  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.
