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