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1 Introduction Origins synchronization: C. Huygens, 1650 two identical pendulums attached to a beam control: J. Watt, 1788 steam engine governor, a lift-tenter mechanism chaos: 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.”
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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!