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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

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mini course bifurcation theory

Mini-course bifurcation theory

Part two: equilibria of 2D systems

George van Voorn

two dimensional systems
Two-dimensional systems
  • Consider 2D ODE

α = bifurcation parameter(s)

model analysis
Model analysis
  • Different kinds of analysis for 2D ODE systems
    • Equilibria: determine type(s)
    • Transient behaviour
    • Long term behaviour
equilibria types
Equilibria: types
  • Different types of equilibria
  • Stability
    • Stable
    • Unstable
    • Saddle
  • Convergence type
    • Node
    • Spiral (or focus)
equilibria nodes
Equilibria: nodes

Ws

Wu

Stable node

Unstable node

Node has two (un)stable manifolds

equilibria saddle
Equilibria: saddle

Wu

Ws

Saddle point

Saddle has one stable & one unstable manifold

equilibria foci
Equilibria: foci

Ws

Wu

Stable spiral

Unstable spiral

Spiral has one (un)stable (complex) manifold

equilibria determination
Equilibria: determination
  • How do we determine the type of equilibrium?
  • Linearisation of point
  • Eigenfunction
jacobian matrix
Jacobian matrix
  • Linearisation of equilibrium in more than one dimension  partial derivatives
eigenfunction
Eigenfunction
  • Determine eigenvalues (λ) and eigenvectors (v) from Jacobian

Of course there are two solutions for a 2D system

eigenfunction11
Eigenfunction

If λ < 0  stable, λ > 0  unstable

If twoλ complex pair  spiral

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

Saddle

Stable node

Stable spiral

Unstable spiral

Unstable node

example
Example
  • 2D ODE Rosenzweig-MacArthur (1963)

R = intrinsic growth rate

K = carrying capacity

A/B = searching and handling

C = yield

D = death rate

example15
Example
  • System equilibria
    • E1 (0,0)
    • E2 (K,0)
    • E3 Non-trivial
example16
Example
  • Jacobian matrix
  • Substitute the point of interest, e.g. an equilibrium
  • Determine det(J) and tr(J)
example17
Example

Substitution E2

Result: stable node

example18
Example

Substitution E3

Result: stable node, near spiral

example19
Example

Substitution E3

Result: unstable spiral

one parameter diagram
One parameter diagram

1

2

3

  • Stable node
  • Stable node/focus
  • Unstable focus
isoclines
Isoclines
  • Isoclines: one equation equal to zero
  • Give information on system dynamics
  • Example: RM model
manifolds orbits
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 orbits26
Manifolds & orbits

y

E3

Ws

Wu

E1

E2

x

D < 0  stable manifold E1 is separatrix

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