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Mini-course bifurcation theory

Mini-course bifurcation theory. Part two: equilibria of 2D systems. George van Voorn. Two-dimensional systems. Consider 2D ODE. α = bifurcation parameter(s). Model analysis. Different kinds of analysis for 2D ODE systems Equilibria: determine type(s) Transient behaviour

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Mini-course bifurcation theory

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  1. Mini-course bifurcation theory Part two: equilibria of 2D systems George van Voorn

  2. Two-dimensional systems • Consider 2D ODE α = bifurcation parameter(s)

  3. Model analysis • Different kinds of analysis for 2D ODE systems • Equilibria: determine type(s) • Transient behaviour • Long term behaviour

  4. Equilibria: types • Different types of equilibria • Stability • Stable • Unstable • Saddle • Convergence type • Node • Spiral (or focus)

  5. Equilibria: nodes Ws Wu Stable node Unstable node Node has two (un)stable manifolds

  6. Equilibria: saddle Wu Ws Saddle point Saddle has one stable & one unstable manifold

  7. Equilibria: foci Ws Wu Stable spiral Unstable spiral Spiral has one (un)stable (complex) manifold

  8. Equilibria: determination • How do we determine the type of equilibrium? • Linearisation of point • Eigenfunction

  9. Jacobian matrix • Linearisation of equilibrium in more than one dimension  partial derivatives

  10. Eigenfunction • Determine eigenvalues (λ) and eigenvectors (v) from Jacobian Of course there are two solutions for a 2D system

  11. Eigenfunction If λ < 0  stable, λ > 0  unstable If twoλ complex pair  spiral

  12. Determinant & trace • Alternative in 2D to determine equilibrium type (much less computation)

  13. Diagram Saddle Stable node Stable spiral Unstable spiral Unstable node

  14. Example • 2D ODE Rosenzweig-MacArthur (1963) R = intrinsic growth rate K = carrying capacity A/B = searching and handling C = yield D = death rate

  15. Example • System equilibria • E1 (0,0) • E2 (K,0) • E3 Non-trivial

  16. Example • Jacobian matrix • Substitute the point of interest, e.g. an equilibrium • Determine det(J) and tr(J)

  17. Example Substitution E2 Result: stable node

  18. Example Substitution E3 Result: stable node, near spiral

  19. Example Substitution E3 Result: unstable spiral

  20. One parameter diagram 1 2 3 • Stable node • Stable node/focus • Unstable focus

  21. Isoclines • Isoclines: one equation equal to zero • Give information on system dynamics • Example: RM model

  22. Isoclines

  23. Isoclines

  24. Manifolds & orbits • Manifolds: orbits starting like eigenvectors • Give other information on system dynamics • E.g. discrimination spiral or periodic solution (not possible with isoclines) • Separatrices (unstable manifolds)

  25. Isoclines & manifolds Ws

  26. Manifolds & orbits y E3 Ws Wu E1 E2 x D < 0  stable manifold E1 is separatrix

  27. Continue • In part three: • Bifurcations in 2D ODE systems • Global bifurcations • In part four: • Demonstration: 3D RM model • Chaos

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