V4 cccn stable chaos

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?

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 has an equilibrium

solution , if

Finding such equilibria, by solving the

equation analytically, is easy only in a few

special cases.3

( )x f x

ex ( ) 0ef x

( ) 0f x

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:

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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.

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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.

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Equilibrium in 3D:

3 real eigenvalues

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Equilibrium in 3D:

1 real + 2 complex-conjugates

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Illustration of typical homoclinic

and heteroclinic orbits

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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

equilibria12

Equilibria and eigenvalues of

several typical systems

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

130-141, 1963.

Lorenz System

,

)(

bzxyz

yxzcxy

xyax

28,3/8,10 cba

Untable saddle-focus is

important for generating chaos

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Chen System

28;3;35 cba

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.

,

)(

)(

bzxyz

cyxzxacy

xyax

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http://www.youtube.com/watch?v=owwI2ICB-G4

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Rossler System

Do these two theorems

prevent “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.

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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 …

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Try to find a chaotic system

with a stable Equilibrium

Some criterions for the new system:

1. Simple algebraic equations

2. One stable equilibrium

To start with, let us first review some of the

simple Sprott chaotic systems with only one

equilibrium …

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Some Sprott systems

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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.

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 …

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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

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Equilibria and eigenvalues of

the new system

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The largest Lyapunov

exponent

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The new system:

chaotic attractor with a = 0.006

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Bifurcation diagrama period-doubling route to chaos

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Phase portraits and frequency

spectra

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a = 0.006 a = 0.02

Phase portraits and frequency

spectra

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a = 0.03 a = 0.05

Attracting basins of the

equilibra

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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.

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Theoretical challenges

To be further considered:

Shilnikov homoclinic criterion?

not applicable for this case

Rigorous proof of the existence?

Horseshoe?

Coexistence of point attractor and strange

attractor?

Inflation of attracting basin of the equilibrium?

Coexisting of point, cycle and

strange attractor

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strange attractor

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strange attractor

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Xiong Wang: Chaotic system with only one

stable equilibrium

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one question answered,

more questions come …

Chaotic system with:

No equilibrium?

Two stable equilibria?

Three stable equilibria?

Any number of equilibria?

Tunable stability of equilibria?

Chaotic system with one stable equilibrium

Chaotic system with no

equilibrium


stable equilibrium

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Chaotic system with one

stable equilibrium


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


stable equilibrium

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The idea of symmetry


stable equilibrium

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nW Z

W planeW = (u,v) = u+viOriginal system

(u,v,w)

Z planeZ = (x,y) = x+yi

Symmetrical system(x,y,z)

symmetry


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


stable equilibrium

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symmetry


stable equilibrium

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a = 0.005 > 0, stable equilibria

symmetry


stable equilibrium

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a = - 0.01 < 0, unstable equilibria

symmetry


stable equilibrium

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Three symmetrical equilibria

with tunable stability


stable equilibrium

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symmetry


stable equilibrium

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a = - 0.01 < 0, unstable equilibria

symmetry


stable equilibrium

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a = 0.005 > 0, stable equilibria

Theoretically we can create

any number of equilibria …


stable equilibrium

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Conclusions


stable equilibrium

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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

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

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Xiong Wang 王雄Centre for Chaos and Complex Networks

City University of Hong Kong

Email: [email protected]

ADDITIONAL BONUS:

ATTRACTOR GALLERY

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http://www.youtube.com/watch?v=eNmr-fewDi4

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http://www.youtube.com/watch?v=6N9jrfrBLPA

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V4 cccn stable chaos