Bifurcation and Resonance Sijbo Holtman Overview Dynamical systems Resonance Bifurcation theory...
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Transcript of Bifurcation and Resonance Sijbo Holtman Overview Dynamical systems Resonance Bifurcation theory...
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