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Some Recent Discoveries and Challenges in Chaos Theory Xiong Wang 王雄 Supervised by: Prof. Guanrong Chen Centre for Chaos and Complex Networks City University of Hong Kong
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Some basic questions? What’s the fundamental mechanism in generating chaos? What kind of systems could generate chaos? Could a system with only one stable equilibrium also generate chaotic dynamics? Generally, what’s the relation between a chaotic system and the stability of its equilibria? 2
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Equilibria An equilibrium (or fixed point) of 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, by solving the f ( x) 0 equation analytically, is easy only in a few special cases. 3
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Jacobian Matrix The stability of typical equilibria of smooth ODEs is determined by the sign of real parts of the system Jacobian eigenvalues. Jacobian matrix: 4
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Hyperbolic Equilibria The eigenvalues of J determine linear stability of the equilibria. 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 have non-zero real parts. 5
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Hartman-Grobman Theorem The local phase portrait of a hyperbolic equilibrium of a nonlinear system is equivalent to that of its linearized system. 6
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Illustration of typical homoclinicand heteroclinic orbits 11
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Review of the two theorems Hartman-Grobman theorem says nonlinear system is the ‘same’ as its linearized model Shilnikov theorem says if saddle-focus + Shilnikov inequalities + homoclinic or heteroclinic orbit, then chaos exists Most classical 3D chaotic systems belong to this type Most chaotic systems have unstable equilibria 12
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Equilibria and eigenvalues ofseveral typical systems 13
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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. 14
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Untable saddle-focus isimportant for generating chaos 15
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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. 16
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Do these two theoremsprevent “stable” chaos? Hartman-Grobman theorem says nonlinear system is the same as its linearized model. But it holds only locally …not necessarily the same globally. Shilnikov theorem says if saddle-focus + Shilnikov inequalities + homoclinic or heteroclinic orbit, then chaos exists. But it is only a sufficient condition, not a necessary one. 19
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Don’t be scarred by theorems So, actually the theorems do not rule out the possibility of finding chaos in a system with a stable equilibrum. Just to grasp the loophole of the theorems … 20
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Try to find a chaotic systemwith a stable Equilibrium Some criterions for the new system:1. Simple algebraic equations2. One stable equilibriumTo start with, let us first review some of the simple Sprott chaotic systems with only one equilibrium … 21
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Idea1. Sprott systems I, J, L, N and R all have only one saddle-focus equilibrium, while systems D and E are both degenerate.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 … 23
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Finally Result 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 becomes stable 24
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Equilibria and eigenvalues ofthe new system 25
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Conclusions We have reported the finding of a simple 3D 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. 32
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Theoretical challengesTo be further considered: Shilnikov homoclinic criterion? not applicable for this case Rigorous proof of the existence? Horseshoe? Coexistence of point attractor and strangeattractor? Inflation of attracting basin of the equilibrium? 33
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Coexisting of point, cycle andstrange attractor 34
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Coexisting of point, cycle andstrange attractor 35
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Coexisting of point, cycle andstrange attractor 36
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one question answered,more questions come …Chaotic system with one stable equilibrium Chaotic system with: No equilibrium? Two stable equilibria? Three stable equilibria? Any number of equilibria? Tunable stability of equilibria? Xiong Wang: Chaotic system with only one 37 stable equilibrium
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Chaotic system with noequilibrium Xiong Wang: Chaotic system with only one 38 stable equilibrium
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Chaotic system with onestable equilibrium Xiong Wang: Chaotic system with only one 39 stable equilibrium
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Idea Really hard to find a chaotic system with a given number of equilibria in the sea of all possibility ODE systems … Try another way… To add symmetry to this one stable system. We can adjust the stability of the equilibria very easily by adjusting one parameter Xiong Wang: Chaotic system with only one 40 stable equilibrium
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The idea of symmetry W Z n W plane Z planeW = (u,v) = u+vi Z = (x,y) = x+yiOriginal system Symmetrical system (u,v,w) (x,y,z) Xiong Wang: Chaotic system with only one 41 stable equilibrium
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symmetry Xiong Wang: Chaotic system with only one 42 stable equilibrium
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Stability of the two equilibria There are two symmetrical equilibria which are independent of the parameter a The eigenvalue of Jacobian So, a > 0 stable; a < 0 unstable Xiong Wang: Chaotic system with only one 43 stable equilibrium
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symmetrya = 0.005 > 0, stable equilibria Xiong Wang: Chaotic system with only one 44 stable equilibrium
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symmetrya = - 0.01 < 0, unstable equilibria Xiong Wang: Chaotic system with only one 45 stable equilibrium
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symmetry Xiong Wang: Chaotic system with only one 46 stable equilibrium
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Three symmetrical equilibriawith tunable stability Xiong Wang: Chaotic system with only one 47 stable equilibrium
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symmetrya = - 0.01 < 0, unstable equilibria Xiong Wang: Chaotic system with only one 48 stable equilibrium
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symmetrya = 0.005 > 0, stable equilibria Xiong Wang: Chaotic system with only one 49 stable equilibrium
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Theoretically we can createany number of equilibria … Xiong Wang: Chaotic system with only one 50 stable equilibrium
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Conclusions Chaotic system with: No equilibrium - found Two stable equilibria - found Three stable equilibria - found Theoretically, we can create any number of equilibria … We can control the stability of equilibria by adjusting one parameter Xiong Wang: Chaotic system with only one 51 stable equilibrium
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Chaos is a global phenomenon A system can be locally stable near the equilibrium, but globally chaotic far from the equilibrium. This interesting phenomenon is worth further studying, both theoretically and experimentally, to further reveal the intrinsic relation between the local stability of an equilibrium and the global complex dynamical behaviors of a chaotic system 52
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Xiong Wang 王雄Centre for Chaos and Complex NetworksCity University of Hong KongEmail: wangxiong8686@gmail.com 53