1. 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

2. 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

3. 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

4. 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

5. 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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6. 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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7. Equilibrium in 3D:
3 real eigenvalues
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8. Equilibrium in 3D:
1 real + 2 complex-conjugates
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10. 10

11. Illustration of typical homoclinic
and heteroclinic orbits
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12. 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

13. Equilibria and eigenvalues of
several typical systems
13

14.  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.
14

15. Untable saddle-focus is
important for generating chaos
15

16.  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. 16

17. 17

18. Rossler System
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19. 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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20. 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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21. 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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22. Some Sprott systems
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23. 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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24. 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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25. Equilibria and eigenvalues of
the new system
25

26. The largest Lyapunov
exponent
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27. The new system:
chaotic attractor with a = 0.006
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28. Bifurcation diagram
a period-doubling route to chaos
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29. Phase portraits and frequency
spectra
a = 0.006 a = 0.02
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30. Phase portraits and frequency
spectra
a = 0.03 a = 0.05
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31. Attracting basins of the
equilibra
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32. 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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33. 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?
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34. Coexisting of point, cycle and
strange attractor
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35. Coexisting of point, cycle and
strange attractor
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36. Coexisting of point, cycle and
strange attractor
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37. 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

38. Chaotic system with no
equilibrium
Xiong Wang: Chaotic system with only one 38
stable equilibrium

39. Chaotic system with one
stable equilibrium
Xiong Wang: Chaotic system with only one 39
stable equilibrium

40. 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

41. The idea of symmetry
W Z n
W plane Z plane
W = (u,v) = u+vi Z = (x,y) = x+yi
Original system Symmetrical system
(u,v,w) (x,y,z)
Xiong Wang: Chaotic system with only one 41
stable equilibrium

42. symmetry
Xiong Wang: Chaotic system with only one 42
stable equilibrium

43. 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

44. symmetry
a = 0.005 > 0, stable equilibria
Xiong Wang: Chaotic system with only one 44
stable equilibrium

45. symmetry
a = - 0.01 < 0, unstable equilibria
Xiong Wang: Chaotic system with only one 45
stable equilibrium

46. symmetry
Xiong Wang: Chaotic system with only one 46
stable equilibrium

47. Three symmetrical equilibria
with tunable stability
Xiong Wang: Chaotic system with only one 47
stable equilibrium

48. symmetry
a = - 0.01 < 0, unstable equilibria
Xiong Wang: Chaotic system with only one 48
stable equilibrium

49. symmetry
a = 0.005 > 0, stable equilibria
Xiong Wang: Chaotic system with only one 49
stable equilibrium

50. Theoretically we can create
any number of equilibria …
Xiong Wang: Chaotic system with only one 50
stable equilibrium

51. 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

52. 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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53. Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email: wangxiong8686@gmail.com
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54. ADDITIONAL BONUS:
ATTRACTOR GALLERY
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