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Bifurcation of periodic solutions of singularly perturbed delay differential equation

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In this dissertation we studied the periodic solutions of the singularly perturbed delay differential equation [epsilon]x(t) = -x(t) + f(x(t - 1),[mu]), t ≥ 0 where [epsilon] > 0 is a small parameter and f is a nonlinear function depending on a parameter [mu];The bifurcation curves are described by studying the characteristic equation of the linearized delay differential equation. A theorem is proved about the pure imaginary roots of the equation (z + a)e[superscript] z + b = 0, where a ≠ 0, b ≠ 0 are real. The result is used to find an equation of the bifurcation curves. Periodic solutions of periods 2, 4 were found. The period-4 solution bifurcates as the nonsymmetric square wave loses its stability at [mu] = √5. Also solutions of period 2/3, 2/5, dots were found. A strange orbit of period 2 consisting of three square waves each of length 2/3 was found as [mu] increased past 2. Splitting of periodic solutions was described at the points where the bifurcation occurs. The boundary layer for the symmetric square wave was studied. And the value of the parameter [mu] at which the wave form is changed were found for the cases of symmetric and nonsymmetric square waves. It is noticed that the symmetric square wave of period 2 stays stable for some values of [mu] > 2. The period of the solution was evaluated numerically and the dependence of the period on [mu] was shown.