2. Bifurcation and Resonance Sijbo Holtman

4. Overview Dynamical systems Resonance Bifurcation theory Bifurcation and resonance Conclusion

5. 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.

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

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

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

9. 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.

10. Bifurcation theory Bifurcation: s mall change of evolution rule causes big change in qualitative behaviour of the system.

11. Bifurcation&Resonance Couple two oscillators with some frequency Resonance if ratio of frequencies is rational number Solution of oscillator is a circle (S 1 ) Solution of two oscillators is on a torus (S 1 XS 1 =T 2 )

12. Bifurcation&resonance Resonance if trajectory closes

13. Bifurcation&resonance

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