Chaotic Systems with Any Number of
Equilibria and Their Hidden Attractors
IFAC CHAOS 2015
Tokyo, 26-28 August 2015
GuanRong Chen
Centre for Chaos and Complex Networks
City University of Hong Kong
Attractor with Multi-Equilibia
Acknowledgements
Xiong Wang
Gennady A. Leonov
Nikolay V. Kuznetsov
2

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.
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Main Characteristics
“Simple”
3D quadratic autonomous
Multiple equilibria
Smooth
After all: Chaotic
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State of the Art
Chaotic systems with
 No equilibrium
 One equilibrium
 Two equilibria
 Three equilibria
• An arbitrary number of equilibria ?
5
Chaotic system with no equilibrium
X Wang and G Chen: Constructing a chaotic system with any number
of equilibria, Nonl Dynam 2013
6
Chaotic system with one stable equilibrium
 x  yz  a

2


y
y
x

 z  1  4 x

X Wang and G Chen: A chaotic system with only one stable equilibrium,
Comm in Nonl Sci and Numer Simul, 2012
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Chaotic systems with two equilibria
Rössler system:
8
Chaotic systems with three equilibria
Lorenz system:
9
How to construct a chaotic system
with any number of equilibria ?
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Idea
 Start from a chaotic system with one stable equilibrium
 Add symmetry by using a suitable local diffeomorphism
 By-product: stability of the equilibria can be easily
adjusted by tuning a single parameter
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Symmetry
W  Zn
W plane
Z plane
W  (u , v )  u  vi
Z  ( x, y )  x  yi
original system:
translated system:
(u , v, w)
( x, y , z )
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Symmetry
local diffeomorphism
One equilibrium
Two equilibria
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Stability of the two equilibria
• Two symmetrical equilibria, independent of
parameter a
• Eigenvalue of Jacobian:
• So, a > 0 – stable; a < 0 – unstable
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Simulation
a  0.005  0
stable equilibria
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Simulation
a  0.01  0
unstable equilibria
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Symmetry
local diffeomorphism
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Three equilibria are
symmetrical with tunable stabilities
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Simulation
a  0.005  0
stable equilibria
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Simulation
a  0.01  0
unstable equilibria
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Now, theoretically one can create
any number of equilibria …
(u  wi )  ( x  yi) n
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For example …
2 
Ry   
5 
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Next
Consider multi-equilibrium chaotic
systems with hidden attractors
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Hidden Attractors
Self-excited attractor: if its basin of attraction intersects with a
small neighborhood of an equilibrium
Hidden attractor: otherwise
Examples of hidden attractors:
in a system without equilibrium
In a system with only one stable equilibrium
How to predict and compute hidden attractors? See:
Survey 1: G.A. Leonov, N.V. Kuznetsov, IJBC, 23(1): 1330002 (69pp), 2013
Survey 2: G.A. Leonov, N.V. Kuznetsov, Euro. Phys. J, 224: 1421-1458, 2015
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Hidden Attractor
of the system with one stable equilibrium
 x  yz  a

2


y
y
x

 z  1  4 x

X Wang and G Chen: A chaotic system with only one stable equilibrium, Comm in
Nonl Sci and Numer Simul, 2012
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Following:
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Recall: A chaotic system with no equilibrium
X Wang and G Chen: Constructing a chaotic system with any number
of equilibria, Nonl Dynam 2013
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More: chaotic systems with no equilibrium
……………
V T Pham et al., IJBC, 24(11): 1450146, 2014
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Consider Chaotic System NE4
Notice: Complex equilibria always exists
Transformation: 3D (real)  6D (complex)
x  x r  xi
y  y r  yi
z  zr  z i
Take and then slightly vary some initial conditions ( i,  ,  )
( ,  ,  ) are real parameters
Some transient chaotic behaviors can be observed,
but the original attractor(s) cannot be found.
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System NE4
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The Chaotic System with One Stable Equilibrium
Notice: Complex equilibria always exists
Transformation: 3D (real)  6D (complex)
x  xr  xi
y  y r  yi
z  zr  z i
Take and then slightly vary some initial conditions ( i,  ,  )
( ,  ,  ) are real parameters
Some transient chaotic behaviors can be observed, but the original
attractor(s) cannot be found 
“Yes, that is really hidden; among many other
mysterious secrets in the hidden oscillation world.”
V T Pham et al., IJBC, 24(11): 1450146, 2014
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Chaoticity of Systems with Multi-Equilibia
How to Verify (“Prove”) it ?
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Existence of Chaotic Attractors
Shilnikov Theorem (1967):
If a 3D autonomous system has two distinct saddle fixed
points and there exists a heteroclinic orbit connecting them,
and if the eigenvalues of the Jacobin of the system at these
fixed points are
 k ,  k  j k (k  1,2)
Satisfying |  k |  |  k |  0 (k  1,2) and 1  2  0 or 1 2  0
then the system has infinitely many Smale horseshoes and
hence has horseshoe chaos.
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EU2  j2 ( P2( 2) )
EU2  j2 ( P1( 2) ) ES2 ( P1( 2 ) )
ES2 ( P2( 2 ) )
Q2( 2)
(2)
2
P
L1( 2 )
(2)
2
L
P1( 2 )
Q1( 2)
V2
S
V1
(a) Single heteroclinic orbit
(b) Double-scroll attractor
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EU2  j2 ( P1( 2) )
EU2  j2 ( P3( 2) )
EU2  j 2 ( P2( 2 ) )
ES2 ( P1( 2 ) )
P1( 2 )
Q
L(22 )
L1( 2 )
L(32 )
(2)
2
P
Q6( 2 )
( 2)
4
L(62)
( 2)
3
(2)
5
P
L
V2
P4( 2 )
Q5( 2)
Q3( 2)
S1
ES2 ( P4( 2 ) )
(2)
4
L
Q1( 2)
V1
ES2 ( P3( 2 ) )
ES2 ( P2( 2 ) )
Q2( 2)
EU2  j 2 ( P4( 2 ) )
S2
V3
S3
V4
(a) Multiple heteroclinic orbit
(b) Multi-equilibrium attractors
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Multi‐Equilibrium Attractors in Nature
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Conclusions
In a typical 3D quadratic autonomous chaotic
system, such as the Lorenz and Rössler systems,
the number of equilibrium points is three or less
and the number of scrolls in their attractors is two
or less.
Today, we are able to construct a simple 3D
quadratic autonomous chaotic system that can
have any desired number of equilibrium points.
Such multi-equilibrium chaotic systems all have
hidden attractors.
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