Complex dynamics of a microwave time-delayed feedback loop
Hien DaoSeptember 4th , 2013
PhD Thesis Defense
Chemical Physics Graduate Program
Prof. Thomas Murphy - ChairProf. Rajarshi Roy Dr. John RodgersProf. Michelle GirvanProf. Brian Hunt – Dean Representative
Committee:
Outline• Introduction:
- Deterministic chaos- Deterministic Brownian motion- Delay differential equations
• Microwave time-delayed feedback loop:- Experimental setup- Mathematical model- Complex dynamics: - The loop with sinusoidal nonlinearity: bounded and unbounded dynamics regimes- The loop with Boolean nonlinearity
• Potential applications: - Range and velocity sensing
• Conclusion
• Future works
Introduction : Deterministic chaos Deterministic Brownian motion Delay differential equations
Lorenz attractorWikipedia Motion of double compound pendulumThe distribution of dye in a fluid
http://www.chaos.umd.edu/gallery.html
Wikipedia
• ‘‘An aperiodic long term behavior of a bounded deterministic system that exhibits sensitive dependence on initial conditions’’ – J. C. Sprott, Chaos and Time-series Analysis
• Universality
• Applications: - Communication G. D. VanWiggeren, and R. Roy, Science 20, 1198 (1998)
- Encryption L. Kocarev, IEEE Circ. Syst. Mag 3, 6 (2001)
- Sensing, radar systems J. N. Blakely et al., Proc. SPIE 8021, 80211H (2011)
- Random number generation A. Uchida et al., Nature Photon. 2, 728 (2008)
-…
Chaos Quantifying chaos Type of chaotic signal Microwave chaos
• Lyapunov exponents and
- The quantity whose sign indicates chaos and its value measures the rate at which initial nearby
trajectories exponentially diverge.
- A positive maximal Lyapunov exponent is a signature of chaos.
• Power spectrum
- Broadband behavior
Power spectrum of a damp, driven pendulum’s aperiodic motion
Introduction : Deterministic chaos Deterministic Brownian motion Delay differential equations
Chaos Quantifying chaos Type of chaotic signal Microwave chaos
• Kaplan – Yorke dimensionality
Kaplan-Yorke dimension: fractal dimensionality
Chaotic signal
0 5 10 15 20 25 30-20
0
20
time(s)
x
Chaos in amplitude or envelope
Chaos in phase or frequency!!
A.B. Cohen et al, PRL 101, 154102 (2008)
Lorenz system’s chaotic solution
Deterministic chaos Deterministic Brownian motion Delay differential equations
Chaos Quantifying chaos Type of chaotic signals Microwave chaos
x (t)
Time
Introduction :
Demonstration of a frequency-modulated signal
• Modern communication: cell-phones, Wi-Fi, GPS, radar, satellite TV, etc…
• Advantages of chaotic microwave signal:– Wider bandwidth and better ambiguity diagram
– Reduced interference with existing channels
– Less susceptible to noise or jamming
Global Positioning Systemhttp://www.colorado.edu/geography/gcraft/notes/gps/gps_f.html
Deterministic chaos Deterministic Brownian motion Delay differential equations
Chaos Quantifying chaos Type of chaotic signals Microwave chaos
Introduction :
Frequency modulated chaotic microwave signal.
Deterministic chaos Deterministic Brownian motion Delay differential equations
Definition Properties Hurst exponents
Brownian motion:
Deterministic Brownian motion:
- A random movement of microscopic particles suspended in liquids or gases resulting from the impact of molecules of the surrounding medium
- A macroscopic manifestation of the molecular motion of the liquid
Simulation of Brownian motion - Wikipedia
Introduction :
A Brownian motion produced from a deterministic process without the addition of noise
Deterministic chaos Deterministic Brownian motion Delay differential equations
Definition Properties Hurst exponents
Gaussian distribution of the displacement over a given time interval.
