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

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 –Long term behaviour

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

Equilibria: nodes Stable nodeUnstable node WsWs Node has two (un)stable manifolds WuWu

Equilibria: saddle Saddle point WsWs Saddle has one stable & one unstable manifold WuWu

Equilibria: foci Stable spiralUnstable spiral Spiral has one (un)stable (complex) manifold WsWs WuWu

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

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

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

Eigenfunction If λ 0  unstable If two λ complex pair  spiral

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

Diagram Saddle Stable node Stable spiral Unstable spiral Unstable node

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

Example System equilibria –E 1 (0,0) –E 2 (K,0) –E 3 Non-trivial

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

Example Result: stable node Substitution E 2

Example Result: stable node, near spiral Substitution E 3

Example Result: unstable spiral Substitution E 3

One parameter diagram Stable node 2.Stable node/focus 3.Unstable focus

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

Isoclines

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)

Isoclines & manifolds WsWs

Manifolds & orbits D < 0  stable manifold E 1 is separatrix WsWs WuWu E2E2 E3E3 E1E1 x y

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