spectrumamperefore the state of system (10) maybe determined based on the power spectrum When theamplitude f0 is adjusted to 08747909 the phase diagram andpower spectrum of system (10) are shown in Figure 8According to Figure 8 the x-z phase diagram of system (10)is in a chaotic state and the power spectrum is continuoustherefore system (10) is in a chaotic state

At this time after a weak signal with a frequency ω 1initial phase angle θ 0 and amplitude b 1 times 10minus 6 is added tosystem (10) Lyapunov index values of the system areLE1 minus0014575LE2 0LE3 minus0251969LE4 minus0253323and LE5 minus0490388 ampe phase diagram and the powerspectrum for system (10) are shown in Figure 9 As seen fromFigure 9 when a weak signal with an amplitude 1 times 10minus 6 isadded to system (10) the x-z phase diagramof system (10) is in aperiodic state and the power spectrum is a discrete spectrumand the state of system (10) becomes periodic

Because system (10) can only detect weak signals in thesame or similar interval as the driving force frequency andinitial phase angle the research system can detect the fre-quency range of the weak signal and take the driving forcefrequency ω0 1 and the initial phase angle θ0 0 the

initial phase angle of the weak signal θ 0 and the amplitudeb 1 times 10minus 6 By calculating ω in [06 13] we get system (10)Lyapunov exponent data when the frequency ω changes inTable 1 As seen in Table 1 when the frequency ω of the weaksignal is within [08 11] the dynamic state of system (10)will change

ampe studied system (10) can detect the range of the initialphase angle θ of a weak signal taking the systemrsquos drivingforce frequency ω0 1 the initial phase angle θ0 0 theweak signal frequency ω 1 and the amplitude b 1 times 10minus 6ampe driving force f0 of the system and the weak signal b areto be measured as a whole when A is input into the system

A(t) f0 cos ω0t + θ0( 1113857 + b cos(ωt + θ)

f0 cosω0t cos θ0 minus f0 sinω0t sin θ0 + b cosωt cos θ

minus b sinωt sin θ

(11)

When ω0 ω

A(t) f0 cos θ0 + b cos θ( 1113857cosωt

minus f0 sin θ0 + b sin θ( 1113857sinωt

F(t)cos(ωt + φ(t))

(12)

In equation (10) F(t)

f20 + 2f0b cos(θ0 minus θ) + b2

1113969

φ(t) arctan((f0 sin θ0 + b sin θ)(f0 cos θ0 + b cos θ))When F(t)ge 08747919 the state of system (10) will

change from a chaotic state to a periodic state Whenb 1 times 10minus 6 the frequency of the driving force ω 1 andthe initial phase angle θ0 1deg to obtain 091deg le θle 109degAs seen from the research conclusion of the initial phaseangle of the driving force of system (2) the threshold ofsystem f0 does not change when the initial phase anglechanges and the weak signals of different initial phase anglescan be detected by adjusting the initial phase angle of thesystemrsquos driving force According to the above analysis thesystem has a higher detection accuracy for the unknownweak signal and at the moment the relative error of theinitial phase angle measurement does not exceed plusmn009

52 Research on the Antinoise Performance of System (10)ampe antinoise performance of system (10) was studied byadding Gaussian white noise When excessive noise wasadded the phase diagram of system (10) converging in theperiodic state will also become disorderly amperefore theantinoise performance of system (10) can be studied bydrawing the phase diagram of system (10) under noiseconditions ampe regulation system is in a critical state AfterGaussian white noise power ps(t) 00035 was added tosystem (10) the system dynamic state did not change asshown in Figure 10(a) ampen a weak signal was added tosystem (10) and the state of the system changes at this timeas shown in Figure 10(b) As seen in Figure 10(b) the addednoise has no relevant impact on the stability of the systemamperefore the system signal-to-noise ratio SRN is


ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

z (0)

Figure 7 ampe x (0)minusz (0) basins of attraction diagram for system(2)


SRN 10lgpb

pS(t)

1113888 1113889

10lg 051 times 10minus 6

1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)

In [53] only a signal with a size of 001 can be detectedand the signal-to-noise ratio can only reach minus4585 dB ampesystem in [54] can detect signals with the size of 1 times 10minus 9 butthe noise signal that the system can be immune to can onlyreach a size of 1 times 10minus 9 ampe system in [55] can detect signalsof a larger frequency range but the fluctuation range of the

initial phase angle from the measured signal is too large andthe error range is plusmn606 which is not conducive to con-firmation of signal parameters ampe system proposed in thispaper can detect signals sized 1 times 10minus 6 and the signal-to-noise ratio reaches -73892 dB Moreover the immune signalreached the order of 35 times 10minus 3 and the error range of theinitial phase angle is plusmn009 In summary the system in-troduced in this paper has superior detection performance

6 Circuit Design and Realization of System

61 Circuit Design of System According to the circuitprinciple and the characteristics of electronic components

4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)

Figure 8 (a) Phase diagram of system (10) when a weak signal is not added (b) Power spectrum diagram of system (10) when a weak signalis not added

4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)

Figure 9 (a) Phase diagram of system (10) after a weak signal was added (b) Power spectrum of system (10) after a weak signal is added

