Existence of dichotomies and invariant splittings for linear differential systems I

This paper is concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy for a class of equations which includes those with Bohr almost-periodic coefficients. The problem is treated in the context of linear skew-product flows, where it becomes clear how to generalize to the case of fiber-preserving flows on vector bundles. Both continuous and discrete flows are treated and the results apply to the linearized variational equation for a time-varying vector field on a manifold as well as the linearization of a diffeomorphism acting on a manifold. Sufficient conditions are given for a diffeomorphism on a manifold to be an Anosov diffeomorphism. For linear skew-product flows arising from ordinary differential equations our theory is a partial generalization of Floquet theory to the almost-periodic case.     

Existence of dichotomies and invariant splittings for linear differential systems,III

This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of a “trichotomy,” i.e., a continuous decomposition of Rⁿ into stable, unstable and neutral subspaces. For constant coefficients it reduces to the usual (Jordan) decomposition of Rⁿ into subspaces corresponding to eigenvalues with negative, positive, and zero real parts, respectively, but only in the case in which the eigenvalues with zero real parts occur with simple elementary divisors. The conditions are related to those used by Favard in his study of almost periodic equations. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. In the continuous case, sufficient conditions are given for a flow on a compact manifold to be an Anosov flow.     

On Invariant Measures for Diffusions on Banach Spaces

We consider a Banach space valued diffusion process corresponding to a stochastic evolution equation with strongly nonlinear drift. Sufficient conditions are given for the existence of a unique martingale solution and existence of an invariant measure. The resulting diffusion process is shown to be strongly Feller and irreducible. These properties yield uniqueness of invariant measure and ergodicity of the process. We also show that the invariant measure is equivalent to the invariant measure of the diffusion without drift. The main tool to show these results is the Girsanov Transformation.     

Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

This paper deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from the authors?? previous studies where the gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zeroand one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse-Smale diffeomorphism on a closed 3-manifold.     

Pitchfork bifurcations of invariant manifolds

A pitchfork bifurcation of an (m−1)-dimensional invariant submanifold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M₊ and M₋, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example.     

Zero-sum stochastic games with unbounded costs: Discounted and average cost cases

Zero-sum stochastic games with countable state space and with finitely many moves available to each player in a given state are treated. As a function of the current state and the moves chosen, player I incurs a nonnegative cost and player II receives this as a reward. For both the discounted and average cost cases, assumptions are given for the game to have a finite value and for the existence of an optimal randomized stationary strategy pair. In the average cost case, the assumptions generalize those given in Sennott (1993) for the case of a Markov decision chain. Theorems of Hoffman and Karp (1966) and Nowak (1992) are obtained as corollaries. Sufficient conditions are given for the assumptions to hold. A flow control example illustrates the results.     

On the existence of tori in Asada’s two-regional model

A two-regional five-dimensional model describing the development of income, capital stock and money stock, which was introduced by Asada in (2004) [2] is analysed. Sufficient conditions for the existence of two pairs of purely imaginary eigenvalues and the last one being negative in the linear approximation matrix of the model are found. Formulae for the calculation of the coefficients in the bifurcation equation of the model are derived. The theorem on the existence of invariant tori is presented. A numerical example illustrating the gained results is given.     

Inverse problems for stationary Navier-Stokes systems

An inverse problem for a nonlinear equation in a Hilbert space is considered in which the right-hand side that is a linear combination of given functionals is found from given values of these functionals on the solution. Sufficient conditions for the existence of a solution are established, and the solution set is shown to be homeomorphic to a finite-dimensional compact set. A boundary inverse problem for the three-dimensional thermal convection equations for a viscous incompressible fluid and an inverse magnetohydrodynamics problem are considered as applications.     

Impulsive Hopfield-type neural network system with piecewise constant argument

In this paper we introduce an impulsive Hopfield-type neural network system with piecewise constant argument of generalized type. Sufficient conditions for the existence of the unique equilibrium are obtained. Existence and uniqueness of solutions of such systems are established. Stability criterion based on linear approximation is proposed. Some sufficient conditions for the existence and stability of periodic solutions are derived. An example with numerical simulations is given to illustrate our results.     

Numbertheoretical endomorphisms with σ-finite invariant measure

A class of measurable transformations which serves as a model for severalf-expansions is discussed. Sufficient conditions for ergodicity and the existence of a σ-finite invariant measure are given.     

On solutions defined on an axis for differential equations with shifts of the argument

We consider linear first-order differential equations with shifts of the argument for functions with values in a Banach space. Sufficient conditions for the existence of nontrivial solutions of homogeneous equations are obtained. Ordinary differential equations are constructed for which all solutions defined on the entire axis are solutions of a given equation with shifts of the argument.     

Deformations of functionals and bifurcations of extremals

We study homology characteristics of critical values and extremals of Lipschitz functionals defined on bounded closed convex subsets of a reflexive space that are invariant under deformations. Sufficient conditions for the existence of a bifurcation point of a multivalued potential operator (the switch principle for the typical number of an extremal) are established.     

Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces

The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used.     

GUARANTEED COST CONTROL OF CONTINUOUS-TIME UNCERTAIN SINGULAR SYSTEM WITH BOTH STATE AND INPUT DELAYS

The guaranteed cost control problem for a continuous-time uncertain singular system with state and control delays, and a given quadratic cost function is studied in this paper. Sufficient conditions for the existence of the guaranteed cost controller are derived based on the linear inequality (LMI) approach. A parameterized characterization of the guaranteed cost laws is given in terms of the feasible solutions to a certain LMI, and the cost function of guaranteed cost controller exists an upper bound.     

A projectively natural flow for circle diffeomorphisms

Summary Motivated by a simple iteration on polygons, we study a projectively natural flow on the space of diffeomorphisms of the circle. This flow, which is given by a nonlinear fourth order PDE, has long term existence an uniqueness, and evolves an arbitrary diffeomorphism into a projective transformation. The flow can have radically different behavior, e.g. finite time blow-ups, when it is defined relative to different projective structures on the circle.Oblatum 12-II-1992     

Determination of the right-hand side of the Navier-Stokes system of equations and inverse problems for the thermal convection equations

Inverse problem for an evolution equation with a quadratic nonlinearity in the Hilbert space is considered. The problem is, given the values of certain functionals of the solution, to find at each point in time the right-hand side that is a linear combination of those functionals. Sufficient conditions for the nonlocal (in time) existence of a solution (on the whole time interval) are established. An application to the inverse problems for the three-dimensional thermal convection equations of viscous incompressible fluid is considered. Unique nonlocal (in terms of time) solvability of the problem of determining the density of heat sources under the regularity condition of the initial data and sufficiently large dimension of the observation space is proved.     

On the existence of a cycle in an asymmetric model of a molecular repressilator

In this paper, a nonlinear six-dimensional dynamic system, which is a model of functioning of a simple molecular repressilator, is considered. Sufficient conditions for the existence of a cycle C in the phase portrait of this system are found. An invariant neighborhood of C, which retracts to C, is constructed.     

Weak Convergence of Measures in Conservative Systems

Families of probability measures on the phase space of a dynamical system are considered. These measures are obtained as shifts of a given measure by the phase flow. Sufficient conditions for the existence of the weak convergence of the measures as the rate of the shift tends to infinity are suggested. The existence of such a limit leads to a new interpretation of the second law of thermodynamics. Bibliography: 5 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 194–205     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN R~n

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs. The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.