SOME NOTABLE OBSERVATIONS BY OUR GROUP

SOME NOTABLE OBSERVATIONS BY OUR GROUP

K. ThamilmaranCentre for Nonlinear Dynamics

Bharathidasan UniversityTiruchirappalli, India – 620 024

CNLD, School of Physics, Bharathidasan University

Nonlinear Electronics Research Group

K. Suresh: Strange nonchaos, Amplitude death and time series analysis

R. Jothimurugan : Design of simple nonlinear circuits, Resonance in electronic systems

S. Sabarathinam: Dynamics of coupled dissipative and nearly conservative systems

Dr. K. Thamilmaran: Nonlinear Electronic circuits: Experiment, Numeric and Theoretical study

A. Arulgnanam: Simplification of Chaotic circuits : Analytical and Experimental

P. S. Swathy: Cellular Nonlinear Networks (CNN) through electronic circuits

Top most accessed article in the year 2007 in IJBC

Hyperchaos in modified canonical Chua’s circuit . . .

Fig. (a) Canonical Chua’s circuit (b) Modified canonical Chua’s circuit (c) Negative conductance (d) Nonlinear resistor NR

(e) (v-i) characteristic of NR


Fig. Experimental phase portraits in the (v1-v2) plane of the modified canonical Chua’s circuit. Period-3 doubling sequence to hyperchaos via chaos. (a) period-3 limit cycle (R = 730 Ω) (b) period-6 limit cycle (R = 696 Ω) (c) period-12 limit cycle (R = 685Ω) (d) chaos (R = 600 Ω) (e) hyperchaos (R = 285 Ω) and (f) outer limit cycle (R = 150 Ω). Scale: v1 = 0.2v/div; v2 = 1.0v/div

(a) (b) (c)

(d) (e) (f)


a(i) a(ii) c(i)

b(i) b(ii) c(ii)

Fig. Experimentally observed (a) phase portraits in the (v2-iL) plane, (b) Poincare maps in the (v2-iL) plane and (c) power spectra of the signal v2(t) for (i) chaotic and (ii) hyperchaotic attractors for the specific choice of control parameter, R = 450 Ω and R = 285 Ω.


a(i) a(ii)c(i)

b(i) b(ii)c(ii)

Fig. Pspice simulation of phase portraits in the (a) (v1-v2) palne and (b) (v2-iL) plane, and (c) power spectra of the signal v2(t) for (i) chaotic and (ii) hyperchaotic attractors for the specific choice of control parameter, R = 450 Ω and R = 285 Ω.


Fig. Computer simulation of (a) phase portraits and (b) poincare maps in the (v2-iL) plane and (c) power spectra of the signal v2(t) for (i) chaotic and (ii) hyperchaotic attractors for the specific choice of control parameter, R = 450 Ω and R = 285 Ω.

a(i) a(ii) c(i)

b(i) b(ii) c(ii)


Fig : One parameter bifurcation diagram

Fig : First two Lyapunov exponents spectrum


Fig : One parameter bifurcation diagram

Fig : First two Lyapunov exponents spectrum

Fig. The circuit realization of the proposed simple nonautonomous chaotic circuit. (a) The nonlinear resistor (NR) using one op-amp and three linear resistors and (b) the (v-i) characteristic of the nonlinear element.

( ) 0.5( )a a b P Pg v G G G v B v B

The mathematical form

In this case Ga is negative and Gb is positive

The values of the resistors are chosen as R1 = 1.990 k, R2 = 1.981k and R3 = 1.989 k

The value of the slopes are Ga = -0.56 mS and Gb = 2.5 mS

The value of the breakpoint Bp is ±3.8 V

Chaotic dynamics with high complexity in a simplified new nonautonomous nonlinear electronic circuit . . .

Chaotic dynamics with high complexity in a simplified new nonautonomous nonlinear electronic circuit . . .

EXPERIEMENT

SPICE SIMULATION

NUMERICAL SIMULATION

Parameters:a = -1.148, b = 5.125, β = 0.9865, ω = 0.7084 and f = 1.187

Parameters:C = 10.32 nF, L = 42.60 mH,R = 2.05 k, υ = 5500 Hz and f = 4.9 V

ANALYTICAL

Fig. One parameter bifurcation diagram in the (f-x) plane and (b) the corresponding maximal Lyapunov spectrum in the (f-λmax) plane for a = -1.148, b = 5.125, β = 0.9865 and ω = 0.7084.

