Energy determines multiple stability in time-delayed systems
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ORIGINAL PAPER
Energy determines multiple stability in time-delayed systems Yao Yan Jian Xu
· Shu Zhang · Qing Guo · · Kyung Chun Kim
Received: 30 July 2020 / Accepted: 24 October 2020 © Springer Nature B.V. 2020
Abstract Infinite dimensions always challenge the analysis of multiple stability in nonlinear time-delayed systems, as the computation and visualization of conventional basin of attraction are hampered by the increase in systems’ dimensions. To address this issue, this paper introduces an orthonormal basis to approximate the delayed states, uses their signal energy to represent them, and generalises the concept of basin of attraction into stochastic, where each pixel of the basin has the same energy level for each delayed state but corresponds to many initial conditions. Thus, the probabilities are estimated by Monte Carlo method, which is then extensively boosted by artificial neural networks including both classification and regression types. This procedure has been successively applied in the analysis of multiple stability in three typical time-delayed systems, which are a two-dimensional autonomous cutting process, a three-dimensional autonomous neural system, and a two-dimensional non-autonomous forced vibration isolator. They, respectively, have one, two, Y. Yan · Q. Guo School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu 611731, China S. Zhang · J. Xu (B) School of Aerospace Engineering and Applied Mechanics, Tongji University, 1239 Siping Road, Shanghai, China e-mail: [email protected] K. C. Kim School of Mechanical Engineering, Pusan National University, Busan 609-735, South Korea
and two delayed states, with two, three, and five coexisting attractors. It is found that the energy distribution in the delayed state determines both the convergence of Monte Carlo simulation and sensitivity of the classification neural network. It is also seen that the performance of classification neural networks decreases with respect to the increase in the number of attractors, but the regression neural networks show a robuster performance. As a result, the stochastic basin of attraction can be accurately and efficiently computed to reveal the multiple stability in various time-delayed systems. Keywords Nonlinear time-delayed systems · Multiple stability · Stochastic basin of attraction · Artificial neural networks · Machine learning
1 Introduction Time delay comprehensively exists in many nonlinear dynamic systems, such as regenerative cutting [1], neural system [2], vibration absorber [3], human balancing [4], and internet dynamics [5]. Delays map these systems into infinite-dimensional functional spaces [6], which hampers both analytical and numerical investigations of their dynamics. When these systems exhibit multiple stability, their long-term dynamics becomes even more complicated than those in ordinary differential systems, in which the conventional basin of attraction (BoA) is applicable [7]. Typical BoA relates initial conditions (ICs) with the
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