Deterministic Chaos

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Deterministic Chaos
caoz

• PHYS 306/638
• University of Delaware

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Chaos vs. Randomness

• Do not confuse chaotic with random
• Random
• irreproducible and unpredictable
• Chaotic
• deterministic - same initial conditions lead to
  same final state but the final state is very
  different for small changes to initial conditions
• difficult or impossible to make long-term
  predictions

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Clockwork (Newton) vs. Chaotic (Poincaré) Universe

• Suppose the Universe is made of particles of
  matter interacting according to Newton laws ?
  this is just a dynamical system governed by a
  (very large though) set of differential equations
• Given the starting positions and velocities of
  all particles, there is a unique outcome ? P.
  Laplaces Clockwork Universe (XVIII Century)!

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Can Chaos be Exploited?
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Brief Chaotic History Poincaré
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Brief Chaotic History Lorenz
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Chaos in the Brave New World of Computers

• Poincaré created an original method to understand
  such systems, and discovered a very complicated
  dynamics,but
• "It is so complicated that I cannot even draw
  the figure."

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An Example A Pendulum

• starting at 1, 1.001, and 1.000001 rad

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Changing the Driving Force

• f 1, 1.07, 1.15, 1.35, 1.45

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Chaos in Physics

• Chaos is seen in many physical systems
• Fluid dynamics (weather patterns),
• some chemical reactions,
• Lasers,
• Particle accelerators,
• Conditions necessary for chaos
• system has 3 independent dynamical variables
• the equations of motion are non-linear

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Why Nonlinearity and 3D Phase Space?
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Dynamical Systems

• A dynamical system is defined as a deterministic
  mathematical prescription for evolving the state
  of a system forward in time
• Example A system of N first-order, autonomous
  ODE

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Damped Driven Pendulum Part I

• This system demonstrates features of chaotic
  motion
• Convert equation to a dimensionless form

q
q
2
d
d
f
w
q




)
cos(

sin

t
A
mg
c
ml
D
2
dt
dt
q
w
d
d
w
q



)
cos(

sin
t
f
q
0
D
dt
dt
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Damped Driven Pendulum Part II

• 3 dynamic variables ?, ?, t
• the non-linear term sin ?
• this system is chaotic only for certain values of
  q, f0 , and wD
• In these examples
• wD 2/3, q 1/2, and f0 near 1

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Damped Driven Pendulum Part III

• to watch the onset of chaos (as f0 is increased)
  we look at the motion of the system in phase
  space, once transients die away
• Pay close attention to the period doubling that
  precedes the onset of chaos...

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07
.
1
f
0

15
.
1
f
0
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f0 1.35
f0 1.45
f0 1.47
f0 1.48
f0 1.49
f0 1.50
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Forget About Solving Equations!

• New Language for Chaos
• Attractors (Dissipative Chaos)
• KAM torus (Hamiltonian Chaos)
• Poincare sections
• Lyapunov exponents and Kolmogorov entropy
• Fourier spectrum and autocorrelation functions

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Poincaré Section
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Poincaré Section Examples
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Poincaré Section Pendulum

• The Poincaré section is a slice of the 3D phase
  space at a fixed value of ?Dt mod 2?
• This is analogous to viewing the phase space
  development with a strobe light in phase with the
  driving force. Periodic motion results in a
  single point, period doubling results in two
  points...

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Poincaré Movie

• To visualize the 3D surface that the chaotic
  pendulum follows, a movie can be made in which
  each frame consists of a Poincaré section at a
  different phase...
• Poincare Map Continuous time evolution is
  replace by a discrete map

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f0 1.07
f0 1.48
f0 1.50
f0 1.15 q 0.25
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Attractors

• The surfaces in phase space along which the
  pendulum follows (after transient motion decays)
  are called attractors
• Examples
• for a damped undriven pendulum, attractor is just
  a point at ???0. (0D in 2D phase space)
• for an undamped pendulum, attractor is a curve
  (1D attractor)

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Strange Attractors

• Chaotic attractors of dissipative systems
  (strange attractors) are fractals
• Our Pendulum 2 lt dim lt 3
• The fine structure is quite complex and similar
  to the gross structure self-similarity.

non-integer dimension
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What is Dimension?

