30 years of chaos research
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30 years of chaos research. from a personal perspective. 不来梅大学物理系. Peter H. Richter. 中国科学院 — 马普学会 计算生物学伙伴研究所 CAS-MPG Partner Institute for Computational Biology. 上海 2007 年 3 月 29 日 Shanghai, March 29, 2007. Outline. History Dynamical systems: general perspective

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30 years of chaos research

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30 years of chaos research

from a personal perspective


Peter H. Richter

中国科学院 — 马普学会计算生物学伙伴研究所

CAS-MPG Partner Institute for Computational Biology

上海 2007年3 月29 日Shanghai, March 29, 2007

Peter H. Richter


  • History

  • Dynamical systems: general perspective

  • Deterministic chaos: regular vs. chaotic dynamics

  • Example I: the Lorenz system

  • Example II: the double pendulum

  • Scenarios of transition to chaos: universality

  • Chaos and fractals: dynamics and geometry

  • Hamiltonian systems: entanglement of order and chaos

  • Other developments and summary

Peter H. Richter

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 Berry‘s review on regular and irregular motion

  • 1981 Bremen conference on invariant sets in chaotic dynamics

  • 1985 Exhibition „Frontiers of Chaos“

Peter H. Richter

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)

Peter H. Richter

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

Peter H. Richter

strange attractor

(r,s)-bifurcation diagrams

4. Example I: the Lorenz system

standard parameter values

s = 10, r = 28, b = 8/3


Peter H. Richter

r = 178

r = 178, upper parts, scaled


cubic iteration


(s,x)-bifurcation diagrams

r = 178

Peter H. Richter

exponential divergence

E =4




5. Example II: the double pendulum

Peter H. Richter

E = 10

E = 9

E =

E = 1

E = 2

Stability of the golden KAM torus

E = 20

E = 10

Peter H. Richter

  • period doubling: „Feigenbaum“

    • universal constants d, a

    • inverse cascade

x → x2 + c

  • Intermittency

    • onset of turbulence

6. Scenarios of transition to chaos: universality

  • through quasi-periodicity =

    break-up of irrational tori

Peter H. Richter

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“


Peter H. Richter

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“

Peter H. Richter

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)

Peter H. Richter

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


Peter H. Richter

Chaos in the 3-body problem may help to establish order in solar systems

Peter H. Richter

Chaotic scattering

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

Peter H. Richter

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

  • ……

Peter H. Richter


  • 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


Peter H. Richter

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