Primary resonant optimal control for nested homoclinic and heteroclinic bifurcations in single-dof nonlinear oscillators |
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| Institution: | 1. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, PR China;2. Department of Mathematics, Chuxiong Normal University, 461 Lucheng South Road, Chuxiong City 675000, Yunnan, PR China;1. São Paulo State University (UNESP), Câmpus de São João da Boa Vista, São João da Boa Vista, São Paulo, Brazil;2. Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA;1. School of Electrical and Electronic Engineering, Wuhan Polytechnic University, Wuhan 430023, People?s Republic of China;2. School of Physics and Electronics, Henan University, Kaifeng 475004, People?s Republic of China;3. School of Science, Hubei Province Key Laboratory of Intelligent Robot, Wuhan Institute of Technology, Wuhan 430073, People?s Republic of China;4. Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People?s Republic of China;1. Science and Technology on Power Beam Processes Laboratory, Beijing Aeronautical Manufacturing Technology Research Institute, Beijing 100024, China;2. Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK;3. Huazhong University of Science and Technology, Wuhan 430074, China;4. Department of Materials, Imperial College London, London SW7 2AZ, UK;1. King Abdulaziz University, Faculty of Science, Mathematics Department, P.O. Box 80203, Jeddah 21589, Saudi Arabia;2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt |
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| Abstract: | A unified control theorem is presented in this paper, whose aim is to suppress the transversal intersections of stable and unstable manifolds of homoclinic and heteroclinic orbits in the Poincarè map embedding in system dynamics. Based on the control theorem, a primary resonant optimal control technique (PROCT for short) is applied to a general single-dof nonlinear oscillator. The novelty of this technique is able to obtain the unified analytical expressions of the control gain and the control parameters for suppressing the homoclinic and heteroclinic bifurcations, where the control gain can guarantee that the control region where the homoclinic and heteroclinic bifurcations do not occur can be enlarged as much as possible at least cost. The technique is applied to a nonlinear oscillator with a pair of nested homoclinic and heteroclinic orbits. By the PROCT, the transversal intersections of homoclinic and heteroclinic orbits can be suppressed, respectively. The hopping phenomenon that there coexist two kinds of chaotic attractors of Duffing-type and pendulum-type can be suppressed. On the contrary, if the first amplitude coefficient is greater than the critical heteroclinic bifurcation value, then another degenerate hopping behavior of chaos will take place again. Therefore, the phenomenon of hopping is the dominant type of chaos in this oscillator, whose suppressing or inducing is admissible from the points of practical and theoretical view. |
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