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Visualizations:
Taylor approximation of invariant fiber bundles (cf. Pötzsche and Rasmussen: Taylor Approximation of Invariant Fiber Bundles for Nonautonomous Difference Equations, to appear in: Nonlinear Analysis. Theory, Methods & Applications):
Kuang and Cushing (Global stability in a nonlinear difference-delay equation model
of flour beetle population growth, Journal of Difference Equations and Applications 2(1)
(1996), 31-37) consider the following difference equation
to describe a flour beetle population. We fix parameter values a=678559/891000, b=11/10, and cannibalism rates
βk=1-(1/π)arctan(k) and δk=1+(1/π)arctan(k). The following animation shows
the Taylor approximation (up to order six) of the stable (dimension 2) and unstable fiber bundles (dimension 1)
corresponding to the trivial
solution, where we work with the equivalent system with decoupled linear part. The animation parameter is the time
k=-20...20. We have computed the invariant fiber bundles with our Maple program IFB_Comp.
Taylor Approximation of integral manifolds (cf. Pötzsche and Rasmussen: Taylor Approximation of Integral Manifolds):
We consider the following FitzHugh-Nagumo-like equation with quadratic coupling
The problem to find solutions of the form (U,V)(x+ct)=(u,v)(x,t) leads to the ordinary differential equation
We set c=√2. The animation below shows the Taylor approximation (up to order 4) of its center-stable and center-unstable
fiber bundel corresponding with the trivial solution. We work with the equivalent system with decoupled linear part. The animation
parameter is the time t=-8...8 and we have a(t)=arctan(t).
Possibly existing traveling waves solutions are contained in the intersection of the two integral manifolds.
Martin Rasmussen, October 2004
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