complex dynamics of a microwave time-delayed feedback loop

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


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


Chaos Quantifying chaos Type of chaotic signals Microwave chaos

Introduction :

Frequency modulated chaotic microwave signal.


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



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



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:


• 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.


• 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+


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


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


• Field Programmable Gate Array board

• Sampling rate: Fs = 75.75 Msample/s• 2 phase-locked loop built in• 8-bit ADC• 10-bit DAC


Altera Cyclone II

FX2 USB port

Output

Input

DAC

FPGA chip ADC

• Memory buffer with length N to create delay


• 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



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


• Low feedback strength generated periodic behavior.

• Period: 4 (6.25kHz)

R = /2


• Intermediate feedback strength generated: More complicated but still periodic behavior.

R = 4.1


• High feedback strength: Chaotic behavior.

• Irregular, aperiodic but still deterministic.

• lmax = +5.316/t , DK-Y = 2.15

R = 4.176


basebandmicrowave


Power spectra

Period-doubling route to chaos


Bifurcation diagrams


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


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)


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).



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


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!


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


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


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


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


• Broadband behavior at microwave frequency

Experiment Simulation

Spectrum of FM microwave chaotic signal

2.9 GHZ

52 MHz

15dB/div


• 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 ?!

complex dynamics of a microwave time-delayed feedback loop

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