Approximate limit cycles of coupled nonlinear oscillators with fractional derivatives |
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| Institution: | 1. Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300350, China;2. Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin University, Tianjin 300372, China;1. School of Science, University of New South Wales, Canberra, Australia;2. I.E. Tamm Theory Department, P.N. Lebedev Physical Institute of the Russian Academy Sciences, 53 Leninsky Prospect, 119991 Moscow, Russia;1. College of Electronic and Optical Engineering & College of Microelectronic, Nanjing University of Posts and Telecommunications, Nanjing 210003, China;2. National and local Joint Engineering Laboratory of RF Integration and Micro-Assembly Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China;3. School of Mechanical Engineering, Southeast University, Nanjing 211189, China;1. School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, China;2. Departments of Civil and Environmental Engineering, Mechanical Engineering, and Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA;3. AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China;1. Laboratory of Nano- and Microfluidics and Microsystems - LabMEMS, Mechanical Engineering Department and Nanoengineering Department, POLI & COPPE and Interdisciplinary Nucleus for Social Development - NIDES/CT, Universidade Federal do Rio de Janeiro, Cidade Universitária, Cx. Postal 68503, Rio de Janeiro, RJ CEP 21945-970, Brazil;2. General Directorate of Nuclear and Technological Development - DGDNTM, Brazilian Navy, Rio de Janeiro, RJ, Brazil |
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| Abstract: | In this paper, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled nonlinear oscillators with fractional derivatives. Approximate limit cycles (LCs) of two systems of the coupled fractional van der Pol (VDP) oscillators and the fractional damped Duffing resonator driven by a fractional VDP oscillator are exampled for illustrating the validity and great potential of the HAM. The presented approach can provide approximate LCs very accurately and efficiently compared with some direct simulation results. This method can keep high accuracy and efficiency for both weakly and strongly nonlinear problems with any given fractional order. Furthermore, it is capable of tracking unstable LCs which cannot be generated by some time-marching numerical algorithm. Based on the obtained results, we analyze effect of different fractional orders, coupling coefficient, and nonlinear coefficient of the coupled equations on amplitudes and frequencies of the LCs. |
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