Equations of Advanced–Retarded Type and Solutions of Traveling-Wave Type for Infinite-Dimensional Dynamic Systems

In the paper we study infinite-dimensional dynamic systems with the Frenkel–Kontorova potentials. For such systems we describe their traveling-wave-type solutions, which are solutions for the corresponding boundary-value problem with nonlocal conditions. Describing the mentioned solutions is equivalent to describing the space of solutions for a functional differential equation that can be canonically derived from the original dynamic system. The stability of traveling-wave-type solutions is also investigated.     

Differential operators for systems of linear partial differential equations

The aim of this paper is the representation of solutions of systems of formally hyperbolic differential equations of second order. I. N.Vekua gave a representation of the solutions using the Riemann-matrix-function. Here we introduce special differential operators which map holomorphic functions into the set of solutions. An existence theorem for such operators is proved which gives a necessary and sufficient condition on the coefficients of a system. These operators are represented explicitly and several properties of them are investigated. We give different representations of the solutions of such systems and discuss the relation between the integral operator method and the method using differential operators which leads to an explicit representation of the Riemann-matrix-function by means of the differential operators. Two examples of special systems with differential operators are given.     

On stochastic and chaotic motion

In this paper we prove results regarding certain precise relationships between random motion and chaotic motion. In particular we prove a strong invariance principle for smooth functions of certain chaotic dynamical systems, and show that solutions of dynamical systems which are coupled to such chaotic systems may be approximated by solutions of stochastic differential equations     

Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities

Q-conditional symmetries (nonclassical symmetries) for the general class of two-component reaction-diffusion systems with non-constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type, an exhausted list of reaction-diffusion systems admitting such symmetry is derived. The results obtained for the reaction-diffusion systems are compared with those for the scalar reaction-diffusion equations. The symmetries found for reducing reaction-diffusion systems to two-dimensional dynamical systems, i.e., ODE systems, and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for a nonlinear reaction-diffusion system with an arbitrary diffusivity are constructed. Finally, the application of the exact solutions for solving a biologically and physically motivated system is presented.     

The New Method for the Searching Periodic Solutions of Periodic Differential Systems

In the paper we are giving the new method for searching periodic solutions of periodic differential systems. For this we construct a differential system with the same Reflecting Function as the Reflecting Function of the given system and with a known periodic solution. Then the initial data of the periodic solutions of this two systems coincide. In such a way the problem of existance periodic solutions goes to the Cauchy problem.     

Strongly flow-invariant sets

Some previously established existence theorems for periodic solutions of nonlinear, two-diminsional systems contain a hypothesis which requires inversion of the given system and is generally difficult to establish. This difficulty is eliminated for certain subclasses of systems. Further, some results due to Cronin concerning the number of such solutions and the characterization of hyperbolic sectors about critical points are shown to apply to the systems considered     

OPTIMAL CONTROL OF A CLASS OF DISTRIBUTED PARAMETER DELAY SYSTEMS

In this paper we consider the optimal control problem for a class of infinite dimensional delay evolution systems whose principal operator is the infinitesimal generator of an analytic semigroup. We give an existence result of α - solutions of the controlled systems and prove the existence of solutions for an extremal problem subject to such systems. In particular, the necessary conditions of optimality for the same problem are presented.     

Fixed poles in the model matching problem for systems over semirings

In this paper, solution existence conditions for the model matching problem are studied for systems over semirings, which are used in many applications, such as queueing systems, communication networks, and manufacturing systems. The main contribution is the discovery of fixed pole structure in solutions to the model matching problem. This fixed pole structure provides essential information contained in all the solutions to the model matching problem. For a discrete-event dynamic system example, a common Petri net component in the solutions of the model matching problem can be discovered from the fixed pole structure.     

On the holomorphic regularization of singularly perturbed systems of differential equations

A method for constructing pseudo-holomorphic solutions to strongly nonlinear singularly perturbed systems of differential equations, which a logical continuation of the Lomov regularization method, is proposed. The existence of integrals of such systems, holomorphic in the small parameter, is proven, and sufficient conditions for the convergence of expansion of solutions to these systems in powers of the small parameter in the usual sense are obtained.     

