Homoclinic orbits of the Kovalevskaya top with perturbations
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Summary: The instability issue of the permanent rotation of a heavy top is revisited and the analytical characteristic equation for the particular solution is derived. The homoclinic orbits of the Kovalevskaya top are formulated from the Kovalevskaya fundamental equation and the Kotter transformation. Some integrable motions of the undisturbed Kovalevskaya top are obtained by means of the Jacobian elliptic integrals. The criteria for judging the onset of homoclinic transversal intersections of the stable and unstable manifolds at a saddle in the Poincaré map when the Kovalevskaya top is disturbed by a small external torque are established via the Melnikov integral due to \textit{P. Holmes} and \textit{J. Marsden} [Arch. Ration. Mech. Anal. 76, 135--165 (1981; Zbl 0507.58031)]. This theoretical achievement is crosschecked by the fourth-order Runge-Kutta algorithms and by the Poincaré section to investigate the long-term behaviors of the Euler-Poisson equations with small forced torques. This also gives a theoretical and numerical evidence for the nonintegrability of the disturbed Kovalevskaya top.
- City University of Hong Kong China (People's Republic of)
Perturbation methods for rigid body dynamics, Hamilton's equations, Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics, homoclinic orbits, chaos, Poincaré section, Dynamical systems in classical and celestial mechanics, Kovalevskaya fundamental equation, Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods, Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol'd diffusion, Kotter transformation, Kovalevskaya top, Integrable cases of motion in rigid body dynamics, Homoclinic and heteroclinic orbits for dynamical systems, Melnikov integral
Perturbation methods for rigid body dynamics, Hamilton's equations, Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics, homoclinic orbits, chaos, Poincaré section, Dynamical systems in classical and celestial mechanics, Kovalevskaya fundamental equation, Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods, Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol'd diffusion, Kotter transformation, Kovalevskaya top, Integrable cases of motion in rigid body dynamics, Homoclinic and heteroclinic orbits for dynamical systems, Melnikov integral
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