The familiar period doubling route to chaos is found in this circuit

Apart from that the periodic windows, intermittent bursts are also observed

Strange nonchaotic behaviour is also observed when an additional force is added by keeping the frequencies are incommensurate

The positive value confirms the chaotic nature of this circuit

Its high complexity is verified by comparing the value of Lyapunov exponent with Driven Chua’s circuit, Murali – Lakshaman – Chua’s circuit and variant of MLC circuits.

Bubble doubling route to strange nonchaotic attractor in quasiperiodically driven Chua’s circuitK. Suresh, K. Thamilmaran and A. Awadhesh Prasad

This work is submitted to Physics Letters A (under revision)

11 2 1 1

22 1 2

2 1 1 2 2

1 1 1 1

( ) ( )

( )

sin( ) sin( )

( ) 0.5( )

CC C C

CC C L

LC

C b C a b C P C P

dvC G v v g v

dtdv

C G v v idt

diL v F t F t

dt

g v G v G G v B v B

Fig. Schematic diagram of quasiperiodically driven Chua’s circuit

Fig. Projection of Poincare surface section in the (i) (φ-x) plane with φ modulo 2π and (ii) the corresponding blow-ups as a function of amplitude of the forcing f1. (a) period-1 torus (f1 = 0.7040), (b) period-2 torus (f1 = 0.7296), (c) bubbled strands of period-2 torus (f1 = 0.7744), (d) 2-bubble in the strands of the period-2 torus (f1 = 0.7790) and (e) 4-bubble in the strands of the period-2 torus (f1 = 0.7803).

Poincare surface section

Bubble doubling route to strange nonchaotic attractor in quasiperiodically driven Chua’s circuit . . .

Poincare surface section Rational Approximation

Fig. (a) Two band SNA for the value of f1 = 0.7835, (b) merged one band SNA and (c) wrinkling of merged band SNA f1 = 0.8210 and (d) chaotic attractor f1 = 0.9728.

Fig. Bifurcation diagram in the (θ-x) plane for the rational approximation of the order k = 3 and the rational frequency Ωk = 4/5. (a) period -4 orbit formation of (b) period-1bubble, (c) period-2 bubble and (d) chaoic band and band merging.

Fig. Bifurcation diagram in the (θ-x) plane for the rational approximation of the order k = 3 and the rational frequency Ωk = 4/5. (a) period -4 orbit for f1 = 0.7744 and (b) varied dynamical behaviours as a function of formation of θ, (c) a blown up portion of (b) and (d) spectrum of maximal Lyapunov exponent of (b) in the (φ-λ) plane for f1 = 0.7790.


Rational Approximation

Rational Approximation

Fig. Bifurcation diagram in the (θ-x) plane for the three different orders of rational approximation (namely with k = 1 with Ωk = 2/3, k = 3 with Ωk = 2/3.(a) period -4 orbit for f1 = 0.7744 and (b) varied dynamical behaviours as a function of formation of θ, (c) a blown up portion of (b), (d) spectrum of maximal Lyapunov exponent of (b) in the (φ-λ) plane for f1 = 0.7790.

Fig. The fraction of the chaotic components obtained using rational approximation with orders varying in the range (3 < k < 21) for torus, SNA and chaotic behaviours.


Phase sensitivity Exponent

Distribution of finite time LE

Maximal LE and its variance

Fig. Transition from two torus to chaos through SNA. (a) the largest Lyapunov exponent Λ and (b) its variance σ as a function of control parameter f1.

Fig. Distribution of finite time Lyapunov exponent for f1 = 0.7835, obtained for three different finite time intervals M = 1000 (solid line-black), M = 2000 (dashed line-red) and M = 3000 (dotted line-blue).

Fig. Phase sensitivity exponent curve for (a) Torus at f1 = 0.7296, (b) SNA at f1 = 0.7835 and (c) chaos at f1 = 0.9728.

An experimental study on SC-CNN based canonical Chua’s circuitP. S. Swathy and K. Thamilmaran

This work is submitted to Nonlinear Dynamics (under revision)

Single generalized CNN cell

Cell connection scheme

Fig. Schematic diagram of canonical Chua’s circuit.

Fig. The CNN output nonlinearity (a) Numerical and (b) Experimental

(a)(b)

An experimental study on SC-CNN based canonical Chua’s circuit …

Fig. The full realization of the proposed SC-CNN based canonical Chua’s circuit with three CNN cells.