Table 1 Lyapunov exponent statistics table of frequency ω withing the interval [07 13]

ω LE1 LE2 LE3 LE4 LE5 System state

06 0054782 0 minus0253654 minus0264728 minus0542015 Chaotic state07 0001082 0 minus0253727 minus0259955 minus0506094 Chaotic state08 0 minus0010830 minus0254174 minus0253927 minus0493878 Periodic state09 0 minus0025991 minus0250020 minus0251349 minus0478236 Periodic state11 0 minus0029113 minus0251380 minus0250912 minus0474047 Periodic state12 0069098 0 minus0272891 minus0274010 minus0577603 Chaotic state13 0015809 0 minus0259136 minus0261232 minus0521500 Chaotic state


400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)

Figure 10 (a) Phase diagram of system (10) after adding Gaussian white noise (b) Phase diagram of system (10) after adding a weak signal inthe presence of Gaussian white noise

cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2

Figure 11 Circuit diagram schematic for system (2)


the circuit equation (14) as shown in system (2) can beobtained as follows

dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

ampe circuit for equation (14) is shown in Figure 11System (2) is compared with equation (14) to make theircoefficients equal and the selected device parametersC1 C2 C3 C4 47nF R2 R3 R5 R6 R14 R15

R18 R24 R25 1kΩ R1 R4 R13 R23 10kΩ R11 R12 12kΩ R21 R22 14kΩ R7 24kΩ R8 2kΩR9 6Ω R10 3kΩ R16 175KΩ R17 175ΩR19 14Ω and R20 35kΩ ampe operational amplifier

V (x)

ndash20V

ndash10V

0V

10V

20V

10 ms 20 ms 30 ms 40 ms 50 ms 60 ms 70 ms 80 ms0 sTime

(a)

V (z)

ndash20V

ndash10V

0V

10V

20V

ndash8V ndash6V ndash4V ndash2V 0V 2V 4V 6V 8V 10Vndash10VV (x)

(b)

Figure 12 (a) Oscillogram of system (2) variable x (b) x-z phase diagram of system (2)

ndash20V

ndash10V

0V

10V

20V

ndash8V ndash6V ndash4V ndash2V 0V 2V 4V 6V 8V 10V 12Vndash10VV (x)

V (z)

(a)

ndash10V

ndash5V

0V

5V

10V

ndash60V ndash40V ndash20V 0V 20V 40V 60V 80Vndash80VV(x)

V (z)

(b)

Figure 13 (a) x-z phase diagram of (10) without a weak signal (b) x-z phase diagram of system (10) with a weak signal


model selected by the circuit is TL082 and the multipliermodel is AD633

62 System Circuit Simulation and FPGA ImplementationIn this paper OrCAD was used to simulate the systemcircuit the amplitude was set as f 8V and the frequencyφ 400Hz with the waveform diagram of the state variablex and the phase diagram of x-z shown in Figure 12

Test the detection performance of system (10) take thefrequency φ 400Hz and adjust the amplitudef 135664362V with system (10) in a chaotic state as shown inFigure 13(a) When the driving force amplitude f 135664372is adjusted the state of system (10) changes to a periodic stateas shown in Figure 13(b)amperefore system (10) can effectivelydetect the sine signal with an amplitude of 1 times 10minus 6 V

Based on FPGA technology and fixed-point numbermethod hardware experiment is carried out on system (10)We use Xilinx zynq-7000 XC7Z020 FPGA chip and AD9767dual-port parallel 14-bit AD module with the highestconversion rate of 125MHz ampe software used is Vivado174 and system generator compiler is used to realize thejoint debugging of MATLAB and FPGA In addition we alsouse SIGLENT 1202X digital oscilloscope to visualize analogoutput After compiling the automatically generated RTLview is shown in Figure 14

ampe regulation system was in a critical state as shown inFigure 15(a) After adding a weak signal the phase diagramof system (10) is shown in Figure 15(b) and the state ofsystem (10) changes amperefore system (10) can effectivelydetect weak signals

(a) (b)

Figure 15 (a) x-z phase diagram of system (10) before adding a weak signal (b) x-z phase diagram of system (10) after adding a weak signal

rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0

0dac_z_data[130]_OBUF_inst

dac_x_data[130]_OBUF_inst

OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk

Figure 14 RTL view based on FPGA implementation


7 Conclusion

amprough calculation this paper concludes that system (2)has an infinite number of equilibrium points and there is aLyapunov exponent greater than 0amperefore system (2) is ahidden attractor coupled chaotic system ampe analysis of thebasins of attraction shows that system (2) has the charac-teristics of structural symmetry and the state of the attractorchanges synchronously Further research finds that theLyapunov exponent of the hidden attractor is locally un-stable ampe research on the Lyapunov indexes of all pa-rameters of system (10) driving force has proved that system(10) can more accurately detect signals with a size of1 times 10minus 6 and the research on system (10) with noise showsthat system (10) has excellent antinoise performance Finallycircuit simulation and FPGA implementation have dem-onstrated the theoretical analysis and the accuracy of weaksignal detection to be correct

In future research the focus will be on how to detectdifferent types of signals how to improve the systemrsquosantinoise performance and identifying the systemrsquos state Asthere is no unified conclusion on the structure of the basinsof attraction the research results should be further verifiedMoreover no unified results have been achieved in thetheoretical research on the structure of the basins of at-traction and thus the research on the basins of attraction willalso be a hot spot in the future

