State dependent delay maps: numerical algorithms and dynamics of projections

TL;DR

This paper develops numerical algorithms and analyzes the dynamics of state dependent delay maps, a class of delay differential equations where delays depend on the current state, with applications to systems like Ikeda and Mackey-Glass.

Contribution

It introduces a fixed point formulation for state dependent delay maps, providing constructive proofs and numerical methods, and applies topological data analysis to study convergence of the dynamics.

Findings

Numerical procedures successfully implemented for example systems.

Local convergence of the proposed numerical method established.

Quantitative analysis of dynamics convergence using persistent homology.

Abstract

This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state depends only on delayed variables, and not on the un-delayed state. However, we allow that the delay is itself a function of the state variable. A delay map with constant delays can be rewritten explicitly as a discrete time dynamical system on an appropriate function space, and a delay map with small state dependent terms can be viewed as a ``non-autonomous'' perturbation. We develop a fixed point formulation for the Cauchy problem of such perturbations, and under appropriate assumptions obtain the existence of forward iterates of the map. The proof is constructive and leads to numerical procedures which we implement for illustrative examples, including the cubic Ikeda and Mackey-Glass systems with constant and state-dependent delays. After proving a local convergence result for the method, we study more qualitative/global convergence issues using data analytic tools for time series analysis (dimension and topological measures derived from persistent homology). Using these tools we quantify the convergence of the dynamics in the finite dimensional projections to the dynamics of the infinite dimensional system.