A.S. Gorobtzov, O.E. Grygoyeva, E.N. Ryzhov

Аuto oscillatory polynomial systems with limit cycles containing close to linear parts - Lame curves - are considered in the article. Such a geometry provides an auto oscillatory character close to relaxatory. Systems received are enough to control oscillations for a wide class of systems changing geometrical characteristics of limit cycles. Sufficient conditions of Lame surface invariance and stability, the sufficient conditions of an attraction for the trajectories, which lie both at the domain, limited by the surface of Lame and outer adjoining layer, described by equations with natural even degrees, for the systems of nonlinear connected oscillators are received. It is possible to control the geometry of limit cycles by degree in the Lame equation as the control. Equations connecting squares of the domains inside limit circles in four connected systems are received. In this case the attraction of trajectories to the boundary of Lame manifold occurs due to the generation of steady limit cycles. The stabilization of oscillations is ensured by stabilizing odd polynomial controls of the third order. The synthesis of systems is based on the theory of invariant asymptotically steady sets of V.I. Zubov. The results are illustrated by the phase portraits of limit cycles on the multidimensional surfaces of Lame and by the integral tubes of the transient processes of stabilization, obtained with the numerical simulation of the double-connected oscillators. In this case the attraction of trajectories to the boundary of Lame - manifold occurs due to the generation of steady limit cycles.