Konstantin E. Starkov

Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane.

Output maps with associated asymptotically stable zero dynamics

Abstract Conceptions of a relative degree and a minimum phase system are connected with many control problems. In order to apply them it is necessary to know the output map for which the affine system takes on the minimum phaseness property. We present necessary and sufficient conditions of the existence of such outputs. In the case of the relative degree more than one the obtained conditions result in a new setting of the stabilization problem.

Modeling cancer evolution: evolutionary escape under immune system control

It can be expected that adequate immune response should be able to annihilate cancer at a very early stage of its appearance. However, in some cases cancer is able to persist avoiding immune response. One can conject that cancer is able to avoid immune response control due to a succession of mutations leading to the development of immune-resistant cells. In order to illustrate this possibility, in this paper we present a 2n–dimensional mathematical model that describes interaction of n subtypes of tumor cells with corresponding genotype-specific immune response. The model postulates that there is a probability for tumor cells of each of n subtype to produce offsprings of other types. Each of the subtypes activates the genotype-specific immune response with a possibility of suppressing cancer cells of other genotypes (the cross-immunity). Numerical simulations show that if cancer cells are able to mutate comparatively fast and if immune response is not strong enough, then, despite immune system pressure, cancer is able to persist.

Iteration method of the localization of periodic orbits

A method for estimating domains with all periodic orbits of nonlinear systems is developed.

Bounding the domain of some three species food systems

Abstract This paper presents results concerning the solution of the localization problem of any compact invariant set contained in the positive orthant for two 3-species food systems. The final localization domain in both of examples is a tetrahedron which may have linear excisions for some values of parameters. Some sufficient conditions of the nonexistence of compact invariant sets in the positive orthant are given as well. The results have been obtained by using the first order extremum conditions and the iteration theorem. Parameters of all localization sets are computed explicitly.

Examples of localization of periodic orbits of polynomial systems

This paper contains examples of localization of periodic orbits of quadratic right-side systems by using polynomials of degree not exceeded two. Our results are based on the application of the first order extremum conditions.