Data Availability

ampe data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported by the National Natural ScienceFoundation of China under Grants 61504013 and 61561022the Natural Science Foundation of Hunan Province underGrants 2019JJ50648 2020JJ4315 and 2017JJ3254 and Hu-nan Provincial Education Department under Grant 20B216

References

[1] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[2] Z Liu Y Li and G Chen ldquoampe basin of attraction of the Chenattractorrdquo Chaos Solitons amp Fractals vol 34 no 5pp 1696ndash1703 2007

[3] S P Raj S Rajasekar and K Murali ldquoCoexisting chaoticattractors their basin of attractions and synchronization ofchaos in two coupled Duffing oscillatorsrdquo Physics Letters Avol 264 no 4 pp 283ndash288 1999

[4] J A Wright J H B Deane M Bartuccelli et al ldquoBasins ofattraction in forced systems with time-varying dissipationrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 29 no 1-3 pp 72ndash87 2015

[5] G Wang F Yuan G Chen et al ldquoCoexisting multipleattractors and riddled basins of a memristive systemrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 28no 1 Article ID 013125 2018

[6] A Bayani K Rajagopal A J M Khalaf S JafariG D Leutcho and J Kengne ldquoDynamical analysis of a newmultistable chaotic system with hidden attractor anti-monotonicity coexisting multiple attractors and offsetboostingrdquo Physics Letters A vol 383 no 13 pp 1450ndash14562019

[7] L Xiong S Zhang Y Zeng and B Liu ldquoDynamics of a newcomposite four-Scroll chaotic systemrdquo Chinese Journal ofPhysics vol 56 no 5 pp 2381ndash2394 2018

[8] H Jahanshahi A Yousefpour J M Munoz-PachecoI Moroz Z Wei and O Castillo ldquoA new multi-stablefractional-order four-dimensional system with self-excitedand hidden chaotic attractors dynamic analysis and adaptivesynchronization using a novel fuzzy adaptive sliding modecontrol methodrdquo Applied Soft Computing vol 87 p 1059432020

[9] Q Deng C Wang and L Yang ldquoFour-wing hidden attractorswith one stable equilibrium pointrdquo International Journal ofBifurcation and Chaos vol 30 no 06 p 2050086 2020

[10] Sundarapandian Vaidyanathan Aceng Sambas MustafaMamat and W S Mada Sanjaya ldquoA new three-dimensionalchaotic system with a hidden attractor circuit design andapplication in wireless mobile robotrdquo Archives of ControlSciences vol 27 no LXIII pp 541ndash554 2017

[11] Yu Fei Li Liu H Shen et al ldquoDynamic analysis Circuitdesign and Synchronization of a novel 6D memristiv four-wing hyperchaotic system with multiple coexisting attrac-torsrdquo Complexity vol 2020 Article ID 5904607 2020

[12] F Yu L Liu H Shen et al ldquoMultistability analysis coexistingmultiple attractors and FPGA implementation of Yu-Wangfour-wing chaotic systemrdquo Mathematical Problems in Engi-neering vol 2020 Article ID 7530976 2020

[13] A Sambas S Vaidyanathan E Tlelo-Cuautle et al ldquoA novelchaotic system with two circles of equilibrium points mul-tistability electronic circuit and FPGA realizationrdquo Elec-tronics vol 8 no 11 Article ID 1211 2019

[14] C Wang H Xia and L Zhou ldquoA memristive hyperchaoticmultiscroll jerk system with controllable scroll numbersrdquoInternational Journal of Bifurcation and Chaos vol 27 no 06Article ID 1750091 2017

[15] R Wu and C Wang ldquoA new simple chaotic circuit based onmemristorrdquo International Journal of Bifurcation and Chaosvol 26 no 09 Article ID 1650145 2016

[16] C Wang X Hu and L Zhou ldquoImplementation of a newmemristor-based multiscroll hyperchaotic systemrdquo Pramanavol 88 no 2 p 34 2017

[17] H Lin C Wang W Yao and Y Tan ldquoChaotic dynamics in aneural network with different types of external stimulirdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 90 Article ID 105390 2020

[18] W Yao C Wang Y Sun C Zhou and H Lin ldquoExponentialmultistability of memristive Cohen-Grossberg neural net-works with stochastic parameter perturbationsrdquo AppliedMathematics and Computation vol 386 p 125483 2020

[19] H Lin C Wang and Y Tan ldquoHidden extreme multistabilitywith hyperchaos and transient chaos in a Hopfield neuralnetwork affected by electromagnetic radiationrdquo NonlinearDynamics vol 99 pp 2369ndash2386 2020


[20] V T Pham D S Ali N M G Al-Saidi K RajagopalF E Alsaadi and S Jafari ldquoA novel mega-stable chaoticcircuitrdquo Radioengineering vol 29 no 1 pp 140ndash146 2020

[21] V -T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[22] J Jin and L Cui ldquoFully integrated memristor and its appli-cation on the scroll-controllable hyperchaotic systemrdquoComplexity vol 2019 Article ID 4106398 2020

