30 years of chaos research

1. 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

2. 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

3. 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

4. 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

5. 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

6. strange attractor (r,s)-bifurcation diagrams 4. Example I: the Lorenz system standard parameter values s = 10, r = 28, b = 8/3 LP Peter H. Richter

7. r = 178 r = 178, upper parts, scaled x cubic iteration s (s,x)-bifurcation diagrams r = 178 Peter H. Richter

8. exponential divergence E =4 periodic chaotic quasi-periodic 5. Example II: the double pendulum Peter H. Richter

9. E = 10 E = 9 E = E = 1 E = 2 Stability of the golden KAM torus  E = 20 E = 10 Peter H. Richter

10. 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

11. 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 Peter H. Richter

12. 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

13. 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

14. 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 Peter H. Richter

15. Chaos in the 3-body problem may help to establish order in solar systems Peter H. Richter

16. Chaotic scattering • Preimages of unstable hyperbolic periodic orbits in the space of incoming trajectories are Cantor sets Peter H. Richter

17. 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

18. 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 谢谢你们的兴趣 Peter H. Richter