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Delay-induced homoclinic bifurcations in bistable dynamical systems
| Content Provider | Semantic Scholar |
|---|---|
| Author | Janson, Natalia B. Marsden, Christopher J. |
| Copyright Year | 2019 |
| Abstract | This paper is a prequel to our study of the possibility to solve global optimization problem by introducing a time delay into the classical gradient descent setting. Ultimately, we wish to explore the phenomena caused by delay in a dynamical system of the form $\dot{x}$$=$$f(x(t-\tau))$, $\tau$$>$$0$, where $f(z)$ is the negative of the gradient of some cost, or utility, function $V(z)$ with an arbitrary number of local minima of an arbitrary configuration. Using prior knowledge of the typical delay-induced effects in nonlinear dynamical systems, we hypothesize that the increase of delay should eliminate all local attractors and give rise to global chaos embracing all local minima, which could be exploited for optimization. In the subsequent paper studying multi-well $V(z)$ we demonstrate that this general idea has been correct. However, the details of bifurcations involved appear quite intricate and need to be first revealed for the simplest forms of utility functions having only two local minima, which is the goal of this paper. We discover that for a smooth $f$, the key phenomenon induced by the variation of the only control parameter $\tau$ is a homoclinic bifurcation coming in various versions. We reveal both the universal features of bifurcation scenarios realized for general smooth double-well landscapes $V$, and the distinctions between them determined by the local properties of $V$. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/1902.06341v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
