Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Abstract

Abstract. We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation, and the resulting code HHnfDDE is supplied as a supplement to this article. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds (when there is a single unstable Floquet multiplier). This allows us to study transitions through resonance tongues and the breakup of a 1 : 4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.Key words. State-dependent delay differential equations, bifurcation analysis, invariant tori, resonance tongues, Hopf-Hopf bifurcation, normal form computation AMS subject classifications. 34K60, 34K18, 37G05, 37M201. Introduction. Time delays arise naturally in numerous areas of application as an unavoidable phenomenon, for example, in balancing and control [8,19,35,39,64,65,66,67], machining [36], laser physics [40,46,54], agent dynamics [52,53,70,73], neuroscience and biology [1,18,20,42,79], and climate modelling [13,41,48]. Important sources of delays are communication times between components of a system, maturation and reaction times, and the processing time of information received. When they are sufficiently large compared to the relevant internal time scales of the system under consideration, then the delays must be incorporated into its mathematical description. This leads to mathematical models in the form of delay differential equations (DDEs). In many situations the relevant delays can be considered to be fixed; examples are the travel time of light between components of a laser syst...

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