Control and Synchronization of Chaos

Li-Qun Chen

Department of Mechanics,Shanghai UniversityShanghai Institute of Applied Mathematics and MechanicsShanghai Center of Nonlinear Science

Outline

1 Introduction

2 Chaos

3 Control of chaos

4 Synchronization of chaos

5 Summary

25/01

1 IntroductionOrigins synchronization: C. Huygens, 1650 two identical pendulums attached to a beamcontrol: J. Watt, 1788steam engine governor, a lift-tenter mechanismchaos: H. Poincare, 1894“It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible.”

25/02

1 Introduction (cont.)controlling chaos: J. von Neumann, 1950 “As soon as we have some large computers working,the problems of meteorology will be solved. All processes that are stable we shall predict, and all processes that are unstable we shall control.”Active research fieldsince 1990Significances new stage of the development of nonlinear dynamicspowerful stimulation to nonlinear system theorypossible approach to explore complexityfirst step towards application of chaos

25/03

2 Chaos(Liu YZ & Chen LQ, Nonlinear Oscillations. Higher-Education Press, 2001)Descriptions of Chaos motion in a deterministic systemsensitively depending on initial conditions(thus unpredictable in long time) recurrent but without any periodsrandom-like Example: Ueda’s oscillator

txxx cos5.705.0 3 25/04

M=1

-x3x

c=0.05

7.5cost

displacement

2 Chaos (cont.)

sensitivity to initial state Numerical characteristic: the Lyapunov exponents (positive)Time histories: x(t)-t

0.40,0.30 11 xx

02.40,01.30 22 xx

25/05

2 Chaos (cont.)

phase trajectories: x(t)- (t)

butterfly effect: long-time unpredictability25/06

x

2 Chaos (cont.)

recurrent aperiodicity Numerical characteristic: fractal dimensions (non-integer)

Poincare map: X(2)- (2)

Ueda’s attractor

25/07

x

2 Chaos (cont.)

Intrinsic (spontaneous) stochasticity Numerical characteristic: power spectral (continuously distributed)

25/08

3 Control

Definition (Liu YZ & Chen LQ. Nonlinear Dynamics. Shanghai Jiaotong Univ. Press, 2000)

controlled discrete-time system governing equation

with a control input uk, and observable output variable

kkkk ,,1 uxEx

kkk ,xMy 25/09

3 Control (cont.)

For prescribed periodic goal gk

design a control law

such that

0lim kkk

gy

kkkk ,, gxfu

25/10

3 Control (cont.)

specific problems of controlling chaos

Stabilizing chaos unstable periodic orbits embedded in chaos

targeting chaos

Suppressing chaos

0,1 kk gEg

0 j

kjk gx

25/11

3 Control (cont.)

Example 1: control of a discrete-time system (Chen LQ, Physics Letters A, 2001, 281: 327)

hyperchaotic chaotic map

(1) tracking given periodic orbits

nnnnnn yabxyyxax 2,1 122

1

1.0,15.31 yn

xn

xn

xn gggg

2.0,12.31 yn

xn

xn

xn gggg

25/12

3 Control (cont.)

given periodic orbits tracked

2000 2500 3000 3500 4000-1

0

1

n

x n

2000 2500 3000 3500 4000-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

n

y n

25/13

3 Control (cont.)

(2) stabilizing periodic orbits

2000 2500 3000 3500 4000

0

n

x n

2000 2500 3000 3500 4000

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

n

y n

25/14

3 Control (cont.)

Example 2: control of a chaotic oscillator(Chen LQ & Liu YZ, Nonlinear Dynamics, 1999, 20: 309)

desired goals

fixed point

periodic motion

qutqqqq cos3.02.0 3

11R ty

tty 5.0sin4.05.02R 25/15

3 Control (cont.)

Controlled time histories

0 40 80 120 160 200

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

y

t0 40 80 120 160 200

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

y

t

25/16

3 Control (cont.)

Control signals required

0 40 80 120 160 200-40

-20

0

20

40

60

u

t0 40 80 120 160 200

0

200

400

600

800

1000

u

t

25/17

4 Synchronization

Definition (Chen LQ, Chaos, Solitions, & Fractals, 2004, 21: 349)

two coupled systems with control inputsgoverning equation

observable output functions

2,1,,, 21 itii uxxfx

2,1,,, 21 itii uxxhy

25/18

4 Synchronization (cont.)

design a control law

exact synchronization

asymptotic synchronization

approximate synchronization

tt 21 yy

0lim 21

ttt

yy

021 tt yy25/19

21,, xxgu t

4 Synchronization (cont.)

Special types of synchronization coordinate synchronization projective synchronization frequency synchronization phase synchronization generalized synchronization

control of chaos and anti-control of chaos

25/20

4 Synchronization (cont.)

Example: synchronization of chaotic maps(Chen LQ & Liu YZ, International Journal of Bifurcation and Chaos, 2002, 12: 1219)

Gauss map

logistic map

n

x

nn uxx n

2221

1 e2

nnn ggg 10.41

25/21

4 Synchronization (cont.)

Synchronization between Gauss map and logistic mapcontrolled time history the difference

2000 2500 3000 3500 4000

0

2

n

x n

2000 2500 3000 3500 4000-2

0

2

n

x n-gn

25/22

4 Synchronization (cont.)

Synchronization between chaotic orbits starting at different initial conditionscontrolled time history the difference

25/23

2000 2500 3000 3500 4000

0

2

n

x n

2000 2500 3000 3500 4000-2

0

2

n

x n-gn

4 Synchronization (cont.)

control signalsGauss map-logistic map Gauss map

25/24

2000 2500 3000 3500 4000

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

n

u n

2000 2500 3000 3500 4000-2

0

n

u n

5 Summarychaosa deterministic recurrent aperiodic motionsensitive to its initial conditions

control of chaosdriving asymptotically an output of a chaoticsystem to a prescribed periodic goal

synchronization of chaosadjusting a given property of two chaoticsystems to a common behavior

25/25

Thank You!