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 point's position in its ambient space.”

Interpretation How to describe mathematically any

process involving motion and/or changes.

Dynamical systems Examples

Milky way Solar system Climate on

earth Magma Population Growth Cognitive

theory

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.

Bifurcation&Resonance

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)

Bifurcation&resonance Resonance if trajectory closes

Bifurcation&resonance

Conclusion

Given a dynamical system describing some process Conditions for resonance are known Corresponding bifurcation diagram known