[23] J Jin ldquoProgrammable multi-direction fully integrated chaoticoscillatorrdquo Microelectronics Journal vol 75 pp 27ndash34 2018

[24] J Jin and L Zhao ldquoLow voltage low power fully integratedchaos generatorrdquo Journal of Circuits Systems and Computersvol 27 no 10 Article ID 1850155 2018

[25] J Sun Mu Peng F Liu and C Tang ldquoProtecting compressiveghost imaging with hyper-chaotic system and DNA encod-ingrdquo Complexity vol 2020 Article ID 8815315 2020

[26] C Xu J Sun and C Wang ldquoA novel image encryption al-gorithm based on bit-plane matrix rotation and hyper chaoticsystemsrdquoMultimedia Tools and Applications vol 79 no 9-10pp 5573ndash5593 2020

[27] Xi Chen S Qian Yu Fei et al ldquoPseudorandom numbergenerator based on three kinds of four-wing memristivehyperchaotic system and its application in image encryptionrdquoComplexity vol 2020 Article ID 8274685 2020

[28] S Wang C Wang and C Xu ldquoAn image encryption algo-rithm based on a hidden attractor chaos system and theKnuth-Durstenfeld algorithmrdquo Optics and Lasers in Engi-neering vol 128 p 105995 2020

[29] A Sambas S Vaidyanathan E Tlelo-Cuautle et al ldquoA 3-Dmulti-stable system with a peanut-shaped equilibrium curvecircuit design FPGA realization and an application to imageencryptionrdquo Ge Institute of Electrical and Electronics Engi-neers Access vol 8 pp 137116ndash137132 2020

[30] Y Huang L Huang Y Wang Y Peng and F Yu ldquoShapesynchronization in driver-response of 4-D chaotic system andits application in image encryptionrdquoGe Institute of Electricaland Electronics Engineers Access vol 8 pp 135308ndash1353192020

[31] L Zhou F Tan and F Yu ldquoA robust synchronization-basedchaotic secure communication scheme with double-layeredand multiple hybrid networksrdquo Ge Institute of Electrical andElectronics Engineers Systems Journal vol 14 no 2pp 2508ndash2519 2020

[32] L Zhou F Tan F Yu and W Liu ldquoCluster synchronizationof two-layer nonlinearly coupled multiplex networks withmulti-links and time-delaysrdquo Neurocomputing vol 359pp 264ndash275 2019

[33] F Peng X Zhu and M Long ldquoAn ROI privacy protectionscheme for H 264 video based on FMO and chaosrdquo GeInstitute of Electrical and Electronics Engineers Transactions onInformation Forensics and Security vol 8 no 10 pp 1688ndash1699 2013

[34] F Yu S Qian X Chen et al ldquoA new 4D four-wing mem-ristive hyperchaotic system dynamical analysis electroniccircuit design shape synchronization and secure communi-cationrdquo International Journal of Bifurcation and Chaosvol 30 no 10 Article ID 2050147 2020

[35] J Jin L Zhao M Li F Yu and Z Xi ldquoImproved zeroingneural networks for finite time solving nonlinear equationsrdquoNeural Computing and Applications vol 32 no 9pp 4151ndash4160 2020

[36] F Yu L Liu L Xiao K Li and S Cai ldquoA robust and fixed-time zeroing neural dynamics for computing time-variantnonlinear equation using a novel nonlinear activation func-tionrdquo Neurocomputing vol 350 pp 108ndash116 2019

[37] J Jin ldquoA robust zeroing neural network for solving dynamicnonlinear equations and its application to kinematic controlof mobile manipulatorrdquo Complex amp Intelligent Systems 2020

[38] J Jin and J Gong ldquoAn interference-tolerant fast convergencezeroing neural network for dynamic matrix inversion and itsapplication to mobile manipulator path trackingrdquo AlexandriaEngineering Journal 2020

[39] F Yu L Liu S Qian et al ldquoChaos-based application of anovel multistable 5D memristive hyperchaotic system withcoexisting multiple attractorsrdquo Complexity vol 2020 ArticleID 8034196 2020

[40] F Yu QWan J Jin et al ldquoDesign and FPGA implementationof a pseudorandom number generator based on a four-wingmemristive hyperchaotic system and Bernoulli maprdquo GeInstitute of Electrical and Electronics Engineers Access vol 7pp 181884ndash181898 2019

[41] F Yu L Li Q Tang et al ldquoA survey on true random numbergenerators based on chaosrdquo Discrete Dynamics in Nature andSociety vol 2019 Article ID 2545123 2019

[42] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detection methodology and its application onincipient fault diagnosis and ECG signal analysisrdquo SignalProcessing vol 98 pp 74ndash87 2014

[43] Y Wang G Xu L Liang and K Jiang ldquoDetection of weaktransient signals based on wavelet packet transform andmanifold learning for rolling element bearing fault diagnosisrdquoMechanical Systems and Signal Processing vol 54-55pp 259ndash276 2015

[44] L Lu J Yan and C W de Silva ldquoDominant feature selectionfor the fault diagnosis of rotary machines using modifiedgenetic algorithm and empirical mode decompositionrdquoJournal of Sound and Vibration vol 344 pp 464ndash483 2015

