Bifurcations and resonance分岔和共振

1、Bifurcation and ResonanceSijbo Holtman OverviewnDynamical systemsnResonance nBifurcation theorynBifurcation and resonancenConclusionDynamical systemsnWikipedian“Mathematical formalization for a fixed rule which describes the time dependence of a points position in its ambient space.”nInterpretationn

2、How to describe mathematically any process involving motion and/or changes.Dynamical systemsnExamplesnMilky waynSolar systemnClimate on earthnMagmaPopulation nGrowthnCognitive theoryDynamical systemsnEvolution rule usually given implicitly by how a system changes at any time (e.g. by a differential

3、equation). Dynamical systemsnFor simple systems knowing trajectories is enoughnMore complex systemsnStabilitynType of orbit: e.g. periodic or chaoticResonancenTypes of dynamicsnChaosnTwo points that start close do not stay close nResonancenMarching soldiers on bridgenTwo Clocks on wall (Christiaan H

4、uygens)nMoon-earth 1:1 resonancenElectrical circuitsnEtc.Bifurcation theorynBifurcation: small change of evolution rule causes big change in qualitative behaviour of the system.Bifurcation&ResonancenCouple two oscillators with some frequencynResonance if ratio of frequencies is rational numbernSolution of oscillator is a circle (S1)Solution of two oscillators is on a torus (S1XS1=T2)Bifurcation&resonancenResonance if trajectory closesBifurcation&resonanceConclusionnGiven a dynamical system describing some process