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# Stable chaos

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### Stable chaos

1. 1. Stable ChaosStudy the relation between the chaotic dynamic system and the stability of equilibrium Xiong Wang 王雄Supervised by: Chair Prof. Guanrong Chen Centre for Chaos and Complex Networks City University of Hong Kong
2. 2. Some interestingquestions? Ifyou are given a simple 3D ODE system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Could such a system also generate chaotic dynamic? Generally, what’s the relation between the chaotic dynamic system and the stability of equilibrium? Xiong Wang:Email:wangxiong8686@gmail.com 2
3. 3. PART 1INTRODUCTION
4. 4. Equilibrium An equilibrium (or fixed point) of a dynamical system generated by an autonomous system of ordinary differential equations (ODEs) is a solution that does not change with time. The ODE x = f ( x ) has an equilibrium & solution xe , if f ( xe ) = 0 Finding such equilibria, i.e., solving the f ( x) = 0 equation is easy only in a few special cases. Xiong Wang:Email:wangxiong8686@gmail.com 4
5. 5. Jacobian Matrix Thestability of typical equilibria of smooth ODEs is determined by the sign of real part of eigenvalues of the Jacobian matrix. Xiong Wang:Email:wangxiong8686@gmail.com 5
6. 6. Hyperbolic Equilibria The eigenvalues of J determine linear stability properties of the equilibrium. An equilibrium is stable if all eigenvalues have negative real parts; it is unstable if at least one eigenvalue has positive real part. The equilibrium is said to be hyperbolic if all eigenvalues of the Jacobian matrix have non- zero real parts. Xiong Wang:Email:wangxiong8686@gmail.com 6
7. 7. Hartman-Grobman Theorem Hyperbolic equilibria are robust: Small perturbations do not change qualitatively the phase portrait near the equilibria. The local phase portrait of a hyperbolic equilibrium of a nonlinear system is equivalent to that of its linearization. Xiong Wang:Email:wangxiong8686@gmail.com 7
8. 8. Equilibrium in 3D3 real eigenvalues case Xiong Wang:Email:wangxiong8686@gmail.com 8
9. 9. Equilibrium in 3D1 real+ 2 complex-conjugate Xiong Wang:Email:wangxiong8686@gmail.com 9
10. 10. Non-hyperbolic equilibria Ifat least one eigenvalue of the Jacobian matrix is zero or has a zero real part, then the equilibrium is said to be non-hyperbolic. Non-hyperbolic equilibria are not robust (i.e., the system is not structurally stable) Xiong Wang:Email:wangxiong8686@gmail.com 10
11. 11.  Lorenz System  x = a( y − x) &   y = cx − xz − y &  z = xy − bz , & a = 10, b = 8 / 3, c = 28E. N. Lorenz, “Deterministic non-periodic flow,” J. Atmos. Sci., 20, 130-141, 1963. Xiong Wang:Email:wangxiong8686@gmail.com 11
12. 12.  Chen System  x = a ( y − x) &   y = (c − a ) x − xz + cy &  z = xy − bz , & a = 35; b = 3; c = 28G. Chen and T. Ueta, “Yet another chaotic attractor,” Int J. of Bifurcation and Chaos, 9(7), 1465-1466, 1999.T. Ueta and G. Chen, “Bifurcation analysis of Chen’s equation,” Int J. of Bifurcation and Chaos, 10(8), 1917-1931, 2000.T. S. Zhou, G. Chen and Y. Tang, “Chens attractor exists,” Int. J. of Bifurcation and Chaos, 14, 3167-3178, 2004. Xiong Wang:Email:wangxiong8686@gmail.com 12
13. 13. Xiong Wang:Email:wangxiong8686@gmail.com 13
14. 14. Rossler System Xiong Wang:Email:wangxiong8686@gmail.com 14
15. 15. PART 2TWO STABLE
