Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems |
| |
| Institution: | 1. School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China;2. School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China;1. Université Paris-Est, Laboratoire Navier (UMR 8205), CNRS, Ecole des Ponts ParisTech, IFSTTAR, F-77455 Marne La Vallée, France;2. Université Paris-Est, MAST, SDOA, IFSTTAR, F-77447 Marne La Vallée, France;1. Department of Mathematics G. Castelnuovo, Sapienza Roma University, Roma, Italy;2. In Unam Sapientiam, Roma, Italy;3. IAPS, Istituto Nazionale di Astrofisica INAF, Roma, Italy;4. IASF, Istituto Nazionale di Astrofisica INAF, Palermo, Italy;5. INFN, Sezione Roma1, Roma, Italy;1. Instituto de Física da USP, Rua do Matão, Travessa R, 187, Cidade Universitária, 05508-090 São Paulo, SP, Brazil;2. School of Mathematics, University of Bristol, Bristol, United Kingdom;3. Departamento de Física, UNESP – Univ Estadual Paulista, Av.24A, 1515, 13506-900 Rio Claro, SP, Brazil;1. DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa;2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052 Australia |
| |
| Abstract: | In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm. |
| |
| Keywords: | Vibro-impact Nonlinear dynamics Unstable periodic orbit Unstable limit cycle Poincaré map |
| 本文献已被 ScienceDirect 等数据库收录！ |
|
