170 likes | 536 Vues
Mini-course bifurcation theory. Part four: chaos. George van Voorn. Bifurcations. Bifurcations in 3 and higher D ODE models Chaos (requires at least 3D) Example: 3D RM model. Rosenzweig-MacArthur. The 3D RM model is written as. Where X = prey, Y = predator, Z = top predator.
E N D
Mini-course bifurcation theory Part four: chaos George van Voorn
Bifurcations • Bifurcations in 3 and higher D ODE models • Chaos (requires at least 3D) • Example: 3D RM model
Rosenzweig-MacArthur The 3D RM model is written as Where X = prey, Y = predator, Z = top predator
3D R: rescaling The rescaled version is written as Scaled functional responses
3D RM: bifurcations • Primary bifurcation parameters d1 and d2 • Displays a whole range of bifurcation curves • Point M of higher co-dimension • Tangent of equilibrium (Te) • Transcritical of equilibrium (TCe) • Hopf of 2D system equilibrium (Hp) • Hopf of non-trivial equilibrium (H+) • Transcritical of limit cycle (TCc)
3D RM: bifurcations Maximum x3 Minimum x3 d1 = 0.5
3D RM: bifurcations Separatrix (3D) 2 attractors
3D RM: chaos • Flip bifurcations after each other • Period doubling 1,2,4,8,16 to infinity
3D RM: chaos d1 = 0.5
3D RM: chaos Pattern *2 *4 *8
unstable equilibrium X3 Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map Minima x3 cycles
X3 Boundaries of chaos Example: Rozenzweig-MacArthur next-minimum map Possible existence x3 No existence x3
Boundaries of chaos • Chaos born through flip bifurcations (possible route) • Chaos bounded by global bifurcations (work by Martin Boer)
The end (for now) Any questions?