16. 16. Chaotic system with two stableequilibriaWhen r<0.05, there are one saddle and two stable node-foci http://arxiv.org/abs/1101.4262 Xiong Wang:Email:wangxiong8686@gmail.com 16
17. 17. Xiong Wang:Email:wangxiong8686@gmail.com 17
18. 18. LLE for 0<r<0.05 Xiong Wang:Email:wangxiong8686@gmail.com 18
19. 19. Chaotic transient1 r=0 Xiong Wang:Email:wangxiong8686@gmail.com 19
20. 20. Change the initial condition alittle bit … Xiong Wang:Email:wangxiong8686@gmail.com 20
21. 21. r=0.01, converge at time 6000 Xiong Wang:Email:wangxiong8686@gmail.com 21
22. 22. r=0.015, converge at time17000 Xiong Wang:Email:wangxiong8686@gmail.com 22
23. 23. r=0.02 Xiong Wang:Email:wangxiong8686@gmail.com 23
24. 24. r=0.02, time 610000 Xiong Wang:Email:wangxiong8686@gmail.com 24
25. 25. Question Is this chaos? If so, How to prove the existence of chaos when r is around 0.2~0.5 When the equilibria are stable… while the numerical LLE is positive…. Xiong Wang:Email:wangxiong8686@gmail.com 25
26. 26. PART 3ONE STABLE
27. 27. Try to find chaotic systemwith stable Equilibrium Some criterions for the new system:1. One equilibrium2. Equation algebraic simple3. StableTo start with, let us ﬁrst review some of the simple Sprott chaotic systems with only one equilibrium… Xiong Wang:Email:wangxiong8686@gmail.com 27
28. 28. Some Sprott systems Xiong Wang:Email:wangxiong8686@gmail.com 28
29. 29. Idea1. Sprott systems I, J, L, N and R all have only one saddle-focus equilibrium, while systems D and E are both degenerate case.2. A tiny perturbation to the system may be able to change such a degenerate equilibrium to a stable one.3. Hope it will work… Xiong Wang:Email:wangxiong8686@gmail.com 29
30. 30. Result (very lucky) When a = 0, it is the Sprott E system; when a>0, however, the stability of the single equilibrium is fundamentally different The single equilibrium become stable Xiong Wang:Email:wangxiong8686@gmail.com 30
31. 31. Equilibria and eigenvalues ofthe new system Xiong Wang:Email:wangxiong8686@gmail.com 31
32. 32. The largest Lyapunovexponent Xiong Wang:Email:wangxiong8686@gmail.com 32
33. 33. The new system:chaotic attractor with a = 0.006 Xiong Wang:Email:wangxiong8686@gmail.com 33
34. 34. Bifurcation diagrama period-doubling route to chaos Xiong Wang:Email:wangxiong8686@gmail.com 34
35. 35. Phase portraits and frequencyspectrums a=0.0 a=0.0 06 2 Xiong Wang:Email:wangxiong8686@gmail.com 35
36. 36. Phase portraits and frequencyspectrums a=0.0 a=0.0 3 5 Xiong Wang:Email:wangxiong8686@gmail.com 36
37. 37. Attracting basin of theequilibrium Xiong Wang:Email:wangxiong8686@gmail.com 37
38. 38. Conclusion We reported the ﬁnding of a simple three- dimensional autonomous chaotic system which, very surprisingly, has only one stable node-focus equilibrium. It has been veriﬁed to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous frequency spectrum, and a period-doubling route to chaos. Xiong Wang:Email:wangxiong8686@gmail.com 38
39. 39. Open questionsTo be further considered:Ši’lnikov homoclinic criterion? not applicable for this caseRigorous proof of the existence? Horseshoe ? Coexistence of point attractor and strangeattractor…Inflation of attraction basin of the equilibrium… Xiong Wang:Email:wangxiong8686@gmail.com 39
40. 40. Xiong Wang 王雄Centre for Chaos and Complex NetworksCity University of Hong KongEmail: wangxiong8686@gmail.com Xiong Wang:Email:wangxiong8686@gmail.com 40