NDLI: Bifurcations of Heteroclinic Orbits

Content Provider	Springer Nature Link
Author	McSwiggen, Patrick Hou, Xiaojie Meyer, Kenneth R.
Copyright Year	2010
Abstract	The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ $_{1}$ as t → −∞ and u(t) → γ $_{2}$ as t → ∞ where γ $_{1}$, γ $_{2}$ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.
Ending Page	380
Page Count	14
Starting Page	367
File Format	PDF
ISSN	10407294
e-ISSN	15729222
Journal	Journal of Dynamics and Differential Equations
Issue Number	3
Volume Number	22
Language	English
Publisher	Springer US
Publisher Date	2010-07-15
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Homoclinic and heteroclinic solutions Ordinary Differential Equations Heteroclinic orbits Bifurcation Traveling wave Applications of Mathematics Partial Differential Equations
Content Type	Text
Resource Type	Article
Subject	Analysis

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