[45] D L Birx and S J Pipenberg ldquoChaotic oscillators andcomplex mapping feed forward networks (CMFFNS) forsignal detection in noisy environmentsrdquo Ge Institute ofElectrical and Electronics Engineers in Proceedings 1992]IJCNN International Joint Conference on Neural Networksvol 2 pp 881ndash888 Baltimore MD USA January 1992

[46] Z-H Lai and Y-G Leng ldquoWeak-signal detection based onthe stochastic resonance of bistable Duffing oscillator and itsapplication in incipient fault diagnosisrdquo Mechanical Systemsand Signal Processing vol 81 pp 60ndash74 2016

[47] Y Jiang H Zhu and Z Li ldquoA new compound faults detectionmethod for rolling bearings based on empirical wavelettransform and chaotic oscillatorrdquo Chaos Solitons amp Fractalsvol 89 pp 8ndash19 2016

[48] X Liu and X Liu ldquoWeak signal detection research based onduffing oscillator used for downhole communicationrdquo Jour-nal of Computers vol 6 no 2 pp 359ndash367 2011

[49] F Yang L Jing W Zhang Y Yan and H Ma ldquoExperimentaland numerical studies of the oblique defects in the pipes usinga chaotic oscillator based on ultrasonic guided wavesrdquo Journalof Sound and Vibration vol 347 pp 218ndash231 2015

[50] P Kumar S Narayanan and S Gupta ldquoInvestigations on thebifurcation of a noisy Duffing-van der Pol oscillatorrdquo Prob-abilistic Engineering Mechanics vol 45 pp 70ndash86 2016

[51] N Chunyan and S Yaowu ldquoampe research of Chaotic char-acteristic identification base on weak signal detectionrdquo ACTAMetrological Sinica vol 10 no 4 pp 308ndash313 2000


[52] T Xie X Wei and R Yu ldquoNoise immunity analysis in ex-ternal excitation chaotic oscillator detecting system[C]2010International Conference on Intelligent System Design andEngineering Applicationrdquo Ge Institute of Electrical andElectronics Engineers vol 1 pp 1013ndash1016 2010

[53] Z Zhao and S Yang ldquoApplication of vanderPol-Duffingoscillator in weak signal Detectionrdquo Computers and ElectricalEngineering vol 41 pp 1ndash8 2015

[54] A Gokyildirim Y Uyaroglu and I Pehlivan ldquoA novel chaoticattractor and its weak signal detection applicationrdquo Optikvol 127 no 19 pp 7889ndash7895 2016

[55] Y Zhang H Mao H Mao and Z Huang ldquoDetection thenonlinear ultrasonic signals based on modified Duffingequationsrdquo Results in Physics vol 7 pp 3243ndash3250 2017

[56] U Chaudhuri and A Prasad ldquoComplicated basins and thephenomenon of amplitude death in coupled hidden attrac-torsrdquo Physics Letters A vol 378 no 9 pp 713ndash718 2014

[57] S Camargo R L Viana and C Anteneodo ldquoIntermingledbasins in coupled Lorenz systemsrdquo Physical Review E vol 85no 3 Article ID 036207 2012

[58] B L Lan and C Yapp ldquoDissipative relativistic standard mapperiodic attractors and basins of attractionrdquo Chaos Solitons ampFractals vol 37 no 5 pp 1300ndash1304 2008

[59] C Chen J Chen H Bao M Chen and B Bao ldquoCoexistingmulti-stable patterns in memristor synapse-coupled Hopfieldneural network with two neuronsrdquo Nonlinear Dynamicsvol 95 no 4 pp 3385ndash3399 2019


chaotic systems Nowadays chaotic systems based onmemristors [14ndash16] and chaotic neural networks [17ndash19]have received great attention from researchers

With the in-depth theoretical research of chaotic sys-tems their engineering application has become more ma-ture In recent years the application of chaotic systems hasspread throughout many fields including electronic circuit[20ndash24] image processing [25ndash30] secure communication[31ndash34] complex networks [35ndash38] and random signalgeneration [39ndash41] Traditional stochastic resonance [42]wavelet transform [43] and empirical mode decomposition[44] are widely used in early fault signal detection ampestochastic resonance method enhances the signal utilizingnoise destroying the original information of the signalImmune to certain noises and sensitive to specific signalsthe chaotic system is featured by simple detection methodsand retention of the original signal information In 1992Birx and Pipenberg first proposed detecting weak signals byusing chaotic oscillators [45] and since then the chaoticsystems have been represented by the Duffing system and thevan der Pol system [46 47] began to be widely applied inweak signal detection ampis research based on the timedomain and frequency domain detection methods for weaksignals [48 49] and the antinoise of chaotic system [50]enables chaotic system detection methods to be more ma-ture ampe criteria for judging the state of chaotic systems willdirectly affect the accuracy of detection thus scholars havealso conducted in-depth studies on the calculation of thethreshold of chaotic systems In [51] Melnikov method isused to identify chaotic features In [52] the threshold of thesystem is determined by solving the Lyapunov exponent Atpresent most of the researches on weak signals based onchaos theory use a single chaotic system for detection ampesystem proposed in [53] can only detect signals with a size of001 and the SNR can only reach minus4585 dB ampe system in[54] can detect signals with a size of 1 times 10minus 9 but a noisesignal that can be immune to it can only reach the order of1 times 10minus 9 ampe system in [55] can detect a signal with a widerfrequency range but the detection range of the initial phaseangle of the signal is too large and thus is unfavorable fordetermining the signal parameters At present most of theresearches on weak signals based on chaos theory use a singlechaotic system for detection Compared with the singlechaotic system the coupled system has excellent stabilityhigh antinoise performance and low detection thresholdamperefore the coupled chaotic system has high theoreticalresearch value and practical application value

