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 node Unstable node

Ws

Node has two (un)stable manifolds

Wu

Equilibria: saddle

Saddle point

Ws

Saddle has one stable & one unstable manifold

Wu

Equilibria: foci

Stable spiral Unstable spiral

Spiral has one (un)stable (complex) manifold

Ws Wu

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 stable, λ > 0 unstableIf two λ complex pair spiral

Determinant & trace

• Alternative in 2D to determine equilibrium type (much less computation)

Diagram

SaddleStable nodeStable spiralUnstable spiralUnstable node

Example

• 2D ODE Rosenzweig-MacArthur (1963)

R = intrinsic growth rateK = carrying capacityA/B = searching and handlingC = yieldD = death rate

Example

• System equilibria– E1 (0,0)

– E2 (K,0)

– E3 Non-trivial

Example

• Jacobian matrix

Substitute the point of interest, e.g. an equilibriumDetermine det(J) and tr(J)

Example

Result: stable node

Substitution E2

Example

Result: stable node, near spiral

Substitution E3

Example

Result: unstable spiral

Substitution E3

One parameter diagram1 2 3

1. Stable node2. Stable node/focus3. Unstable focus

Isoclines

• Isoclines: one equation equal to zero

• Give information on system dynamics

• Example: RM model

Isoclines

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

Ws

Manifolds & orbits

D < 0 stable manifold E1 is separatrix

Ws WuE2

E3

E1 x

y

Continue

• In part three:– Bifurcations in 2D ODE systems– Global bifurcations

• In part four:– Demonstration: 3D RM model– Chaos