11

1

22

2

2 1

1 1 1 1

1( ( ))

1( )

1( )

( ) 0.5( )

L

N L

LL

b a b P P

dvi f v

dt C

dvG v i

dt C

div v Ri

dt L

f v G v G G v B v B

The state equations of the original circuit:

Normalized state equations of the CNN circuit

1 1 11 1 11 1 13 3

2 2 22 2 23 3

3 3 31 1 32 2 33 3

1 1 1 1( ) 0.5( 1 1)

x x a y s x s x

x x s x s x

x x s x s x s x

y f x x x

Normalized state equations

1 2

2

1 1

( ( ))

( )

( ) 0.5 ( 1 1)

x z h x

y y z

z y x z

h x bx a b x x

Generalized SC-CNN equation is

; ; ;( ) ( ) ( ) ( ) ( ) ( )

j j k k j k k j k k jC k N j C k N j C k N j

x x A y B u C x I


c(i)

c(ii)

Fig. Experimentally observed (i) phase portraits in the (v1-v2) plane [scale: horizontal axis (vc1) = 5v/div, vertical axis (vc2) = 0.5v/div] (ii) corresponding time waveforms of vc1(t) and vc2(t) and (iii) the power spectrum of vc1(t) of (a) single and (b) double band chaos.


Fig. Oscilloscope observations for phase portraits in the (vc1-vc2) planes shows period doubling bifurcation sequence; horizontal axis (vc1) 5v/div, vertical axis (vc2) 0.5v/div. (a) period-1 (R19 = 990 Ω), (b) period-2 (R19 = 1010 Ω), (c) period-4 (R19 = 1038 Ω), (d) one band chaos (R19 = 1076 Ω), (e) double scroll chaos (R19 = 1150 Ω) and (f) boundary (R19 = 1200 Ω).

Fig. Period doubling sequences of the numerically simulated phase portraits in the (x1-x2) plane as a function of S32 including the transition from period to boundary through chaos.(a) period-1 (S32 = 0.5350), (b) period-2 (S32 = 0.5118), (c) period-4 (S32 = 0.4937), (d) one band chaos (S32 = 0.3449) (e) double band chaos (S32 = 0.3143) and (f) boundary (S32 = 0.2850).

(a) (b)

(c) (d)

(e) (f)


Fig. Numerically simulated (i) Phase portraits in the (x1-x2) plane and (ii) corresponding power spectrum of x1(t). (a) one band chaotic attractor for S32 = 0.3449 and (b) double band chaotic attractor for for S32 = 0.3143.

Fig. Projections of double band chaotic attractor in the (a) (x2-x3) and (b) (x1-x3) planes by numerical simulation.

Fig. (a) One parameter bifurcation diagram in the (S32 – x1) plane and (b) the corresponding Lyapunov spectrum in the plane as a function of S32 for fixed other parameters obtained through numerical analysis.

Transient chaos in two coupled, dissipatively perturbed Hamiltonian Duffing-Oscillators S. Sabarathinam, K. Thamilmaran, L. Borkowski, P. Perlikowski, P. Brzeski, A. Stefanski and

T. Kapitaniak

This work is submitted to Communications in Nonlinear Science and Numeric Simulation

Poincare map

Fig. Poincare maps for three different energy spaces, k = 0.08; (a) x1(0)=0.5021, dx1(0)/dt = -0.17606, x2(0) = -0.96946, dx2(0)/dt = 0.34206, ʜ = 0.105156, (b) x1(0)=-0.7, dx1(0)/dt = -0.513927, x2(0) = 0.3859, dx2(0)/dt = 0.2363, ʜ = 0.167042 and (c) x1(0)=2.0, dx1(0)/dt = -1.5, x2(0) = -0.7, dx2(0)/dt = 1.3, ʜ = 4.81512.

Transient chaos in two coupled, dissipatively perturbed Hamiltonian Duffing-Oscillators …

Fig. Time plots of conservative system. (a) x1(t), (b) transient Lyapunov exponents, (c) enlargement of (a). Initial condition: x1(0)=-0.5021, , dx1(0)/dt =-0.17606, x2(0) =-0.96946, , dx2(0)/dt =0.34206 and energy level ʜ = 0.105156.

Fig. Time series of dissipatively perturbed system for negative damping b = -0.0001. (a) x1(t), (b) trnsient Lyapunov exponents, (c) enlargement of (a) and for positive damping b = 0.0001; (d) x1(t), (e) transient Lyapunov exponents, (f) enlargement of (d).


Fig. (a) Schematic and (b) photo of the analyzed of electrical circuit of two coupled Duffing systems.


Fig. Time series x1(t) of nearly Hamiltonian system with positive damping (absence of R1) in (a) experimental results and in (c) numerical results. In (b, d) zoom of chaotic motion from experiment and numeric respectively.


