30 years of chaos research

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30 years of chaos research
from a personal perspective

• Peter H. Richter

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????? ?????????????? CAS-MPG Partner Institute
for Computational Biology
?? 2007 ? 3 ? 29 ?
Shanghai, March 29, 2007
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Outline

1. History
2. Dynamical systems general perspective
3. Deterministic chaos regular vs. chaotic dynamics
4. Example I the Lorenz system
5. Example II the double pendulum
6. Scenarios of transition to chaos universality
7. Chaos and fractals dynamics and geometry
8. Hamiltonian systems entanglement of order and
   chaos
9. Other developments and summary

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1. History

• 1890 Poincaré Méthodes nouvelles de la mécanique
  céleste
• 1925 Strömgren numerical determination of
  periodic orbits
• 1963 Kolmogorov, Arnold, Moser invariant
  irrational tori
• 1963 Lorenz period doubling scenario and
  butterfly effect
• 1967 Smale horseshoes contain invariant Cantor
  sets
• 1970 Kadanoff, Wilson renormalization scaling,
  universality
• 1975 Mandelbrot fractal geometry
• 1975 Aspen conference on network dynamics
• 1975 Li Yorke period three implies chaos
• 1977 Großmann Thomae analysis of period
  doubling
• 1978 Feigenbaum universality of period doubling
• 1978 Berrys review on regular and irregular
  motion
• 1981 Bremen conference on invariant sets in
  chaotic dynamics
• 1985 Exhibition Frontiers of Chaos

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2. Dynamical Systems general perspective

• systems live in phase space
• of low dimension, compact or open
• or of high dimension (infinite in case of PDEs)
• and develop in time
• continuous time differential equations
• discrete time difference equations
• the dynamical laws may be
• deterministic (no uncertainty in the laws)
• stochastic (due to fluctuations)
• the phase space flow may be
• dissipative (contracting due to friction or other
  losses)
• conservative (no friction, no expansion)
• expansive (due to autocatalysis or other positive
  feedback)

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3. Deterministic chaos regular vs. chaotic
dynamics

• dynamical point of view long term
  (un)predictability
• regular motion points that lie initially close
  together tend to stay together or increase their
  distance at most linearly with time
• chaos sensitive dependence on initial
  conditions points that lie initially close
  together get separated exponentially in time
  (Lyapunov exponents)
• geometric point of view
• regular motion the phase space is foliated by
  low-dimensional sets given an initial condition,
  the possible future is strongly restricted
• chaotic motion given an initial condition,
  relatively large portions of phase space may be
  visited though not necessarily the entire space
• symbolic point of view
• regular motion generates regular sequences of
  numbers
• chaotic motion generates random sequences of
  numbers

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4. Example I the Lorenz system
standard parameter values s 10, r 28, b
8/3
LP
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(s,x)-bifurcation diagrams
r 178
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5. Example II the double pendulum
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Stability of the golden KAM torus
?
E 10
E 20
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6. Scenarios of transition to chaos universality

• through quasi-periodicity
• break-up of irrational tori

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Complexification universality of higher degree
x ? x2 c, x and c complex

• c inside the Mandelbrot set
• ? finite attractors exists, domains of
  attraction bounded by Julia sets
• c outside the Mandelbrot set
• ? no finite attractor chaos

JMN
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7. Chaos and fractals dynamics and geometry

• dissipative systems
• chaotic ( strange) attractors have fractal
  dimensions
• meromorphic systems
• chaotic repellors ( Julia sets) have
  fractal dimension
• Hamiltonian systems
• chaotic regions are fat fractals

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8. Hamiltonian systems entanglement
f degrees of freedom if f independent constants
of motion exist, the phase space is foliated by
(rational and irrational) invariant f-tori
Liouville-Arnold integrability

• When there are less than f integrals, the system
  tends to be chaotic
• all rational tori break up (Poincaré-Birkhoff)
  into an alternation of islands of stability with
  elliptic centers, and chaotic bands with
  hyperbolic centers containing Smale-horseshoes
• sufficiently irrational tori survive mild
  perturbations of integrable limiting cases
  noble tori (winding numbers related to the
  golden mean) are the most robust (KAM)

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Poincaré sections of the restricted 3-body system
Section condition local maximum or minimum
distance from the main body (sun), with one of
the two possible angular velocities
3-B
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Chaos in the 3-body problem may help to establish
order in solar systems
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Chaotic scattering

• Preimages of unstable hyperbolic periodic orbits
  in the space of incoming trajectories are Cantor
  sets

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9. Other developments and summary

• from celestial mechanics to molecular dynamics
• quantum chaos level statistics, scars,
  quasi-classical quantization
• rigid body dynamics
• more than 2 degrees of freedom
• theory of turbulence (many degrees of freedom)
• influence of stochastic elements in the dynamics
• fractal growth patterns
• synchronization of non-linear oscillators
• neurodynamics
• econophysics

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Summary

• Chaos theory has deep roots in science.
• It emerged from questions on stability and
  predictability of systems,
• is founded on solid mathematical insight,
• but was boosted by the development of computer
  technology.
• The identification of universal scenarios came as
  an exciting surprise
• As chaos is the rule rather than the exception,
  there are many discoveries yet to be made

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