The effect of oscillations in the dynamics of differential equations with delay

Abstract

The purpose of this paper is to study the dynamic behavior of delay differential equations of the form x ˙ ( t ) = f ( x ( t − 1 ) ; a ( ε sin ( ν t ) , ε cos ( ν t ) ) ; α ) , ε , ν , α ∈ R , provided that a and f meet some hypotheses. By augmenting the above equation, the explicit time-dependent terms are replaced by state-dependent terms. The augmented system is autonomous and has a pair of purely imaginary and simple zero eigenvalues. Applying the center manifold reduction, the existence of an attractive integral manifold with periodic structure for the original equation is shown. Furthermore, we give a description of the flow on the obtained manifold. This allows us to determine the sufficient conditions for existence of saddle-node bifurcation. To illustrate our results, we consider an autonomous equation perturbed by a periodic function.