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Global bifurcations of a taut string with 1:2 internal resonance
Affiliation:1. Department of Mechanics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China;2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China;1. School of Computer & Communication, Lanzhou University of Technology, Lanzhou 730050, China;2. Department of Automation, Tsinghua University, Beijing 100084, China;1. Centro de Estadística y Software Matemático, Departamento de Cómputo Científico, Universidad Simón Bolívar, Sartenejas, Venezuela;2. Laboratorio de Fenómenos no Lineales, Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, Venezuela;3. Red de Estudios Interdisciplinarios, Academia Nacional de Ciencias Físicas, Matemáticas y Naturales, Venezuela;1. USP – Univ São Paulo, Departamento de Física, Rua do Matão, Cidade Universitária, Travessa R 187, 05508-090 São Paulo, SP, Brazil;2. School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom;3. UNESP – Univ Estadual Paulista, Departamento de Estatística, Matemática Aplicada e Computação, Av. 24A 1515, Bela Vista, 13506-900 Rio Claro, SP, Brazil;4. UNESP – Univ Estadual Paulista, Departamento de Física, Av. 24A 1515, Bela Vista, 13506-900 Rio Claro, SP, Brazil;5. The Abdus Salam – ICTP, Strada Costiera, 11, 34151 Trieste, Italy
Abstract:The global bifurcations of a taut string are investigated with the case of 1:2 internal resonance. The method of multiple scales is applied to obtain a system of autonomous ordinary differential equations. Based on the normal form theory, the desired form for the global perturbation method is obtained. Then the method developed by Kovacic and Wiggins is used to find explicit sufficient conditions for chaos to occur by identifying the existence of a Silnikov-type homoclinic orbit. Finally, numerical results obtained by using fourth-order Runge–Kutta method agree with the theoretical analysis at least qualitatively.
Keywords:Taut string  Normal form  Bifurcation  Homoclinic orbit  Melnikov function
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