arXiv subjects
Publications and source records attributed to A. V. Romanov.
We present a class of one-dimensional systems of nonlinear parabolic equations for which long-time phase dynamics can be described by an ODE with a Lipschitz vector field in R^n. In the considered case of the Dirichlet boundary value problem sufficient conditions for a finite-dimensional reduction turn out to be much wider than the known conditions of this kind for a periodic situation.
We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamics of a system can be described by an ODE with Lipschitz vector field in $\mathbb{R}^{N}$. Precisely in this class, the first example of a parabolic problem of mathematical physics without the indicated property was recently constructed.
We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in ${\mathbb R}^{n}$ permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare--Bendixson theory. In the case $n=3$ we implement such a scenario for a model of a satellite rotation around a celestial body of small mass and for a biochemical model.
We study the problem on the weak-star decomposability of a topological $\mathbb{N}_{0}$-dynamical system $(Ω,φ)$, where $φ$ is an endomorphism of a metric compact set $Ω$, into ergodic components in terms of the associated enveloping semigroups. In the tame case (where the Ellis semigroup $E(Ω,φ)$ consists of $B_{1}$-transformations $Ω\rightarrow Ω$), we show that (i) the desired decomposition exists for an appropriate choice of the generalized sequential averaging method; (ii) every sequence of weighted ergodic means for the shift operator $x\rightarrow x\circφ$, $x\in C(Ω)$, contains a pointwise convergent subsequence. We also discuss the relationship between the statistical properties of $(Ω,φ)$ and the mutual structure of minimal sets and ergodic measures.
For 3D reaction--diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an inertial manifold and a normally hyperbolic inertial manifold is constructed. Similar questions concerning absolutely normally hyperbolic inertial manifolds are discussed.
For a continuous semicascade on a metrizable compact set $Ω$, we consider the weak$^{*}$ convergence of generalized operator ergodic means in ${\rm End}\, \, C^{*} (Ω)$. We discuss conditions on the dynamical system under which (a) every ergodic net contains a convergent subsequence; (b) all ergodic nets converge; (c) all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on $Ω$, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three algebraic-topological objects associated with the dynamical system: the Ellis enveloping semigroup, the Köhler operator semigroup $Γ$, and the semigroup $G$ that is the weak$^{*} $ closure of the convex hull of $Γ$ in ${\rm End}\, C^{*} (Ω)$. The main results are stated for ordinary semicascades (whose Ellis semigroup is metrizable) and tame semicascades. For a dynamics, being ordinary is equivalent to being "nonchaotic" in an appropriate sense. We present a classification of compact dynamical systems in terms of topological properties of the above-mentioned semigroups.