Differential Equations with Bifocal Homoclinic Orbits

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Abstract
Global bifurcation theory can be used to understand complicated bifurcation phenomena in families of differential equations. There are many theoretical results relating to systems having a homoclinic orbit biasymptotic to a stationary point at some value of the parameters, and these results depend upon the eigenvalues of the Jacobian matrix of the flow evaluated at the stationary point. Three important cases arise in the theoretical analysis, and there are many examples of systems which illustrate two of these three cases. We describe a construction which can be used to produce examples of the third case (bifocal homoclinic orbits), and use this construction to prove the existence of a bifocal homoclinic orbit in a simple piecewise linear differential equation.