Introduction :
0
40
80
120
4-4 0Bins width
Prob
abili
ty d
istri
butio
n
Deterministic chaos Deterministic Brownian motion Delay differential equations
Definition Properties Hurst exponents
Introduction :
H = 0.5 regular Brownian motion
H < 0.5 anti-persistence Brownian motion
H > 0.5 persistence Brownian motion1.6 2 2.4 2.8
H = 0.57
0.4
0.8
1.2
slog T
log P t
sP t P t T P t
HsP ~ T
H: Hurst exponent 0 < H < 1
• Fractional Brownian motions:
Introduction : Deterministic chaos Deterministic Brownian motion Delay differential equations
• Ikeda system
• Mackey-Glass system
• Optoelectronic system A.B. Cohen et al, PRL 101, 154102 (2008)Y. C. Kouomou et al, PRL 95, 203903 (2005)
K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987)
M. C. Mackey and L. Glass, Science 197, 287 (1977)
History System realization
Chaos is created by nonlinearly mixing one physical variable with its own history.
Introduction : Deterministic chaos Deterministic Brownian motion Delay differential equations
• Nonlinearity• Delay• Filter function
Nonlinearity
FilterGain Delay
x(t)
History System realization
,x t f x t x t
“…To calculate x(t) for times greater than t, a function x(t) over the interval (t, t - ) must be given. Thus, equations of this type are infinite dimensional…”
J. Farmer et al, Physica D 4, 366 (1982)
Time-delayed feedback loop
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
• Voltage Controlled Oscillator
Baseband signal FM Microwave signal
0 tuned 2 v tdt
tunev t
0 2.56GHz2
180 MHz / Volt
0j t tE t 2Ae
Mini-circuit VCOSOS-3065-119+
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
02 j t tE t Ae
*1 Re2mixer dv E t E t
varies slowly on the time scale t d
0 0cos cos 2mixer d d d tune dv t A t A v t
• A homodyne microwave phase discriminator
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
2
E t
2
dE t
Nonlinear function
• A printed- circuit board microwave generator
0
2
cos tunemixer d
v tv t A
v
120.2 2 0.5dA V V V
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
• Field Programmable Gate Array board
• Sampling rate: Fs = 75.75 Msample/s• 2 phase-locked loop built in• 8-bit ADC• 10-bit DAC
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Altera Cyclone II
FX2 USB port
Output
Input
DAC
FPGA chip ADC
• Memory buffer with length N to create delay
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
• Discrete map equation for filter functionH(s) H(z) Discrete map equation
T: the integration time constant
11tune tune mixers
v n v n v n NTF
s
kF
' '1 t
tune mixerv t v t dtT
0
2
cos tunemixer d
v tv t A
v
0cos2
tunetuned
v tdv Adt T V
0
2
2
2tune
d
v tx t
vAR
v Ttt
sin 1x t R x t
M. Schanz et al., PRE 67, 056205 (2003)J. C. Sprott, PLA 366, 397 (2007)
The ‘simplest’ time-delayed differential equation
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
sin 1x t R x t
Experimental setup Mathematical model
• Simulation
sin 1x t R x t
– 5th order Dormand-Prince method– Random initial conditions– Pre-iterated to eliminate transient – = 40 ms– R is range from 1.5 to 4.2
Parameter Valuesampling rate 15 MS/s
N 600A 0.2Vv2 0.5V 180 MHz/V
0/2 2.92 GHz
a (40-bit) 0.0067-0.0175
scope
• Experiment
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
• Low feedback strength generated periodic behavior.
• Period: 4 (6.25kHz)
R = /2
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
• Intermediate feedback strength generated: More complicated but still periodic behavior.
R = 4.1
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
• High feedback strength: Chaotic behavior.
• Irregular, aperiodic but still deterministic.
• lmax = +5.316/t , DK-Y = 2.15
R = 4.176
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
basebandmicrowave
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Power spectra
Period-doubling route to chaos
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Bifurcation diagrams
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Positive lmax indicates chaos.
Maximum Lyapunov exponents
0
2
cos tunedmixer d
v tv t Asgn
v
02
cos tunemixer d
v tv t A
v
sgn sin 1x t R x t sin 1x t R x t
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Another nonlinearity
Time traces and time-embedding plot
• No fixed point solution• Always periodic• Amplitudes are linearly
dependence on system gain R• R >3/2, the random walk
behavior occurs (not shown)
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Bifurcation diagrams
Periodic, but self-similar!
(c) is a zoomed in version of the rectangle in (b)
(d) Is a zoomed in version of the rectangle in (c).
