Bifurcations and resonance分岔和共振.pptVIP

Bifurcation and Resonance Sijbo Holtman Overview Dynamical systems Resonance Bifurcation theory Bifurcation and resonance Conclusion Dynamical systems Wikipedia “Mathematical formalization for a fixed rule which describes the time dependence of a points position in its ambient space.” Dynamical systems Examples Milky way Solar system Climate on earth Magma Dynamical systems Evolution rule usually given implicitly by how a system changes at any time (e.g. by a differential equation). Dynamical systems For simple systems knowing trajectories is enough More complex systems Stability Type of orbit: e.g. periodic or chaotic Resonance Types of dynamics Chaos Two points that start close do not stay close Resonance Marching soldiers on bridge Two Clocks on wall (Christiaan Huygens) Moon-earth 1:1 resonance Electrical circuits Etc. Bifurcation theory Bifurcation: small change of evolution rule causes big change in qualitative behaviour of the system. BifurcationResonance Couple two oscillators with some frequency Resonance if ratio of frequencies is rational number Solution of oscillator is a circle (S1) Solution of two oscillators is on a torus (S1XS1=T2) Bifurcationresonance Resonance if trajectory closes Bifurcationresonance Conclusion Given a dynamical system describing some process Conditions for resonance are known Corresponding bifurcation diagram known * * Interpretation How to describe mathematically any process involving motion and/or changes. Population Growth Cognitive theory