ON INVESTIGATION OF QUASILINEAR SYSTEMS WITH TWO DEGREES OF FREEDOM
Quasilinear systems with two degrees of freedom in resonant and nonresonant cases are considered. Shortened (averaged) systems (three-dimensional in a resonant case and two-dimensional in a nonresonant case) are given. These systems describe the behaviour of the initial four-dimensional system with accuracy up to small members. The consideration is illustrated by the example of two weakly coupled van der Pole equations. For the case of the main resonance, a three-dimensional averaged system is found which depends on two parameters, and the study of this system is carried out. Bifurcation curves found on the parameter plane are related to: 1) the change of number of equilibrium states, 2) the birth of a limit cycle (Andronov-Hopf bifurcation), 3) the birth of a limit cycle of the second kind. In case of higher-order resonances and also in nonresonant cases there exists a two-dimensional stable invariant torus in four-dimensional space of the initial system.