Chemical Physics Graduate
Program
Complex dynamics
of a microwave time-delayed feedback loop
Hien Dao
PhD Thesis Defense
September 4th , 2013
Committee:
Prof. Thomas Murphy - Chair
Prof. Rajarshi Roy
Dr. John Rodgers
Prof. Michelle Girvan
Prof. Brian Hunt – Dean Representative
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 :
 Chaos
 Deterministic chaos
•
‘‘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
Wikipedia
Lorenz attractor
Wikipedia
•
The distribution of dye in a fluid
Motion of double compound pendulum
http://www.chaos.umd.edu/gallery.html
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)
-…
Introduction :
 Deterministic chaos
 Quantifying 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.
• Kaplan – Yorke dimensionality
Kaplan-Yorke dimension: fractal dimensionality
• Power spectrum
- Broadband behavior
Power spectrum of a damp, driven pendulum’s aperiodic motion
 Deterministic chaos
Introduction :
 Type of chaotic signals
Chaotic signal
x
20
Lorenz system’s chaotic solution
0
-20
0
5
10
15
time(s)
20
25
30
Chaos in amplitude or envelope
A.B. Cohen et al, PRL 101, 154102 (2008)
Chaos in phase or frequency!!
Demonstration of a frequency-modulated signal
x (t)
Time
Introduction :
 Deterministic chaos
 Microwave chaos
• Modern communication: cell-phones, Wi-Fi, GPS,
radar, satellite TV, etc…
Global Positioning System
http://www.colorado.edu/geography/gcraft/notes/gps/gps_f.html
• Advantages of chaotic microwave signal:
– Wider bandwidth and better ambiguity diagram
– Reduced interference with existing channels
– Less susceptible to noise or jamming
Frequency modulated chaotic microwave signal.
Introduction :
 Deterministic Brownian motion
 Definition
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
Deterministic Brownian motion:
Simulation of Brownian motion - Wikipedia
A Brownian motion produced from a deterministic process without the addition of noise
Introduction :
 Deterministic Brownian motion  Properties
Gaussian distribution of the
displacement over a given time
interval.
Probability distribution
120
80
40
0
-4
0
Bins width
4
Introduction :
 Deterministic Brownian motion
 Hurst exponents
 P ~ Ts
H: Hurst exponent
0<H<1
 P  t   P  t  Ts   P  t 
• Fractional Brownian motions:
H = 0.5 regular Brownian motion
H < 0.5 anti-persistence Brownian motion
H > 0.5 persistence Brownian motion
H
1.2
log

P  t 

0.8
H = 0.57
0.4
1.6
2
2.4
log  Ts 
2.8
Introduction :
 Delay differential equations
 History
• Ikeda system
K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987)
• Mackey-Glass system
M. C. Mackey and L. Glass, Science 197, 287 (1977)
• Optoelectronic system
A.B. Cohen et al, PRL 101, 154102 (2008)
Y. C. Kouomou et al, PRL 95, 203903 (2005)
Chaos is created by nonlinearly mixing one physical variable with its own history.
Introduction :
 Delay differential equations
x t   f
 System realization
 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
• Nonlinearity
• Delay
• Filter function
Nonlinearity
x(t)
Gain
Filter
Delay
 Experimental setup
Microwave time-delayed feedback loop:
 Experimental setup
Microwave time-delayed feedback loop:
• Voltage Controlled Oscillator
Baseband signal
v tune  t 
FM Microwave signal
E t 
2A e
Mini-circuit VCO
SOS-3065-119+
j   0 t    t  
d
dt
0
  0  2  v tune  t 
2
 2.56 G H z
  180 M H z / V olt
 Experimental setup
Microwave time-delayed feedback loop:
• A homodyne microwave phase discriminator
E t   d

E t  
2
E t 
2
v m ixer 
  t  varies
1
2
ReE t  E
slowly on the time scale  d
v m ixer  t   A cos  d   t    0 d   A cos  2  d v tune  t    0 d 
*
 t   d 
2 Ae
j   0 t    t  
 Experimental setup
Microwave time-delayed feedback loop:
• A printed- circuit board microwave generator
A  0.2V
Nonlinear function
v m ixer
 v tune  t 

