1. Control and Synchronization of Chaos Li-Qun Chen Department of Mechanics, Shanghai University Shanghai Institute of Applied Mathematics and Mechanics Shanghai Center of Nonlinear Science

2. Outline 1Introduction 2Chaos 3Control of chaos 4Synchronization of chaos 5Summary

3. 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.”

4. 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 field since 1990 Significances new stage of the development of nonlinear dynamics powerful stimulation to nonlinear system theory possible approach to explore complexity first step towards application of chaos

5. 2 Chaos ( Liu YZ & Chen LQ, Nonlinear Oscillations. Higher-Education Press, 2001 ) Descriptions of Chaos motion in a deterministic system sensitively depending on initial conditions (thus unpredictable in long time) recurrent but without any periods random-like Example: Ueda’s oscillator M=1 -x 3 x c=0.05 7.5cost displacement

6. 2 Chaos (cont.) sensitivity to initial state Numerical characteristic: the Lyapunov exponents (positive) Time histories: x(t)-t

7. 2 Chaos (cont.) phase trajectories: x(t)- (t) butterfly effect: long-time unpredictability

8. 2 Chaos (cont.) recurrent aperiodicity Numerical characteristic: fractal dimensions (non-integer) Poincare map: X(2  )- (2  ) Ueda’s attractor

9. 2 Chaos (cont.) Intrinsic (spontaneous) stochasticity Numerical characteristic: power spectral (continuously distributed)

10. 3 Control Definition (Liu YZ & Chen LQ. Nonlinear Dynamics. Shanghai Jiaotong Univ. Press, 2000) controlled discrete-time system governing equation with a control input u k, and observable output variable

11. 3 Control (cont.) For prescribed periodic goal g k design a control law such that

12. 3 Control (cont.) specific problems of controlling chaos Stabilizing chaos unstable periodic orbits embedded in chaos targeting chaos Suppressing chaos

13. 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

14. 3 Control (cont.) given periodic orbits tracked

15. 3 Control (cont.) (2) stabilizing periodic orbits

16. 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

17. 3 Control (cont.) Controlled time histories

18. 3 Control (cont.) Control signals required

19. 4 Synchronization Definition (Chen LQ, Chaos, Solitions, & Fractals, 2004, 21: 349) two coupled systems with control inputs governing equation observable output functions

20. 4 Synchronization (cont.) design a control law exact synchronization asymptotic synchronization approximate synchronization

21. 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

22. 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

23. 4 Synchronization (cont.) Synchronization between Gauss map and logistic map controlled time history the difference

24. 4 Synchronization (cont.) Synchronization between chaotic orbits starting at different initial conditions controlled time history the difference

25. 4 Synchronization (cont.) control signals Gauss map-logistic map Gauss map

26. 5 Summary chaos a deterministic recurrent aperiodic motion sensitive to its initial conditions control of chaos driving asymptotically an output of a chaotic system to a prescribed periodic goal synchronization of chaos adjusting a given property of two chaotic systems to a common behavior

27. Thank You!