# American Institute of Mathematical Sciences

May  2012, 32(5): 1801-1833. doi: 10.3934/dcds.2012.32.1801

## Multiple periodic solutions of state-dependent threshold delay equations

 1 Department of Mathematics, Gettysburg College, Gettysburg, PA 17325-1484, United States

Received  November 2010 Revised  September 2011 Published  January 2012

We prove the existence of multiple periodic solutions for scalar-valued state-dependent delay equations of the form $x'(t) = f(x(t - d(x_t)))$, where $d(x_t)$ is given by a threshold condition and $f$ is close, in a suitable sense, to the step function $h(x) = -\mbox{sign}(x)$. We construct maps whose fixed points correspond to periodic solutions and show that these maps have nontrivial fixed points via homotopy to constant maps.
We also describe part of the global dynamics of the model equation $x'(t) = h(x(t - d(x_t)))$.
Citation: Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801
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