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Unbounded dynamics regime
sin 1x t R x t • Yttrium iron garnet (YIG) oscillator • Delay d is created using K-band hollow rectangular
wave guide• The system reset whenever the signal is saturated
R > 4.9
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Experimental observed deterministic random motion
(a) Tuning voltage time series
(b) Distribution function of displacement
(c) Hurst exponent estimation
The tuning signal exhibits Brownian motion!
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Numerically computed
Experimental estimatedI*
• The tuning signal could exhibit fractional Brownian motion.
• The system shows the transition from anti-persistence to regular to persistence Brownian motion as the feedback gain R is varied
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Synchronization of deterministic Brownian motions
• Unidirectional coupling in the baseband• System equations
Master
Slave• The systems are allowed to come to
the statistically steady states before the coupling is turned on
m mx t R sin x t 1
s s mx t R 1 sin x t 1 sin x t 1
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Simulation results
• The master system could drives the slave system to behave similarly at different cycle of nonlinearity.
• The synchronization is stable.
Evolution of synchronization perturbation vector
Microwave time-delayed feedback loop: Experimental setup Mathematical model Complex dynamics
Synchronization error s
2
m 2 s 2
2 2m 2 s 2
x t x t,
x t x t
s
m 2 mx t x t mod 2
s 2 sx t x t mod 2
Where:
The synchronization ranges depends on the feedback strength R.
Simulation results
o Range and velocity sensor
o Random number generator
oGPS: using PLL to track FM microwave chaotic signal
Potential Applications
Pulse radar system - Wikipedia Doppler radar- Wikipedia
Objective: Unambiguously determine position and velocity of a target.
Can we use the FM chaotic signal for S(t)?
S(t)S(t)
rS(t-)
Potential Applications: Range and velocity sensing application Ambiguity function Experimental FM chaotic signal
• Formula:
2*, Dopplerj f trange Doppler rangef S t S t e dt
Ideal Ambiguity Function
• Ambiguity function for FM signals- Approximation and normalization arg
0t etv
Doppler fc
f
Fixed Point Periodic Chaotic
Potential Applications: Range and velocity sensing application Ambiguity function Experimental FM chaotic signal
• Broadband behavior at microwave frequency
Experiment Simulation
Spectrum of FM microwave chaotic signal
2.9 GHZ
52 MHz
15dB/div
Potential Applications: Range and velocity sensing application Ambiguity function Experimental FM chaotic signal
• Chaotic FM signals shows significant improvement in range and velocity sensing applications. -3 30
Conclusion (1)Designed and implemented a nonlinear microwave
oscillator as a hybrid discrete/continuous time system
Developed a model for simulation of experiment
Investigated the dynamics of the system with a voltage integrator as a filter function
- A bounded dynamics regime:
a. Sinusoidal nonlinearity: chaos is possible
b. Boolean nonlinearity: self-similarity periodic behavior
- An unbounded dynamics regime: deterministic Brownian motion
Conclusion (2)
Generated FM chaotic signal in frequency range : 2.7-3.5 GHz
Demonstrated the advantage of the frequency-modulated microwave chaotic signal in range finding applications
Future work Frequency locking (phase synchronization) in FM chaotic
signals
Network of periodic oscillators
The feedback loop with multiple time delay functions
Thank you!
