Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.     

Construction of Lyapunov functions by the method of localization of invariant compact sets

We suggest a new method for constructing Lyapunov functions for autonomous systems of differential equations. The method is based on the construction of a family of sets whose boundaries have the properties typical of the level surfaces of Lyapunov functions. These sets are found by the method of localization of invariant compact sets. For the resulting Lyapunov function and its derivative, we find analytical expressions via the localizing functions occurring in the specification of the above-mentioned sets. An example of a system with a degenerate equilibrium is considered.     

Functional method for the localization of invariant compact sets in discrete systems

We suggest a method for the localization of invariant compact sets in discrete autonomous systems. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for invariant compact sets of the discrete Henon system.     

Localization of compact invariant sets and global stability in analysis of one tumor growth model

In this paper, we study some features of global behavior of the four‐dimensional superficial bladder cancer model with Bacillus Calmette‐Guérin (BCG) immunotherapy described by Bunimovich‐Mendrazitsky et al. in 2007 with the help of localization analysis of its compact invariant sets. Its dynamics is defined by the BCG treatment and by densities of three cells populations: effector cells, tumor infected cells by BCG, and tumor uninfected cells. We find upper bounds for ultimate dynamics of the whole state vector in the positive orthant and also under condition that there are no uninfected tumor cells. Further, we prove the existence of the bounded positively invariant domain in both of these two situations. Finally, by using these assertions, we derive our main result: sufficient conditions of global asymptotic stability of the tumor‐free equilibrium point in the positive orthant. Copyright © 2013 John Wiley & Sons, Ltd.     

Localization of invariant compact sets of nonautonomous systems

We generalize the localization method for invariant compact sets of an autonomous dynamical system to the case of a nonautonomous system of differential equations. By using this method, we solve the localization problem for the Vallis third-order dynamical system governing some processes in atmosphere dynamics over the Pacific Ocean. For this system, we construct a one-parameter family of localizing sets bounded by second-order surfaces and find the intersection of all sets of the family.     

On compact sets with a certain affine invariant

We give a complete characterization of finite-dimensional compact sets with the following property: all of their images under affine operators are symmetric (that is, have symmetry planes of certain dimensions). We also study the noncompact case; namely, we reveal a class of unbounded closed sets with this property and conjecture that this class is complete.     

Localization analysis of compact invariant sets of multi-dimensional nonlinear systems and symmetrical prolongations

In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh–Bénard convection for which the symmetrical prolongation with the Lorenz system has been constructed.     

Localization of compact invariant sets of discrete-time control systems

Doklady Mathematics -     

Localization of invariant compact sets in uncertain discrete systems

A functional method for the localization of invariant compact sets in discrete autonomous systems is generalized to discrete systems with uncertainty. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for positively invariant compact sets of the discrete Henon system with uncertainty.     

Localization of invariant compact sets of families of discrete-time systems

The functional method of localization of invariant compact sets developed for continuous- and discrete-time dynamical systems is extended to families of discrete-time systems. Positively invariant compact sets are considered. As an example, the method is applied to the Hénon system with uncertain parameters.     

Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.     

Localization of compact invariant sets of discrete-time systems with disturbances

Doklady Mathematics -     

Structural stability of invariant sets of vibro-impact systems

The problem of persistence of invariant sets of vibro-impact systems under small changes of parameters of the system or passage from one impact model to another one, is studied. It is theoretically provied that the impact may be considered as the limiting case of interaction as the stiffness of the delimiter becomes infinite.     

Localizing sets for invariant compact sets of discrete dynamical systems with perturbation and control

We consider the localization problem for the invariant compact sets of a discrete dynamical system with perturbation and control, that is, the problem of constructing domains in the system state space that contain all invariant compact sets of the system. The problem is solved on the basis of a functional method used earlier in localization problems for time-invariant continuous and discrete systems and also for control systems. The properties of the corresponding localizing sets are described.