Abstract
Nonlinear dynamical systems with time delay are abundant in applications but are notoriously difficult to analyze 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 hypothesize 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 reorganization 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 optimization.
Highlights
Nonlinear dynamical systems with time delay are abundant in applications but are notoriously difficult to analyze and predict because delayinduced effects strongly depend on the form of the nonlinearities involved and on the exact way the delay enters the system
Differential equations with time delay represent a special class of dynamical systems, which are routinely used to model the behavior of both natural systems and artificial devices alongside ordinary differential equations (ODEs)
The fact of the occurrence of homoclinic bifurcations induced by the increase of delay in (2) with a multi-well potential V could be predicted based on the theorems overviewed in Sec
Summary
Differential equations with time delay represent a special class of dynamical systems, which are routinely used to model the behavior of both natural systems and artificial devices alongside ordinary differential equations (ODEs). The effects induced by delay greatly depend on a particular form of the DDE under study and on the exact way the delay is introduced The latter makes it hardly possible to predict, before resorting to numerical analysis and observing the behavior, the dynamics of even a scalar equation with a single delay τ ≥ 0, i.e., of x = f(x, xτ ) with x, f ∈ R and xτ = x(t − τ ) for an arbitrary f. System (1) represents the most basic setting for an optimization problem, which in practical applications is posed for x(t) ∈ RN, V(x) : RN → R with N ≥ 1, where V is the multi-well “cost” function, and x is the vector of parameters in need of optimization.[35,36] Solving this ODE can only deliver a local minimum; to find the global one, this setting is usually extended to enable the particle to overcome the barriers between the minima.
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