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Equilibria in 2D Systems: Types, Stability, and Analysis

This mini-course delves into the equilibria of 2D ODE systems, exploring different types, stability criteria, and analytical methods. Learn how to determine equilibrium types through linearization, eigenvalues, and eigenvectors. Examples, diagrams, and analysis of the Rosenzweig-MacArthur model are included.

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Equilibria in 2D Systems: Types, Stability, and Analysis

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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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