High Dimensional Chaos

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)
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Lecture 2
Objective Construct high-dimensional Lorenz and
Duffing systems and study high-dimensional
chaos (HDC).

• High-dimensional Lorenz models
• HDC in these models
• High-dimensional Duffing systems
• HDC in these systems
• Summary

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2D Rayleigh-Benard Convection
Model
? Continuity equation for an incompressible
fluid ? Navier-Stokes equation with constant
viscosity ? Heat transfer equation with constant
thermal conductivity
Saltzman (1962)
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Saltzmans Equations
stream function temperature
changes due to convection Prandtl number
coefficient of thermal diffusivity
Rayleigh number
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Fourier Expansions

where and are horizontal and vertical
modes, respectively
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Fourier Expansions

where and are horizontal and vertical
modes, respectively
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Criteria for Mode Selection

• Selected modes must lead to equations that
• have bounded solutions (Curry 1978, 1979)

2. Selected modes must lead to a system that
conserves energy in the dissipationless
limit (Treve Manley 1982 Thiffeault
Horton 1996 Roy Musielak 2006)


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From 3D to 5D Lorenz Models
Lorenz minimal truncation
Resulting 3D Lorenz model has bounded solutions
and conserves energy in dissipationless limit

4D model with -
uncoupled !
5D model with
and
- uncoupled !
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6D Lorenz Models I
Selected modes (Humi 2004)

r 28.45
System does not conserve energy in
dissipationless limit !!!
Route to chaos period-doubling (?)
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6D Lorenz Models II
Selected modes (Kennamer 1995)

r 38
System does conserve energy in dissipationless
limit
Route to chaos chaotic transients
Musielak et al. (2005)
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6D and 7D Lorenz Models
Howard Krishnamurti (1986) developed a 6D
Lorenz model that included a shear flow
Thiffeault Horton (1996) showed that this 6D
model does not conserve energy in the
dissipationless limit

To construct an energy conserving system,
Thiffeault Horton had to add another mode and
develop a 7D Lorenz model

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8D Lorenz Model
Selected Fourier modes
3D system
5D system
8D system
Roy Musielak (2007)
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8D Lorenz System
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Energy Conservation
Thieffault Horton (1996)
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Phase portraits
r 26.5
3D Lorenz System
r 38.5
8D System
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Power spectra
8D system
r 28.50
r 29.25
r 35.10
r 38.50
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Lyapunov Exponents
8D System
Onset of chaos at r 36
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Route to Chaos in 8D Model
Ruelle Takens (1971)
Quasi-periodicity is route to chaos for 8D
system, which is different than chaotic
transients observed in 3D Lorenz model
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Other Lorenz Models
14D Lorenz model (Curry 1978) Decay of two-tori
leads to a strange attractor that is similar as
the Lorenz strange attractor Model does not
conserve energy in dissipationless limit
5D Lorenz model (Chen Price 2006) A profile of
the strange attractor in this model is similar
to the Rayleigh-Benard convection problem in a
plane fluid motion Fourier modes describing
shear flows are Included!
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SUMMARY

• Neither 4D nor 5D Lorenz models can be
  constructed.
• Unphysical 6D14D Lorenz models that do not
  conserve
• energy in the dissipationless limit have
  been constructed
• chaotic transients, period-doubling and
  quasi-periodicity
• were identified as routes to chaos in these
  systems.
• The lowest-order, high-dimensional (HD) Lorenz
  model
• that conserves energy in the dissipationless
  limit is an
• 8D model and its route to chaos is
  quasi-periodicity.
• Since the strange attractor of the 8D systems has
  high
• dimension, the chaotic behavior observed in
  this system
• represents high-dimensional chaos (HDC).

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Coupled Duffing Oscillators
Systems considered Symmetric systems (2, 4
and 6-coupled oscillators) Asymmetric systems
(3 and 5-coupled oscillators)
D. Musielak, Z. Musielak J. Benner (2005)
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2-Coupled Duffing Oscillators I


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2-Coupled Duffing Oscillators II
Chaos
B 14.5 19.0
B 23 25.5
Ueda et al (1979, 1980)
Lyapunov exponents
Original Duffing system has 8 regions that
exhibit chaos
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2-Coupled Duffing Oscillators II
B 24
B 19
Routes to chaos Period Doubling and Crisis
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4-Coupled Duffing Systems I
Torus at B 86
B 40.5 42 (period-doubling) B 82
124 (quasi-periodicity)
Role played by crisis!
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4-Coupled Duffing Systems II
B 87
Power spectra for four masses showing a
3-periodic window
B 91.5
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6-Coupled Duffing Oscillators

B 73.8, 74.15 and 76.6
B 72 95 (quasi-periodicity) Other regions -
crisis
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3-Coupled Duffing Systems
B 24.8
B 63 72 (quasi-periodicity) B 74.5 77
(crisis)
No chaos!
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5-Coupled Duffing Oscillators
B 79, 81.3, 81.7 and 85.7
B 75 105 (quasi-periodicity) B 106 120
(crisis)
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Period-Doubling Cascade
Formation of a 2D torus and its decay into a
periodic motion
Locking of two incommensurable frequencies
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SUMMARY

• HD symmetric (2, 4 and 6-coupled Duffing
  oscillators) and asymmetric (3 and 5-coupled
  Duffing oscillators) Duffing systems have been
  constructed
• Chaotic behavior of these systems represents HDC
  and routes to chaos observed in these systems
  range from period-doubling to quasi-periodicity
  and crisis
• All systems have one region that exhibits
  quasi-periodicity. The quasi-periodic torus
  breaks down through a 3-periodic window and
  2-periodic window for the symmetric and
  asymmetric systems, respectively.
• Decay of quasi-periodic torus observed in
  symmetric systems is a new route to chaos