To improve the chaotic system complexity and the ac-curacy of signal detection this paper proposes a new hiddenattractor coupled chaotic system amprough an analysis of thesystemrsquos phase diagram the Lyapunov index and so forththe complex dynamic behavior of the system is explained indetail Moreover the basins of attraction for the system aredrawn using the Lyapunov index to study the characteristicsof the systemrsquos attractor ampeoretical analysis indicates thatthe system is sensitive to initial values and immune to noiseand the coupled chaotic system is applied to weak signaldetection Finally through OrCAD circuit simulation andFPGA implementation the correctness of the theoretical

analysis and the accuracy of signal detection are verifiedamprough data analysis and experimental data the systemproposed in this paper can detect signals sized 1 times 10minus 6 andthe signal-to-noise ratio reaches minus73892 dB Moreover theimmune signal reached the order of 35 times 10minus 3 and the errorrange of the initial phase angle isplusmn 009 amperefore thedetection performance of the coupled system proposed inthis paper is better than the single chaotic system proposedin [45 46 53ndash55]

2 Coupled Chaotic System Model andDynamic Analysis

21 Coupled Chaotic System Model amprough [45ndash47] asingle chaotic system with different nonlinear terms is an-alyzed and the limitations of signal detection performancedetection are studied at the same time For example [53] candetect low signal amplitude and [54] can be immune ampenoise is small and [55] detects that the initial phase angleerror of the signal is too large Based on the above reasonsthis paper proposes a new system

eurox + c1 _x minus x + c2x5

+ k(x minus y) f cos(ωt + θ)

euroy + c3 1 minus y3

1113872 1113873 _y + y + y3

+ k(y minus x) f cos(ωt + θ)

⎧⎨

⎩

(1)

where k is the linear coupling coefficient of system (1)ampe larger the coefficient the higher the degree of cou-pling c1 c2 and c3 respectively represent the coeffi-cients of the nonlinear terms f cos(ωt + θ) is the drivingforce of the system and f ω and θ represent the am-plitude frequency and initial phase angle of the drivingforce respectively

22 Analysis of the Equilibrium Point and LyapunovExponent of a Coupled Chaotic System We obtained system(2) by coupling system (1) and taking the coefficientsc1 05 c2 02 c3 08 and k 04 with the state spacemodel expression of system (2) shown in the followingequation

_x z

_y q

_z minusc1z + x minus c2x5

minus k(x minus y) + f cos(ωt + θ)

_q minusc3 1 minus y2

1113872 1113873q minus y minus y3

minus k(y minus x) + f cos(ωt + θ)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

To simulate the coupled chaotic system (2) take thefrequency ω 1 and the initial phase angle θ 0 ampesystemrsquos initial value is (minus1 minus1 0 0) When the magnitude fof the driving force takes 03 046 08 and 12 thehomoclinic orbit the period-doubling bifurcation chaoticstate and periodic state will appear in system (2) ampedriving signal of system (2) is defined as H and the signal asa whole system (2) is analyzed Let the left side of theequation be equal to 0 then


z 0

q 0



minus08 1 minus y2

1113872 1113873q minus y minus y3


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(3)


J

0 0 1 0

0 0 0 1



211113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)



nablaV z _x

zx+

z _y

zy+

z _z

zz+

z _q

zq

minus13 + 08y2

(5)



_x z

_y q




1113872 1113873q minus y minus y3


_β 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)


_X(t) w(X(t) η) (7)





J

0 0 1 0 0

0 0 0 1 0

1 minus x4


04 16qy minus 3y2


1113872 1113873 minusf sin β

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)








f = 08 ω = 1 θ = 0

ndash15

ndash1

ndash05

0

05

1

15

z

(a)

ndash15

ndash1

ndash05

0

05

1

15z


(b)



ndash3

ndash2

ndash1

0

1

2

3

x

02 04 06 08 1812 14 16 20 1w = 1 θ = 0

(a)


00102

Lyap

unov

expo

nent

s

02 04 06 08 1 12 14 16 18 20f

(b)


xy

zq

ndash15ndash1

ndash050

051

152

25

50 100 150 200 250 300 350 4000t

(a)

215


x-zy-q

(x z) = (16299 0)

(y q) = (08060 0)

01020304050

t

2

z q

10

ndash1ndash2

(b)



















( 1113857gt 0 (9)







_x z

_y q




1113872 1113873q minus y minus y3


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)




ndash3

ndash2

ndash1

0

1

2

3

x

15105 2 25 30ω

(a)

05 1 15 2 25 30ω

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

03 04 0701 05 0602 080ω

(c)


ndash3

ndash2

ndash1

0

1

2

3

x

1412 16 181 2ω

(d)




ndash25ndash2

ndash15ndash1

ndash050

051

152

25x

1 53 4 60 2θ

(a)

