# Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimisation

## TL;DR

This paper investigates how delay-induced homoclinic bifurcations in modified gradient systems can reorganize phase space, potentially aiding in global optimization by enabling trajectories to explore multiple local minima.

## Contribution

It introduces a class of delayed gradient systems with two-well potentials and demonstrates how delay can induce bifurcations that reorganize attractors, offering insights into delay's role in optimization.

## Key findings

- Delay induces a chain of homoclinic bifurcations.
- Bifurcations eliminate local attractors.
- Manifolds reorganize, removing barriers between minima.

## Abstract

Nonlinear dynamical systems with time delay are abundant in applications, but are notoriously difficult to analyse and predict because delay-induced effects strongly depend on the form of the nonlinearities involved, and on the exact way the delay enters the system. We consider a special class of nonlinear systems with delay obtained by taking a gradient dynamical system with a two-well "potential" function and replacing the argument of the right-hand side function with its delayed version. This choice of the system is motivated by the relative ease of its graphical interpretation, and by its relevance to a recent approach to use delay in finding the global minimum of a multi-well function. Here, the simplest type of such systems is explored, for which we hypothesise and verify the possibility to qualitatively predict the delay-induced effects, such as a chain of homoclinic bifurcations one by one eliminating local attractors and enabling the phase trajectory to spontaneously visit vicinities of all local minima. The key phenomenon here is delay-induced reorganisation of manifolds, which cease to serve as barriers between the local minima after homoclinic bifurcations. Despite the general scenario being quite universal in two-well potentials, the homoclinic bifurcation comes in various versions depending on the fine features of the potential. Our results are a pre-requisite for understanding general highly nonlinear multistable systems with delay. They also reveal the mechanisms behind the possible role of delay in optimisation.