Bifurcations in a class of time-delay equations

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Abstract
We investigate the properties of the solutions of a nonlinear time-delayed differential equation (infinite dimension) as a function of two parameters: the time delay τ and the nonlinearity parameter A. Occurrence of a Hopf bifurcation is a necessary condition for obtaining a period-doubling cascade. It is also a sufficient condition when the parameter τ is fixed and A is varied. However, it is not a sufficient condition when A is fixed and τ is varied. When period doubling occurs, a strange attractor is obtained and the behavior of the largest Lyapunov exponent as a function of τ is different from its behavior as a function of A.