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

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.