• Capacity dimension of a line and square

e

d
d
)
/
1
(
)
(
e
L
N
e
e

)
/
1
log(

/
)
(
log

lim
N
d
c
e

0
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Trivial Example Point, Line, Surface,
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Non-Trivial Example Cantor Set

• The Cantor set is produced as follows

N ?
1 1
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Lyapunov Exponents Part I

• The fractional dimension of a chaotic attractor
  is a result of the extreme sensitivity to initial
  conditions.
• Lyapunov exponents are a measure of the average
  rate of divergence of neighbouring trajectories
  on an attractor.

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Lyapunov Exponents Part II

• Consider a small sphere in phase space after a
  short time the sphere will evolve into an
  ellipsoid

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Lyapunov Exponents Part III

• The average rate of expansion along the principle
  axes are the Lyapunov exponents
• Chaos implies that at least one is gt 0
• For the pendulum ??i - q (damp coeff.)
• no contraction or expansion along t direction so
  that exponent is zero
• can be shown that the dimension of the attractor
  is d 2 - ?1 / ?2

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Dissipative vs Hamiltonian Chaos

• Attractor An attractor is a set of states
  (points in the phase space), invariant under the
  dynamics, towards which neighboring states in a
  given basin of attraction asymptotically approach
  in the course of dynamic evolution. An attractor
  is defined as the smallest unit which cannot be
  itself decomposed into two or more attractors
  with distinct basins of attraction. This
  restriction is necessary since a dynamical system
  may have multiple attractors, each with its own
  basin of attraction.
• Conservative systems do not have attractors,
  since the motion is periodic. For dissipative
  dynamical systems, however, volumes shrink
  exponentially so attractors have 0 volume in
  n-dimensional phase space.
• Strange Attractors Bounded regions of phase
  space (corresponding to positive Lyapunov
  characteristic exponents) having zero measure in
  the embedding phase space and a fractal
  dimension. Trajectories within a strange
  attractor appear to skip around randomly

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Dissipative vs Conservative Chaos Lyapunov
Exponent Properties

• For Hamiltonian systems, the Lyapunov exponents
  exist in additive inverse pairs, while one of
  them is always 0.
• In dissipative systems in an arbitrary
  n-dimensional phase space, there must always be
  one Lyapunov exponent equal to 0, since a
  perturbation along the path results in no
  divergence.

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Logistic Map Part I

• The logistic map describes a simpler system that
  exhibits similar chaotic behavior
• Can be used to model population growth
• For some values of ?, x tends to a fixed point,
  for other values, x oscillates between two points
  (period doubling) and for other values, x becomes
  chaotic.

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Logistic Map Part II

• To demonstrate

x
n
x
n
-
1
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Bifurcation Diagrams Part I

• Bifurcation a change in the number of solutions
  to a differential equation when a parameter is
  varied
• To observe bifurcatons, plot long term values of
  ?, at a fixed value of ?Dt mod 2? as a function
  of the force term f0

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Bifurcation Diagrams Part II

• If periodic ? single value
• Periodic with two solutions (left or right
  moving) ? 2 values
• Period doubling ? double the number
• The onset of chaos is often seen as a result of
  successive period doublings...

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Bifurcation of the Logistic Map
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Bifurcation of Pendulum
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Feigenbaum Number

• The ratio of spacings between consecutive values
  of ? at the bifurcations approaches a universal
  constant, the Feigenbaum number.
• This is universal to all differential equations
  (within certain limits) and applies to the
  pendulum. By using the first few bifurcation
  points, one can predict the onset of chaos.

-
m
m


d
-
k
k
...
669201
.
4
lim
1
-
m
m


k

k
k
1
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Chaos in PHYS 306/638
Deterministic

• Aperiodic motion confined to strange attractors
  in the phase space
• Points in Poincare section densely fill some
  region
• Autocorrelation function drops to zero, while
  power spectrum goes into a continuum

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Chaos in PHYS 306/638
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Deterministic Chaos in PHYS 306/638
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Deterministic Chaos in PHYS 306/638
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Deterministic Chaos in PHYS 306/638
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Deterministic Chaos in PHYS 306/638
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Chaos in PHYS 306/638
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Deterministic Chaos in PHYS 306/638
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Deterministic Chaos in PHYS 306/638
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Do Computers in Chaos Studies Make any Sense?
Shadowing Theorem Although a numerically
computed chaotic trajectory diverges
exponentially from the true trajectory with the
same initial coordinates, there exists an
errorless trajectory with a slightly different
initial condition that stays near ("shadows") the
numerically computed one. Therefore, the fractal
structure of chaotic trajectories seen in
computer maps is real.