Mini-course bifurcation theory

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# Mini-course bifurcation theory - PowerPoint PPT Presentation

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

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
• Long term behaviour
Equilibria: types
• Different types of equilibria
• Stability
• Stable
• Unstable
• Convergence type
• Node
• Spiral (or focus)
Equilibria: nodes

Ws

Wu

Stable node

Unstable node

Node has two (un)stable manifolds

Wu

Ws

Saddle has one stable & one unstable manifold

Equilibria: foci

Ws

Wu

Stable spiral

Unstable spiral

Spiral has one (un)stable (complex) manifold

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  unstable

If twoλ complex pair  spiral

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

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
• E1 (0,0)
• E2 (K,0)
• E3 Non-trivial
Example
• Jacobian matrix
• Substitute the point of interest, e.g. an equilibrium
• Determine det(J) and tr(J)
Example

Substitution E2

Result: stable node

Example

Substitution E3

Result: stable node, near spiral

Example

Substitution E3

Result: unstable spiral

One parameter diagram

1

2

3

• Stable node
• Stable node/focus
• Unstable focus
Isoclines
• Isoclines: one equation equal to zero
• Give information on system dynamics
• Example: RM model
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)
Manifolds & orbits

y

E3

Ws

Wu

E1

E2

x

D < 0  stable manifold E1 is separatrix

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