1. High Dimensional Chaos
Tutorial Session
IASTED International Workshop on
Modern Nonlinear Theory (Bifurcation and Chaos)
~Montreal 2007~
Zdzislaw Musielak, Ph.D. and Dora Musielak, Ph.D.
University of Texas at Arlington (UTA)
Arlington, Texas (USA)
9/19/2018

2. High Dimensional Chaos (HDC)
Lecture 1: Basic Concepts and Definitions
Lecture 2: Extension of Lorenz and Duffing systems to high dimensions
Lecture 3: Other high-dimensional (HD) systems with chaos and hyperchaos
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3. Lecture 1 Objective: Review basic concepts and define
“high dimensional chaos”
Chaos and Predictability
Basic Techniques
Main Routes to Chaos
Low-Dimensional (LD) Systems
Lorenz and Duffing Systems
High-Dimensional Chaos (HDC)
Summary
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4. Chaos in Deterministic Systems
Deterministic dynamical systems (Newton 1687)
Present and future states of these systems
are fully determined by their initial conditions
Chaos in deterministic systems (Poincaré 1889)
Sensitivity of some systems to small changes
in their initial conditions is so strong that no
reasonable prediction in time is possible
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5. Fundamental Paradox System’s chaotic behavior is generated by
the fixed deterministic rules that do not
themselves involve any element of chance!
Since the system is deterministic, its future is
in principle completely determined by the past.
However, in practice, small uncertainties in the
system’s initial conditions are quickly amplified
and the system’s behavior becomes unpredictable.
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6. Predictability in Time
Distance between two initially close trajectories is
where λ is the Lyapunov exponent.
If λmax is the largest Lyapunov exponent, then the
so-called Lyapunov time TL is given by
where Ds is the characeristic size of the system.
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7. Basic Techniques Phase portraits Poincaré sections Power spectra
Lyapunov exponents
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8. Phase Portraits
Time series and attractors
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9. Low and High Dimensional Systems
Low-dimensional systems:
(i) dissipative and driven systems described in
three-dimensional (3D) phase space
(ii) 1D and 2D iterative maps
High-dimensional systems:
(i) dissipative and driven systems described in
phase space with dimension higher than 3D
(ii) iterative maps with dimensions higher than 2D
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10. Attractors and Lyapunov Exponents
Fixed point: all three Lyapunov exponents are negative
Limit cycle: two Lyapunov exponents are negative and
one is zero
Torus: two Lyapunov exponents are zero and one is
negative
Strange attractor: one Lyapunov exponent is positive,
one is zero and one is negative
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11. Poincaré Sections I
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Thompson & Stewart (1986)

12. Poincaré Sections II Stroboscopic method Thompson & Stewart (1986)
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13. Power Spectra
Roy and Musielak (2007)
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14. Lyapunov Exponents
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Rossler (1983)

15. Routes to Chaos Via local bifurcations: Period-doubling
Quasi-periodicity
Intermittency
Via global bifurcations:
Chaotic transients
Crisis
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16. Period-Doubling Cascade I
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Feigenbaum (1979)

17. Period-Doubling Cascade II
Logistic map
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18. Quasi-Periodicity I Argyris, Faust and Haase (1993)
Landau (1944)
Ruelle & Takens (1971)
Argyris, Faust and Haase (1993)
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19. Quasi-Periodicity II In some systems a third distinct frequency occurs
NRT theorem: no more than 3 frequencies can be observed
Newhouse, Ruelle and Takens (NRT 1978)
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20. Quasi-Periodicity III
Rayleigh-Benard
convection
Swinney and
Gollub (1978)
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21. Quasi-Periodicity IV
Quasi-periodic route to chaos shown by Poincare sections
Transition to chaos
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22. Intermittency Pomeau & Manneville (1980)
Time-history response of the vertical velocity component in
three different Rayleigh-Benard convection experiments
Berge et al. (1980)
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Pomeau & Manneville (1980)

23. Chaotic transients and Crisis
Chaotic transients – system’s trajectories interact with
various unstable fixed points and
limit cycles; homoclinic and
heteroclinic orbits may suddenly
appear and strongly influence the
trajectories
Crisis – a strange attractor may suddenly disappear as
a result of its interaction with an unstable fixed
point or an unstable limit cycle
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24. Low-Dimensional Systems
Lorenz model
Duffing oscillator
Van der Pol oscillator
Dissipative and driven pendulum
Belousov-Zhabotinsky reactions
Rossler model
Logistic map
Hanon map
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25. Lorenz Model  Continuity equation for an incompressible fluid
Convective rolls
HEAT
 Continuity equation for an incompressible fluid
 Navier-Stokes equation with constant viscosity
 Heat transfer equation with constant thermal conductivity
Lorenz (1963)
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26. Mathematical Description
Double Fourier expansion in the vertical (z) and
horizontal (x) direction of the stream function
and the temperature deviation
Modes of expansion:
and
Saltzman (1962)
Lorenz truncation:
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27. Lorenz Strange Attractor
Generated by the “stretch-tear-squeeze” mechanism (Gilmore 1998)
Capacity dimension of the strange attractor dcap = 2.06 (Lorenz 1984)
Rigorous proof of the existence of the attractor (Tucker 1999)
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28. Lorenz Equations
where
and
3D model in phase space
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29. Route to Chaos
Chaotic transients
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Kennamer (1995)

30. Lorenz-like Models
Generalized Lorenz models describing turbulent flows
Lorenz-Maxwell-Haken model describing lasers
Reversal of Earth’s magnetic field
Several models used in communication
Models used to describe chaotic cryptosystems
Models used to control chaos
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31. Duffing Oscillator
Duffing (1918)
Ueda (1979,1980)
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32. Duffing Equations
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33. Lyapunov Exponents
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34. Route to Chaos
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Benner (1997)

35. Duffing Strange Attractor
Generated by the “stretch and roll” mechanism (Gilmore 1998)
Capacity dimension: for ω = 0.04
2.38 for ω = 0.20
2.53 for ω = 0.32
Burke (1998)
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36. Out of Chaos
Crisis route
Benner (1997)
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37. Duffing-like Systems Electric circuits with nonlinear inductance
Nonlinear mechanical oscillators
Mechanical systems that contain gears, backlash
and deadband regions
Large deflections of elastic continua
Buckling of beam-colums
Rigid and flexible missiles
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38. From Low to High-Dimensional Systems
Extending the 3D Lorenz model to higher dimensions
Extending the 3D Duffing system to higher dimensions
Other high-dimensional (HD) dynamical systems
High-dimensional chaos (HDC) in these systems
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39. High-Dimensional Chaos (HDC)
Previously suggested definitions involve:
(a) two or more positive Lyapunov exponents
(also called hyperchaos)
(b) high-dimensional strange attractors
(c) multiple strange attractors
Definition of HDC used in this tutorial:
HDC refers to chaos observed in dynamical
systems with phase space dimensions D > 3
and with strange attractors whose correlation dimension dcor > 3
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40. SUMMARY
Basic techniques: phase portraits, Poincare sections, power spectra and Lyapunov exponents
Routes to chaos: period-doubling, quasi-periodicity, intermittency, chaotic transients and crisis
Chaotic behavior in 3D Lorenz system and chaotic transients as
the system’s route to fully developed chaos
Chaotic behavior in 3D Duffing system and period-doubling as
Methods for constructing high-dimensional (HD) systems and
definition of high dimensional chaos (HDC)
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