Bifurcation from a heteroclinic solution in differential delay equations

Abstract:

We study a class of functional differential equations $\dot x(t) = af(x(t - 1))$ with periodic nonlinearity $f:{\mathbf {R}} \to {\mathbf {R}},0 < f$ in $(A,0)$ and $f < 0$ in $(0,B),f(A) = f(0) = f(B) = 0$ . Such equations describe a state variable on a circle with one attractive rest point (given by the argument $\xi = 0$ of $f$) and with reaction lag $a$ to deviations. We prove that for a certain critical value $a = {a_0}$ there exists a heteroclinic solution going from the equilibrium solution $t \to A$ to the equilibrium $t \to B$. For $a - {a_0} > 0$, this heteroclinic connection is destroyed, and periodic solutions of the second kind bifurcate. These correspond to periodic rotations on the circle.