The theory of nonclassical relaxation oscillations in singularly perturbed delay systems

Some special classes of relaxation systems are introduced, with one slow and one fast variable, in which the evolution of the slow component $x(t)$ in time is described by an ordinary differential equation, while the evolution of the fast component $y(t)$ is described by a Volterra-type differential equation with delay $y(t-h)$, $h=\mathrm{const}>0$, and with a small parameter $\varepsilon>0$ multiplying the time derivative. Questions relating to the existence and stability of impulse-type periodic solutions, in which the $x$-component converges pointwise to a discontinuous function as $\varepsilon\to 0$ and the $y$-component is shaped like a $\delta$-function, are investigated. The results obtained are illustrated by several examples from ecology and laser theory.
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