On non-conservative perturbations of three-dimensional integrable systems

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At present, non-conservative perturbations of two-dimensional nonlinear Hamiltonian systems have been studied quite fully. The purpose of the study is to generalize this theory to the three-dimensional case, when the unperturbed system is nonlinear, integrable and has a region filled with closed phase trajectories. In this paper, autonomous perturbations are considered and the main attention is paid to the problem of limit cycles. Methods. The study is based on the construction of special coordinates in which the variables are divided into two slow and one fast, and in the first approximation with respect to a small parameter the equations for the slow variables are separated. Results. It is shown that hyperbolic equilibrium states of a truncated system determine closed phase trajectories, in the vicinity of which cycles appear under the perturbation. Conclusion. Thus, the problem is reduced to the study of solutions of the “generating” system of two algebraic or transcendental equations, similar to the generating Poincare–Pontryagin equation for two-dimensional systems. As examples, we considere a three-imensional van der Pol type system and the Lorentz system in the case of large Rayleigh numbers.