  0 d 
 t   A cos 
 v2

V 2    2  d

1
 0.5V
 Experimental setup
Microwave time-delayed feedback loop:
• Field Programmable Gate Array board
Output
FX2 USB port
DAC
FPGA chip
ADC
Input
Altera Cyclone II
•
•
•
•
Sampling rate: Fs = 75.75 Msample/s
2 phase-locked loop built in
8-bit ADC
10-bit DAC
 Experimental setup
Microwave time-delayed feedback loop:
• Memory buffer with length N to create delay 
 
k
Fs
• Discrete map equation for filter function
H(s)
H(z)
v tune  n   v tune  n  1  
1
T Fs
Discrete map equation
v m ixer  n  N
T: the integration time constant

 Mathematical model
Microwave time-delayed feedback loop:
 v tune  t 

v m ixer  t   A cos 
  0 d 
 v2

v tune  t  
1
T
t
v
m ixer
t
'
  dt
dv tune
dt
'
 v tune  t  
 cos 
T
 V 2
A

x t  
v tune  t 
v2

  0 d 

  0 d 


2
A
R 
v2 T
t
t

x  t    R sin  x  t  1  
The ‘simplest’ time-delayed differential equation
M. Schanz et al., PRE 67, 056205 (2003)
J. C. Sprott, PLA 366, 397 (2007)
Microwave time-delayed feedback loop:
Experimental setup
 Mathematical model
Mathematical model
x  t    R sin  x  t  1  
Microwave time-delayed feedback loop:
 Complex dynamics
• Simulation
• Experiment
x  t    R sin  x  t  1  
scope
Parameter
Value
sampling rate
15 MS/s
N
600
A
0.2V
v2
0.5V

180 MHz/V
0/2
2.92 GHz
a (40-bit)
0.0067-0.0175
–
–
–
–
–
5th order Dormand-Prince method
Random initial conditions
Pre-iterated to eliminate transient
= 40 ms
R is range from 1.5 to 4.2
Microwave time-delayed feedback loop:
R = /2
• Low feedback strength
generated periodic behavior.
• Period: 4 (6.25kHz)
 Complex dynamics
Microwave time-delayed feedback loop:
R = 4.1
• Intermediate feedback
strength generated:
More complicated but
still periodic behavior.
 Complex dynamics
Microwave time-delayed feedback loop:
R = 4.176
• High feedback strength: Chaotic
behavior.
• Irregular, aperiodic but still
deterministic.
• lmax = +5.316/t , DK-Y = 2.15
 Complex dynamics
Microwave time-delayed feedback loop:
Power spectra
microwave
baseband
 Complex dynamics
Microwave time-delayed feedback loop:
Bifurcation diagrams
Period-doubling route to chaos
 Complex dynamics
Microwave time-delayed feedback loop:
Maximum Lyapunov exponents
Positive lmax indicates chaos.
 Complex dynamics
Microwave time-delayed feedback loop:
 Complex dynamics
Another nonlinearity
 v tune  t 

v m ixer  t   A cos 
  0 d 
 v2

x  t    R sin  x  t  1  
v
d
m ixer
t  

 v tune  t 

A sgn  cos 
  0 d  


 v2



x  t    R sgn sin  x  t  1  

Microwave time-delayed feedback loop:
 Complex dynamics
 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:
 Complex dynamics
 Bifurcation diagrams
(c) is a zoomed in version
of the rectangle in (b)
(d) Is a zoomed in version
of the rectangle in (c).
Periodic, but self-similar!
Microwave time-delayed feedback loop:
 Complex dynamics
Unbounded dynamics regime
•
•
•
Yttrium iron garnet (YIG) oscillator
Delay d is created using K-band hollow rectangular
wave guide
The system reset whenever the signal is saturated
x  t    R sin  x  t  1  
R > 4.9
Microwave time-delayed feedback loop:
 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:
 Complex dynamics
Numerically computed
*I Experimental estimated
• 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:
 Complex dynamics
Synchronization of deterministic Brownian motions
• Unidirectional coupling in the baseband
• System equations
Master
x m  t    R sin  x m  t  1  
x s  t    R   1    sin  x s  t  1     sin  x m  t  1   


