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Bifurcation and Resonance

Bifurcation and Resonance. Sijbo Holtman. Overview. Dynamical systems Resonance Bifurcation theory Bifurcation and resonance Conclusion. Dynamical systems. Wikipedia

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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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