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

1 53 4 60 2θ

(c)

ndash025

ndash02

ndash015

ndash01

ndash005

0

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(d)



ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

y (0)












(11)

When ω0 ω




(12)



1113969





ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

z (0)



SRN 10lgpb

pS(t)

1113888 1113889


1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)





4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)


4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)






400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References































































z 0

q 0



minus08 1 minus y2

1113872 1113873q minus y minus y3


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(3)


J

0 0 1 0

0 0 0 1



211113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)



nablaV z _x

zx+

z _y

zy+

z _z

zz+

z _q

zq

minus13 + 08y2

(5)



_x z

_y q




1113872 1113873q minus y minus y3


_β 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)


_X(t) w(X(t) η) (7)





J

0 0 1 0 0

0 0 0 1 0

1 minus x4


04 16qy minus 3y2


1113872 1113873 minusf sin β

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)








f = 08 ω = 1 θ = 0

ndash15

ndash1

ndash05

0

05

1

15

z

(a)

ndash15

ndash1

ndash05

0

05

1

15z


(b)



ndash3

ndash2

ndash1

0

1

2

3

x

02 04 06 08 1812 14 16 20 1w = 1 θ = 0

(a)


00102

Lyap

unov

expo

nent

s

02 04 06 08 1 12 14 16 18 20f

(b)


xy

zq

ndash15ndash1

ndash050

051

152

25

50 100 150 200 250 300 350 4000t

(a)

215


x-zy-q

(x z) = (16299 0)

(y q) = (08060 0)

01020304050

t

2

z q

10

ndash1ndash2

(b)



















( 1113857gt 0 (9)







_x z

_y q




1113872 1113873q minus y minus y3


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)




ndash3

ndash2

ndash1

0

1

2

3

x

15105 2 25 30ω

(a)

05 1 15 2 25 30ω

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

03 04 0701 05 0602 080ω

(c)


ndash3

ndash2

ndash1

0

1

2

3

x

1412 16 181 2ω

(d)




ndash25ndash2

ndash15ndash1

ndash050

051

152

25x

1 53 4 60 2θ

(a)

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

1 53 4 60 2θ

(c)

ndash025

ndash02

ndash015

ndash01

ndash005

0

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(d)



ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

y (0)












(11)

When ω0 ω




(12)



1113969





ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

z (0)



SRN 10lgpb

pS(t)

1113888 1113889


1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)





4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)


4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)






400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References
































































f = 08 ω = 1 θ = 0

ndash15

ndash1

ndash05

0

05

1

15

z

(a)

ndash15

ndash1

ndash05

0

05

1

15z


(b)



ndash3

ndash2

ndash1

0

1

2

3

x

02 04 06 08 1812 14 16 20 1w = 1 θ = 0

(a)


00102

Lyap

unov

expo

nent

s

02 04 06 08 1 12 14 16 18 20f

(b)


xy

zq

ndash15ndash1

ndash050

051

152

25

50 100 150 200 250 300 350 4000t

(a)

215


x-zy-q

(x z) = (16299 0)

(y q) = (08060 0)

01020304050

t

2

z q

10

ndash1ndash2

(b)



















( 1113857gt 0 (9)







_x z

_y q




1113872 1113873q minus y minus y3


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)




ndash3

ndash2

ndash1

0

1

2

3

x

15105 2 25 30ω

(a)

05 1 15 2 25 30ω

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

03 04 0701 05 0602 080ω

(c)


ndash3

ndash2

ndash1

0

1

2

3

x

1412 16 181 2ω

(d)




ndash25ndash2

ndash15ndash1

ndash050

051

152

25x

1 53 4 60 2θ

(a)

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

1 53 4 60 2θ

(c)

ndash025

ndash02

ndash015

ndash01

ndash005

0

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(d)



ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

y (0)












(11)

When ω0 ω




(12)



1113969





ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

z (0)



SRN 10lgpb

pS(t)

1113888 1113889


1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)





4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)


4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)






400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References















































































( 1113857gt 0 (9)







_x z

_y q




1113872 1113873q minus y minus y3


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)




ndash3

ndash2

ndash1

0

1

2

3

x

15105 2 25 30ω

(a)

05 1 15 2 25 30ω

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

03 04 0701 05 0602 080ω

(c)


ndash3

ndash2

ndash1

0

1

2

3

x

1412 16 181 2ω

(d)




ndash25ndash2

ndash15ndash1

ndash050

051

152

25x

1 53 4 60 2θ

(a)

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

1 53 4 60 2θ

(c)

ndash025

ndash02

ndash015

ndash01

ndash005

0

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(d)



ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

y (0)












(11)

When ω0 ω




(12)



1113969





ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

z (0)



SRN 10lgpb

pS(t)

1113888 1113889


1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)





4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)


4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)






400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References



































































_x z

_y q




1113872 1113873q minus y minus y3


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)




ndash3

ndash2

ndash1

0

1

2

3

x

15105 2 25 30ω

(a)

