修正梯度双稳态系统的时滞诱导同宿分支及其与优化的关系
Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimisation
摘要：
具有时滞的非线性动力系统有着广泛的应用，但由于时滞引起的效应强烈地依赖于所涉及的非线性的形式以及时滞进入系统的确切方式，因此难以分析和预测。我们考虑一类特殊的非线性时滞系统，通过采取一个梯度动力系统的两个井的“潜在”的功能，并取代其延迟版本的右侧功能的参数。该系统的选择是出于其图形解释的相对容易性，以及其与最近的方法的相关性，该方法使用延迟来找到多孔函数的全局最小值。在这里，这种系统的最简单的类型进行了探索，我们假设和验证的可能性，定性预测的延迟引起的影响，如一个链的同宿分支一个接一个消除本地吸引子，使相位轨迹自发访问附近的所有本地最小值。这里的关键现象是延迟引起的流形重组，它不再作为同宿分支后的局部极小值之间的障碍。尽管一般情况下是相当普遍的两阱势，同宿分岔来在不同的版本取决于潜在的精细功能。我们的结果是理解一般的高度非线性多稳态时滞系统的先决条件。它们还揭示了延迟在优化中可能发挥作用的机制。
英文摘要：
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.
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