Fig. Time series x1(t) of nearly Hamiltonian system with negative damping (R1=2.5 MΩ) in (a) experimental results and in (c) numerical results. In (b, d) zoom of chaotic motion from experiment and numeric respectively.

Vibrational resonance and enhanced signal transmission in Chua’s circuitsR. Jothimurugan, K. Thamilmaran, S. Rajasekar and M. A. F. Sanjuan

This work is about to submit.

(a) (b) (c)

Fig. Schematic diagram of Chua’s circuit with biharmonic forcing, (b) the Chua’s diode (NR) and (c) corresponding (v-i) characteristic.

Fig. The power spectrum of the voltage v1 for four values of g. (a) g = 1 V, (b) g = 1.3 V, (c) g = 1.55 V and (d) g = 2 V. The values of other parameters are C1 = 10 nF, C2 = 100 nF, L = 18 mH, R = 1980 Ω , A = 0.3 V, ω = 50 Hz, Ω = 500 Hz..

Vibrational resonance is a phenomenon where the low frequency amplitude resonates with the optimum value of the amplitude of the high frequency drive

Vibrational resonance and enhanced signal transmission in Chua’s circuits …

Fig. (a) Response amplitudes Q at the low-frequency ω (continuous curve) and high-frequency Ω (dotted curve) associated with v1 versus the control parameter g. (b) Q at the low-frequency ω associated with v2 (continuous curve) and iL (dotted curve) versus g.

Fig. Plot of v1(t) (green line) and the low-frequency periodic input signal fsinωt (yellow line) versus t for five values of the amplitude g of the high-frequency periodic signal. (a) & (b) g = 1 V, (c) g = 1.15 V, (d) g = 1.3 V, (e) g = 1.55 V and (f) g = 2 V. In all the subplots the range of v1(t) is [−10 V,10 V]. The low frequency periodic input is 10 times enlarged in all the subplots for clarity.


Fig. The dependence of the response amplitude of the voltage v1(t) versus the parameter g on (a) the high-frequency Ω = 0.5 kHz, 1.5 kHz and 2.5 kHz with f = 0.3 V and ω = 50Hz, (b) the amplitude f = 0.05 V, 0.15 V and 0.25 V with ω = 50 Hz and = 1000 Hz and (c) different combinations of ω and Ω with Ω / ω = 10 where for the curves 1, 2 and 3 the values of (ω, Ω) are (50 Hz , 500 Hz), (150 Hz , 1500 Hz) and (250 Hz , 2500 Hz) while f = 0.5 V. (d) Variation of Q with the parameter for three fixed values of ω with f = 0.3 V and g = 1.75 V.

Fig. Plot of (a) gVR, the critical value of g at which resonance occurs and (b) Qmax, the value of Q at g = gVR versus the high-frequency Ω of the driving force. The values of the parameters are f = 0.3 V and ω = 50 Hz (for the symbol circle), ω = 100 Hz (square) and ω = 150 Hz (triangle).


Fig. The block diagram of “n” unidirectionally coupled Chua’s circuits. In the first circuit (CC1), a

biharmonic signal F(t) is connected in series with current(i1L) and the remaining circuits are driven by

the voltage v1 of the previous circuit. Here B represents a buffer circuit and the arrowhead represents the direction of coupling.

Fig. (a) Qi versus i (the number of the Chua’s circuit) for five values of g with RC = 1.0 kΩ. For the curves 1 − 5 the values of g are 0.6 V, 0.85 V, 1.1 V, 1.15 V and 1.6 V. The values of the other parameters are set as C1 = 10 nF, C2 = 100 nF, L = 18 mH, R = 2150 Ω, ω = 100 Hz, Ω = 1000 Hz and f = 0.3 V. (b) Qi versus i for five values of RC with g = 1.2 V. The values of RC for the curves 1-5 are 1 kΩ, 1.2 kΩ, 1.4 kΩ, 1.8 kΩ, and 2 kΩ.


Fig. Qi as a function of i and g for two values of RC illustrating (a) damped propagation of signal (for RC = 2.165 kΩ) and (b) undamped signal propagation (for RC = 1 kΩ) through the unidirectionaly coupled Chua’s circuits. The thick curve in (b) represents Q1. In (a) Q1 is not shown because Qi’s with i > 1 are much lower than Q1.

Fig. Undamped signal propagation (marked by red color) in the(g − RC) parameters space.


Fig. Evolution of vi1 with time at four different nodes

denoted as i where R = 2.150 kΩ, RC = 1 k and g = 1.1 V. Notice the suppression of high-frequency oscillations as “i” increases.

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SOME NOTABLE OBSERVATIONS BY OUR GROUP