Stable chaos

Stable Chaos
Study 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

Some interesting
questions?
 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

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.


Jacobian Matrix
 Thestability of typical equilibria of smooth
ODEs is determined by the sign of real part of
eigenvalues of the Jacobian matrix.


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.


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.


Equilibrium in 3D
3 real eigenvalues case


Equilibrium in 3D
1 real+ 2 complex-conjugate


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)


 Lorenz System
 x = a( y − x)
&

 y = cx − xz − y
&
 z = xy − bz ,
&
a = 10, b = 8 / 3, c = 28

E. N. Lorenz, “Deterministic non-periodic flow,” J. Atmos. Sci., 20,
130-141, 1963.


 Chen System

 x = a ( y − x)
&

 y = (c − a ) x − xz + cy
&
 z = xy − bz ,
&
a = 35; b = 3; c = 28

G. 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, “Chen's attractor exists,” Int. J. of Bifurcation and Chaos,
14, 3167-3178, 2004. Xiong Wang:Email:wangxiong8686@gmail.com 12

Rossler System


Chaotic system with two stable
equilibria

When r<0.05, there are one saddle and two stable
node-foci
http://arxiv.org/abs/1101.4262

LLE for 0<r<0.05


Chaotic transient1 r=0


Change the initial condition a
little bit …


r=0.01, converge at time 6000


r=0.015, converge at time
17000


r=0.02


r=0.02, time 610000


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….


Try to find chaotic system
with stable Equilibrium
 Some criterions for the new system:
1. One equilibrium
2. Equation algebraic simple
3. Stable

To start with, let us ﬁrst review some of the
simple Sprott chaotic systems with only one
equilibrium…

Some Sprott systems


Idea
1. 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…


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


Equilibria and eigenvalues of
the new system


The largest Lyapunov
exponent


The new system:
chaotic attractor with a = 0.006


Bifurcation diagram
a period-doubling route to chaos


Phase portraits and frequency
spectrums

a=0.0 a=0.0
06 2

Phase portraits and frequency
spectrums

a=0.0 a=0.0
3 5

Attracting basin of the
equilibrium


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.

Open questions
To be further considered:
Ši’lnikov homoclinic criterion?
not applicable for this case

Rigorous proof of the existence?
Horseshoe ?
 Coexistence of point attractor and strange
attractor…

Inflation of attraction basin of the equilibrium…

Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email: wangxiong8686@gmail.com


Stable chaos

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