Supplementary materials
Calculate ambiguity function of Chaos FM signal
• Ambiguity function: the 2-dimensonal function of time delay and Doppler frequency f showing the distortion of the returned signal;
• The value of ambiguity function is given by magnitude of the following integral
* j2 ft,f s t s t e dt
Where s(t) is complex signal, is time delay and f is Doppler frequency
• Chaos FM signal:
j ts t Ae
t
00
tt 2 v t dt
0j tt j 2 v tj2 j2 ft 2 j2 ft,f A e e dt A e e e dt
targetdoppler 0
vff
c
• Approximation:
0d
0 0
1n* n* n4f
00f 2
where
0 / 2
(operating point)
-60
-50
-40
-30
-20
-10
0
0.0 1.0 2.0 3.0 4.0 5.0Frequency [MHz]
Pow
er le
vel [
dB]
L/N L/N L/N
C/2N C/2N C/2N C/2N C/2N C/2N
N units
L=5 mH
C=1nF
u=0.1 ms/unit;
t= 1.2 ms
fcutoff ~ 3 MHz
Loop feedback delay t is built in with transmission line design
Simulation Results
1 2 3 4 65 7b-2
2
0
201000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10-5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time [s]
Vtun
e [V
]
0
-0.4
0.4
20100
Time [ms]
0
-0.6
0.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10-5
-0.4
-0.2
0
0.2
0.4
0.6
Time [s]
Vtun
e [V
]
20
Time [ms]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10-5
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time [s]
Vtun
e [V
]
10
0
-1.5
2
Time [ms]
X(t)
XBifurcation Diagram
b = 1.6 b = 2.7 b = 6.2
-2
2
0V
Bifurcation Diagram
Experiment
Spectral diagram
of microwave signal Freq
uenc
y [G
Hz]
3
3.1
3.2
2.9
2.8
2.7
Coupling and Synchronization
bias
VCO
splitter
d
mixer
H(s)
v1(t)
1(t)
b
bias
VCO
splitter
d
mixer
H(s)
v2(t)
2 (t)
b
: coupling strength
(I) (II)
• Two systems are coupled in microwave band within or outside of filter bandwidth
• Two possible types of synchronization:
- Baseband Envelope Synchronization
1 2v t v t
1 2t t
- Microwave Phase Synchronization
Experimental ResultsUnidirectional coupling, outside filter bandwidth, = 0.25
1 2
2
b = 1.2 b = 5.1
0 5 10 15 20 0 5 10 15 20
V1(t)
V2(t)
V1(t)-V2(t)
V1(t)
V2(t)
V1(t)-V2(t)
1 2
2
0
1
-1
1 2
2
0
-5
5
Time [ms] Time [ms]
Experimental ResultsBidirectional coupling, outside filter bandwidth, = 0.35
b = 1.2 b = 2.1
0 5 10 15 200 5 10 15 20
V1(t)
V2(t)
V1(t)-V2(t)
V1(t)
V2(t)
V1(t)-V2(t)
1 2
2
0
1
-1
1 2
2
0
2
-2
Time [ms]Time [ms]
Transmission line for VCO system?
* Microstrip line with characteristic impedance 50 Ohm
Dielectric material: Roger 4350B with
* Using transmission line to provide certain delay time in RF range
r 3.48 0.05
rL.c
Using HFSS to calculate the width of transmission line and simulate the field on transmission line
Width of trace: 0.044’’ thickness of RO3450 : 0.02”; simulation done with f=5GHz
Printed Circuit Board of VCO system
Distance Radar
o Idea:VCO
integrator
scope
Using microwave signal generated by VCO for detecting position of object in a cavity
o Mathematical model:
Nonlinearity
V2
V0
out o d in 0 dV V cos 2 VIn general
In particular case has been investigated
out o in
2
2V V sin VV
RF delay and nonlinearity
0 / 2
0
2 0d
1V 0 0 d
Transmission line
Gain =2.5
Gain =3.77
Gain =4.137
How much chances we can detect?
VCO
integrator
scope
0d d t
Assumption:d
is in order of 10-9
0dx Rsin x t 1 . t 1dt
Rsin x t 1 . t 1
Approximated equation:
002
VR 2
V T
002
Vx 2 / 2V
0
2 0d
1V 00 0 d
Continuously change d
Normalization:
Watching dynamics of system, can we determine (and then z?)
Using PLL to track chaotic FM signal
VCO
integrator
scope
Chaos Generator
Chaotic FM signal
vp
cj tc t Ae
0cc c
d2 v t
dt
pj t
p t Ae
p 0p p
d2 v t
dt
Mixer output
vpm
p *m p cv t Re tt
p 2m p cv A cos
Always can pick 0 0p c
Integrator equation
2
p 2p c2
d1 1 A cos2 dt T
Or another filter function?
[A2]: voltage as Vp-p
p pm
dv 1 vdt T
PLL equation
2p
p p c2
dcos
dt
b 2
p2 A
Tb
Does solution exist?
2
pp p c2
dcos
dt
b
0cc c
d2 v t
dt
Chaos generator
2
cc
2
dv A 2sin v tdt T v
2
cc c2
2
d 2sin v tdt v
b
2
c2 A
Tb
p c
2
cc c2
2
d 2sin v tdt v
b
Equations:
In general case, bc and bp could be assumed to be different by some scaling factor bc/bp = n
Static = time evolution ?!