05 1 15 2 25 30ω

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

03 04 0701 05 0602 080ω

(c)


ndash3

ndash2

ndash1

0

1

2

3

x

1412 16 181 2ω

(d)




ndash25ndash2

ndash15ndash1

ndash050

051

152

25x

1 53 4 60 2θ

(a)

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

1 53 4 60 2θ

(c)

ndash025

ndash02

ndash015

ndash01

ndash005

0

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(d)



ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

y (0)












(11)

When ω0 ω




(12)



1113969





ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

z (0)



SRN 10lgpb

pS(t)

1113888 1113889


1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)





4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)


4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)






400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References
































































ndash25ndash2

ndash15ndash1

ndash050

051

152

25x

1 53 4 60 2θ

(a)

ndash08

ndash06

ndash04

ndash02

0

02

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(b)


ndash3

ndash2

ndash1

0

1

2

3

x

1 53 4 60 2θ

(c)

ndash025

ndash02

ndash015

ndash01

ndash005

0

Lyap

unov

expo

nent

s

1 2 3 4 5 60θ

(d)



ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

y (0)












(11)

When ω0 ω




(12)



1113969





ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

z (0)



SRN 10lgpb

pS(t)

1113888 1113889


1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)





4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)


4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)






400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References








































































(11)

When ω0 ω




(12)



1113969





ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

z (0)



SRN 10lgpb

pS(t)

1113888 1113889


1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)





4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)


4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)






400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References































































SRN 10lgpb

pS(t)

1113888 1113889


1113872 11138732

(00035)2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ asymp minus 73892 dB

(13)





4002

1t

200x

0ndash1

0 ndash2

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

50 100 150 2000Frequency (Hz)

(b)


4002

t1200

x0ndash1ndash20

ndash2

ndash1

0

1

2

z

(a)

0

02

04

06

08

1

20 40 60 80 100 120 140 160 180 2000Frequency (Hz)

(b)






400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References































































400300 4

2t

200x

0100 ndash20 ndash4

ndash4

ndash2

0

2

4z

(a)

400300 5

t200

x0100

0 ndash5

ndash4

ndash2

0

2

4

z

(b)


cos tR13

R12R8

R9

R7

R10

R11

24k Ω2k Ω

6 Ω

3k Ω12k Ω

C3

ndashzx

y

47nF

zU5 U9

ndashx5 R15

R14

1k Ω

10k Ω

12k Ω

1k Ωndashzndash

ndash

++ U7

R3

U2U1

1k Ω

1k Ω

1k Ω

1k Ω

10k Ω 10k Ω

47nF47nF

R1

C1z

xndashx R4R2 R5

R6C2q

yndashyU3 U4

y

q

x

x2

x3ndashx5

y

ndashy

ndashy2ndashy2 y2y2

R21

R22

175k Ω 14k Ω

cos t

R17

R18

R16

R19

R20

175 Ω

14 Ω

35k Ω

C4 R25

R23 R24

1k Ω

10k Ω1k Ω

ndashq

ndashq

47nF

q

ndashy

x

1k Ω

14k Ω

U8 U9ndashndash

++ U10y2

qy2




dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References
































































dVx

dt minus

1R1C1

Vz

dVy

dt minus

1R2C2

Vq

dVz

dt minus

1R13C3

minusR12

R7Vz +

R12

R8Vx minus

R12

R9Vx5 +

R12

R10Vy +

R12

R11Vcos t1113888 1113889

dVq

dt minus

1R23C4

minusR22

R16Vq +

R22

R17Vqy2 minus

R22

R18Vy minus

R22

R19Vy3 +

R22

R20Vx +

R22

R21Vcos t1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



V (x)

ndash20V

ndash10V

0V

10V

20V


(a)

V (z)

ndash20V

ndash10V

0V

10V

20V


(b)


ndash20V

ndash10V

0V

10V

20V


V (z)

(a)

ndash10V

ndash5V

0V

5V

10V


V (z)

(b)








(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References




































































(a) (b)


rst_n

addra_reg0_i

addra_reg[160]

addra[160]

addra[160]

l1

l0

0[160]l0[160]

l0[3 0]0[3 0]

+

+

RTL_REG

clk_div_reg[3 0]

RTL_REG_SYNC

blk_mem_x

blk_mem_z

douta[150]

OBUF

dac_z_data[130]

dac_x_data[130]0



OBUF

douta[150]

u0_x

u0_z

RTL_ADD

RTL_ADD

RTL_ROM

RST

sys_clk

clk_div_reg0_i

clka

clkadk_div_i

A[30] 0

Q

Q

output_en_iC

CE

D

A[30]

RTL_ROM

RTL_REG

RTL_INV

0

0

Q

output_en_reg

output_en_reg0_i

D

D

C

C

dac_x_dk

dac_x_dk

dac_x_dk

dac_x_dk



7 Conclusion



Data Availability




Acknowledgments


References































































7 Conclusion



Data Availability




Acknowledgments


References


















































































































Documents

AnalysisofaNewHiddenAttractorCoupledChaoticSystemand … · 2020. 10. 1. · KxF Kxz K Kxz KxF /yapunovAonents K KxF Kx[ Kx9 Kx3 z zxF zx[ zx9 zx3 F f (b) Figure 3:(a)ebifurcationdiagramof(f)forsystem(2).(b)elyapunovexponentdiagramof(f)