Hermitian spectral pseudoinversion and its applications

The present paper is devoted to the Hermitian spectral pseudoinversion and its applications to analysis, the solution and reduction of Hermitian differential-algebraic systems. New explicit formulas for the solutions of such systems and the solutions of related generalized Lyapunov equations are proposed. Attainable upper bounds for the norms of the solutions are obtained. A realization of the balanced truncation method not requiring computations involving projections onto deflating subspaces is proposed.     

一类广义四阶非线性Camassa-Holm方程的行波解

用动力系统的分支理论研究了一类广义四阶非线性Camassa-Holm方程的动力学行为和行波解,发现方程存在一些孤立波解,周期波解和一些诸如Compacton类型的非光滑行波解.在不同的参数条件下,给出了这些解存在的条件和一些特殊条件下的精确解.     

Symmetries of compatibility conditions for systems of differential equations

When symmetries of differential equations are applied, various types of associated systems of equations appear. Compatibility conditions of the associated systems expressed in the form of differential equations inherit Lie symmetries of the initial equations. Invariant solutions to compatibility systems are known as orbits of partially invariant and generic solutions involved in the Lie group foliation of differential equations and so on. In some cases Bäcklund transformations and differential substitutions connecting quotient equations for compatibility conditions and initial systems naturally arise. Besides, Ovsiannikov's orbit method for finding partially invariant solutions is essentially based on such symmetries.     

On the relationship between solutions of delay differential equations and infinite-dimensional systems of differential equations

We study the limit properties of solutions for a class of systems of ordinary differential equations as the number of equations and a certain parameter grow unboundedly. We show that the sequence of functions formed by the last components of solutions of such systems has a repeated limit. The limit function is a solution of a delay differential equation. Estimates of the convergence rate are obtained.     

Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.     

Explicit analytical formulation and exact inversion of decomposable fuzzy systems with singleton consequents

This paper is devoted to the inversion of fuzzy systems expressed by fuzzy rules with singleton consequents if input variables are described using strong triangular partitions. As pointed out in recent works, such fuzzy systems can be decomposed into collections of multi-linear subsystems. In this paper, an analytical formulation of the system output is explicitly developed and directly used in order to determine solutions to the inversion problem. Based on this analytical methodology, an algorithm is proposed for computing inverse solutions. As the inversion is handled analytically, the exactness of the obtained solutions is guaranteed. Furthermore, according to the decomposability of the studied fuzzy systems, all inverse solutions are found. Finally, whatever the fuzzy system under consideration, there is no need to study its invertibility beforehand since the algorithm is able to handle all possible situations (no solution, one unique solution, multiple solutions, an infinity of solutions).The proposed approach can be easily extended to other types of fuzzy systems provided that decomposability is preserved. In other words, with regard to exact inversion which often plays a key role in engineering applications such as control or diagnosis, decomposability is probably the first criterion that should be considered when choosing a specific fuzzy system structure.     

Periodic solutions for N-body type dynamical systems

We prove the existence of infinitely many T-periodic solutions of any assigned period T for a class of dynamical systems which contains the N-body one. We also show a sub class for which such solutions are not of simultaneous collision.     

A New Geometric Approach to Lie Systems and Physical Applications

The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be related with equations on a Lie group and with some connections in fiber bundles. We develop two methods for dealing with such systems: the generalized Wei–Norman method and the reduction method, which are very useful when particular solutions of the original problem are known. The theory is illustrated with some applications in both classical and quantum mechanics.     

Periodicity and solutions of some rational difference equations systems

In this paper we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case. Moreover, we study the periodicity of solutions for such systems. Finally, some numerical examples are presented.     

Kleine periodische Lösungen bei nichtlinearen stark-elliptischen Systemen von partiellen Differentialgleichungen I

Strongly elliptic systems of nonlinear partial differential equations are considered in the case when the derivatives of the solutions occuring in the nonlinear terms have the same order as those in the linear principal part. The existence of periodic solutions for such systems is investigated. It is shown that this problem can be reduced to the study of algebraic bifurcation equations, whose small solutions correspond to the classical solutions of the given problem. A discussion of the bifurcation equations will be given in a forthcoming paper.