Mihailova

One of the methods of qualitative analysis of a dynamical system is to estimate the position of its compact invariant sets closely associated with bounded trajectories of the system. As a solution to such a problem, one can use a localizing set, i.e. a set in the phase space which contains all in-variant compact sets of a system. In this paper the discrete-time dynamical Lozi system of second order was considered. This system was proposed as a piecewise linear analogue of the known discrete-time Chenon system which has a chaotic attractor for some parameter values. For positive invariant and negative invariant sets of the Lozi system a family of localizing sets was constructed and their intersections were determined. The results of this investigation were presented in figures.

The three-dimensional polynomial dynamical system dx/dt = y+z, dy/dt = -x+αy, dz/dt = x²-z with complex behavior is considered. In the particular case α = 0,5 this system reduces to one of systems with the chaotic behavior, which was found by J.C. Sprott. For the specified system the problem of localization of invariant compact sets, i.e. a problem of construction of such set in phase space of the system which contains all invariant compact sets of this system is solving. In article  the family of localizing sets for invariant compact sets is received using the functional method of localization proposed by A.P.~Krishchenko. The intersection of this family is found by numerical optimization methods.