Bifurcation and chaos in escape equation model by incremental harmonic balancing |
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Institution: | 1. Department of Mechanical Science and Engineering, University of Illinois at Urbana–Champaign, 1206 West Green Street, Urbana, IL 61822, United States;2. Department of Aerospace Engineering, University of Illinois at Urbana–Champaign, 319L Talbot Lab, 104 S. Wright Street, Urbana, IL 61801, United States;1. Department of Physics, Florida State University, Tallahassee, FL 32306-4350, U.S.A;2. Department of Physics and Physical Science, Marshall University, Huntington, WV 25755, U.S.A;1. Hanyang University, Sa-3dong, Sangrok-gu, Ansan 426-791, South Korea;2. National University of Sciences and Technology, PNEC Campus, Habib Rehmatullah Road, Karachi, Pakistan |
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Abstract: | Periodic motions of the nonlinear system representing the escape equation with cosine and sine parametric excitations and external harmonic excitations are obtained by the incremental harmonic balance (IHB) method. The system contains quadratic stiffness terms. The Jacobian matrix and the residue vector for the type of nonlinearity with parametric excitation are explicitly derived. An arc length path following procedure is used in combination with Floquet theory to trace the response diagram and to investigate the stability of the periodic solutions. The system undergoes chaotic motion for increase in the amplitude of the harmonic excitation which is investigated by numerical integration and represented in terms of phase planes, Poincaré sections and Lyapunov exponents. The interpolated cell mapping (ICM) method is used to obtain the initial condition map corresponding to two coexisting period 1 motions. The periodic motions and bifurcation points obtained by the IHB method compare very well with results of numerical integration. |
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