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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
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