Enhancing subdivision technique with an adaptive interpolation sampling method for global attractors of nonlinear dynami
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Enhancing subdivision technique with an adaptive interpolation sampling method for global attractors of nonlinear dynamical systems Xi Wang1 · Jun Jiang1 · Ling Hong1 Received: 9 August 2020 / Revised: 16 August 2020 / Accepted: 30 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The cell mapping method is a prominent one for global analysis of nonlinear dynamical systems, with which multiple invariant sets can be obtained. However, it is a continuous challenge to enhance the efficiency of the cell mapping method, especially when dealing with high-dimensional nonlinear dynamical systems. In this paper, the subdivision technique commonly used in the cell mapping method is incorporated with an interpolation sampling method, which can further enhance the efficiency over the set-oriented method with subdivision for global attractors of nonlinear dynamical systems. In the present method, a new lattice of interpolating nodes is adopted to fit into the nesting structures of the subdivided cells, portions of which are sequentially removed when resolution goes from low to high. An improved interpolation method is developed to obtain onestep sample mappings, instead of integrations when error bounds are met, in the process of subdivision iterations. Furthermore, a Hash table is introduced in order to fast search and locate the coordinates of cells that cover the invariant sets. Three examples of nonlinear dynamical systems are presented to demonstrate the enhancement in efficiency and effectiveness of the proposed method, indicating that a computational cost is one half down to one sixth of the previous methods. Keywords Cell mapping · Subdivision · Interpolation · Global attractor · High-dimensional system
1 Introduction Plenty of nonlinear problems are presented in a variety of disciplines including physics, chemistry, biology, engineering and even economic sociology. It is inappropriate to describe these problems by building linear dynamical systems. As a result, nonlinear dynamical systems are increasingly considerable for solving these nonlinear problems in real world more accurately. It is seldom that these nonlinear dynamical systems can be solved analytically, therefore several numerical methods are widely utilized in system dynamics. For a continuous system governed by nonlinear differential equations, one of the common methods is numerical integration such as Euler method or Runge–Kutta method, to study the dynamical response of an initial condition. Unfortunately, when there is coexistence of multiple solutions especially unstable solutions in a nonlinear dynamical system, it is
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Ling Hong [email protected] State Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, China
almost impossible to find all solutions by numerical integration. There is another numerical method called the cell mapping method, which is prominent for global analysis of nonlinear dynamical systems. The first version of cell mapping methods called the simple cell m
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