Bifurcation and Resonance

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Bifurcation and Resonance

• Sijbo Holtman

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Overview

• Dynamical systems
• Resonance
• Bifurcation theory
• Bifurcation and resonance
• Conclusion

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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.

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Dynamical systems

• Examples
• Milky way
• Solar system
• Climate on earth
• Magma

• Population
• Growth
• Cognitive theory

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Dynamical systems

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

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Dynamical systems

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

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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 11 resonance
• Electrical circuits
• Etc.

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Bifurcation theory

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

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BifurcationResonance

• 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 (S1XS1T2)

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Bifurcationresonance

• Resonance if trajectory closes

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Bifurcationresonance
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Conclusion

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