“Connecting surfaces” can locate hidden attractors
School of Physics and Opotoelectric Engineering, Xiangtan University, 411105, Xiangtan, Hunan, China
2 School of Automation and Electronic Information, Xiangtan University, 411105, Xiangtan, China
Accepted: 11 December 2022
Published online: 21 February 2023
Hidden attractors are often associated with rare or unusual oscillations in the system, potentially catastrophic hazards for structures such as bridges and aircraft. Due to the lack of reasonable analytical and numerical methods, locating them is one of the most challenging tasks in nonlinear dynamics. In previous work, several localization methods have some limitations, such as low localization efficiency, inaccurate localization, and small localization range. In this study, we observed a new class of critical points named jerk points, where the jerk is zero, but the velocity and acceleration remain non-zero. The results indicate that jerk points are very significant for comprehending the transient dynamics in the phase space. Based on the jerk points, we propose the connecting surface method, which is composed of a set of inflection points, including fixed points, perpetual points and jerk points. It provides as much potential information in system dynamics as possible and can capture attractors more comprehensively, especially multiple stable and hidden attractors. By comparing existing methods and considering various 3D dynamic systems, the results show that the proposed method is effective and superior. The method can be used in the general research of dynamic systems and has high localization efficiency and a wide search range in localization hiding and coexistence attractors.
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