• The systems are allowed to come to
the statistically steady states before
the coupling is turned on
Slave
Microwave time-delayed feedback loop:
 Complex dynamics
Simulation results
Evolution of synchronization perturbation vector
•
The master system could drives the slave system to behave
similarly at different cycle of nonlinearity.
•
The synchronization is stable.
Microwave time-delayed feedback loop:
 Complex dynamics
Simulation results
Synchronization error s
s 





x
m  2
 t   x s  2  t  
2
x m  2  t   x s  2  t 
2
2


,



Where:
x m  2   t   x m  t  m od 2 
x s  2   t   x s  t  m od 2 
The synchronization ranges depends on the feedback strength R.
Potential Applications
o Range and velocity sensor
o Random number generator
o GPS: using PLL to track FM microwave chaotic signal
Potential Applications:
 Range and velocity sensing application
Objective: Unambiguously determine position and velocity of a target.
rS(t-)
S(t)
S(t)
Pulse radar system - Wikipedia
Doppler radar- Wikipedia
Can we use the FM chaotic signal for S(t)?
Potential Applications:
 Ambiguity function
• Formula:
   range , f D oppler  


S  t S
*
t  
range
e
 j 2  f D oppler t
dt

Ideal Ambiguity Function
• Ambiguity function for FM signals
- Approximation and normalization
Fixed Point
Periodic
f
D oppler  f 0
Chaotic
v t arg et
c
Potential Applications:
 Experimental FM chaotic signal
Spectrum of FM microwave chaotic signal
• Broadband behavior at
microwave frequency
52 MHz
15dB/div
2.9 GHZ
• Chaotic FM signals shows
significant improvement in
range and velocity sensing
applications.
Experiment
-3
0
Simulation
3
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

  , f  
 sts t   e
*
 j2  ft
dt

Where s(t) is complex signal,  is time delay and f is Doppler frequency
• Chaos FM signal:
s  t  Ae
j  t 
t
  t    0 t  2   v  t d t
0

  , f   A
2
e
j   t     t  



e
 j2  ft
dt
2
A e
j  0
e
j  2  v  t 
e
 j2  ft
dt

• Approximation:
fd o p p le r
v t arg e t
c
f0
  n * d  n *
0
0
n
1
4 f0
where
f0 
0
2
 0   / 2
(operating point)
Loop feedback delay t is built in with transmission
line design
L/N
C/2N
L/N
L/N
C/2N
C/2N
C/2N
C/2N
C/2N
N units
L=5 mH
0
C=1nF
Power level [dB]
-10
u=0.1 ms/unit;
-20
-30
-40
-50
= 1.2 ms
-60
0.0
1.0
2.0
3.0
Frequency [MHz]
4.0
5.0
fcutoff ~ 3 MHz
Simulation Results
2
Bifurcation Diagram
X
0
-2
1
2
b = 1.6
0.4
0.4
3
4
6
7
b = 2.7
0.6
0.6
0.3
b 5
b = 6.2
2
2
0.4
1.5
0.2
1
Vtune [V]
X(t)
0.1
0
Vtune [V]
0.2
0
Vtune [V]
0
-0.1
0
-0.5
-0.4
-0.3
-0.4
0
0
0
-0.2
-0.2
-0.4
0
0.5
-1
0.2
0.4
0.6
0.8
1
Time [s]
1.2
10
Time [ms]
1.4
1.6
-0.6
0
20
1.8
2
-5
x 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s]
10
Time [ms]
1.6
1.8
2
-1.5
0
20 -1.5
-5
x 10
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s]
10
Time [ms]
1.6
1.8
2
-5
x 10
20
Bifurcation Diagram
2
Experiment
V 0
Spectral diagram
of microwave signal
Frequency [GHz]
-2
3.2
3.1
3
2.9
2.8
2.7
Coupling and Synchronization

