Abstract
We consider a parametrized dynamical system with a saddle-focus equilibrium point and which, for one value of the parameter, has a homoclinic orbit. Conditions on the eigenvalues for the equilibrium point, together with transversality conditions, imply the existence of an infinite discrete set of parameter values for which the system has a homoclinic orbit. Such systems arise in the study of nerve axon equations where a homoclinic orbit corresponds to a finite train of nerve impulses traveling at a velocity identified by the associated parameter value.
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