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Bifurcation and Resonance

This overview explores the concepts of bifurcation and resonance within dynamical systems. Bifurcation theory describes how small changes in the evolution rule can lead to significant shifts in the system's behavior. Resonance occurs when the ratio of frequencies of coupled oscillators is a rational number, leading to predictable behavior in systems like the Milky Way and climate patterns. We will examine examples, including periodic and chaotic dynamics, and conclude with conditions for resonance and the associated bifurcation diagrams.

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Bifurcation and Resonance

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  1. Bifurcation and Resonance Sijbo Holtman

  2. Overview • Dynamical systems • Resonance • Bifurcation theory • Bifurcation and resonance • Conclusion

  3. 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.

  4. Dynamical systems • Examples • Milky way • Solar system • Climate on earth • Magma • Population • Growth • Cognitive theory

  5. Dynamical systems • Evolution rule usually given implicitly by how a system changes at any time (e.g. by a differential equation).

  6. Dynamical systems • For simple systems knowing trajectories is enough • More complex systems • Stability • Type of orbit: e.g. periodic or chaotic

  7. 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.

  8. Bifurcation theory • Bifurcation: small change of evolution rule causes big change in qualitative behaviour of the system.

  9. 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)

  10. Bifurcation&resonance • Resonance if trajectory closes

  11. Bifurcation&resonance

  12. Conclusion • Given a dynamical system describing some process • Conditions for resonance are known • Corresponding bifurcation diagram known

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