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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

Peter H. Richter

Outline

- 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

- 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

standard parameter values

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

LP

Peter H. Richter

r = 178

r = 178, upper parts, scaled

x

cubic iteration

s

r = 178

Peter H. Richter

exponential divergence

E =4

periodic

chaotic

quasi-periodic

Peter H. Richter

E = 10

E = 9

E =

E = 1

E = 2

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

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

Peter H. Richter

- 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

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

Section condition:

local maximum or minimum distance from the

main body (sun), with one of the two possible angular velocities

3-B

Peter H. Richter

Peter H. Richter

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

Peter H. Richter

- 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

谢谢你们的兴趣

Peter H. Richter