 : coupling strength
d
d
VCO
bias
 1 (t)
VCO
mixer
splitter
v1(t)
H(s)
b
bias

(I)
 2 (t)
mixer
splitter
v2(t)
H(s)
b

(II)
• Two systems are coupled in microwave band within or outside of filter bandwidth
• Two possible types of synchronization:
- Baseband Envelope Synchronization
v1  t   v 2  t 
- Microwave Phase Synchronization
1  t    2  t 
Experimental Results
Unidirectional coupling, outside filter bandwidth,  = 0.25
b = 1.2
b = 5.1
V1(t)
V1(t)
V2(t)
V2(t)
1   2
2
V1(t)-V2(t)
V1(t)-V2(t)
5
1
1   2
1   2
2
2
0
0
-1
0
-5
5
15
10
Time [ms]
20
0
5
15
10
Time [ms]
20
Experimental Results
Bidirectional coupling, outside filter bandwidth,  = 0.35
b = 1.2
b = 2.1
V1(t)
V1(t)
V2(t)
V2(t)
V1(t)-V2(t)
V1(t)-V2(t)
1
2
1   2
1   2
2
2
0
0
-1
0
5
15
10
Time [ms]
20
-2
0
5
10
15
Time [ms]
20
Transmission line for VCO system?
* Microstrip line with characteristic impedance 50 Ohm
 r  3.48  0.05
Dielectric material: Roger 4350B with
* Using transmission line to provide certain delay time in RF range
  L.
r
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
Transmission line
Distance Radar
scope
o Idea:
VCO
Using microwave signal generated
by VCO for detecting position of
object in a cavity

integrator
o Mathematical model:
 Nonlinearity
Vo u t  Vo co s  2  d Vin   0 d 
In general
RF delay and nonlinearity
V
0
2

1
 d
0
V2
 0   0 d
In particular case has been investigated
 2

V o u t  V o sin 
V in 
 V2

V0
0    / 2
Gain =2.5
Gain =4.137
Gain =3.77
How much chances we can detect?
Continuously change d
 d  d   t
0
scope
Assumption:
d

VC
O
 is in order of 10-9
V2 
0
1

 0   0 d
0
0
d
integrator

Approximated equation:
dx
dt
  R sin  x  t  1   0  .  t  1 
  R sin  x  t  1   .  t  1 
Normalization:
R  2
V0 
V
0
2
T
x  2
V
0
V2
 0   / 2
Watching dynamics of system, can we determine  (and then z?)
Chaos Generator
Using PLL to track chaotic FM signal
Chaotic FM signal
scope
 c  t  Ae
VC
O
 p  t  Ae
dp
dt
j p  t 
   2  v p  t 
0
p
dc
vpm
vp
integrator
Mixer output
Or another filter function?
v m  t   R e   p  t   c  t 
p
*
0
0
Always can pick  p   c
v m  A co s   p   c 
p
2
[A2]: voltage as Vp-p
Integrator equation
dvp
dt
PLL equation

1
T
1 d p
2
v
p
m
2  d t
d p
2
dt
2
 b p co s   p   c 
2

1
T
A co s   p   c 
2
bp 
2  A
T
2
j c  t 
dt
  c  2  v c  t 
0
Chaos generator
dc
Equations:
dt
  c  2 v c  t 
0
d c
2
dt
2
dvc
dt

 2

sin 
vc t    
T
 v 2

A
2
 2

 b c sin 
vc t    
 v 2

bc 
2  A
2
T
 Does solution  p   c exist?
d p
2
dt
2
d c
2
dt
2
 b p co s   p   c 
 2

 b c sin 
vc t    
 v 2

Static = time evolution ?!
In general case, bc and bp could be assumed to be
different by some scaling factor bc/bp = n