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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.

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## Mini-course bifurcation theory

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**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?

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