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Logistic Differential Equation

N. t. K/2. Logistic Differential Equation. K. K. K/2. Logistic Differential Equation. d/dt N = - d/dN [r N 2 (N/3k -1/2)] = - d/dN V(N). V(N). K. N. 0. K/2. K. 3K/2. What is a bifurcation?. Max & Moritz, W. Busch. A qualitative change in the solution !.

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Logistic Differential Equation

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  1. N t K/2 Logistic Differential Equation K K

  2. K/2 Logistic Differential Equation d/dt N = - d/dN [r N2 (N/3k -1/2)] = - d/dN V(N) V(N) K N 0 K/2 K 3K/2

  3. What is a bifurcation? Max & Moritz, W. Busch A qualitative change in the solution !

  4. Saddle Node Bifurcation (1-dim) Prototypical example:

  5. Transcritical Bifurcatoin Prototypical example:

  6. Pitchfork Bifurcation Prototypical example:

  7. Hopf-Bifurcation Prototypical example:

  8. Bifurcations ? Catastrophes ? • Use simple model from classical mechanics • Particle (m=1) in 1D-potential V(x,λ) • Potential changes very slowly with time • (slow time scale represented by λ) • Equations of motion: Stability of fix points:

  9. Loss-of-Equilibrium (Saddle-Node Bifurcation) Potential, Lyapunov function Loss-of-equilibrium or catastrophe at λ = λc

  10. Transcritical Bifurcation Potential, Lyapunov function Equilibrium branches exchange stability at λ=λc Other possibility: supercitical pitchfork bifurcation

  11. Subcritical Bifurcation

  12. Construct the dynamical System ! d/dt x = - d/dx V(x) V(x) = x2 (b - x2)

  13. Dynamics of Two Dimensional Systems • Find the fixed points in the phase space! • Linearize the system about the fixed points! • Determine the eigenvalues of the Jacobian.

  14. Liapunov Exponents • to determine the phase space extent • These exponents describe the mean exponential rate of divergence of two trajectories initially close to each other in phase space and are defined as: • where and are the initial (t=0) position and deviation, and is the deviation at time t. Such limit is proven to exist and is finite. • the largest exponent is the most important because it results from almost all deviations and therefore is the one found in practice when the initial deviation is chosen randomly.

  15. Liapunov Exponents

  16. Liapunov Exponents • numerical computation of the Liapunov exponents • the finite initial deviation may rapidly grow as large as the size of the orbits themselves -> scaling • Noting the deviation before the first rescaling, this is done by normalising to : every time ti

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