Countable Systems Research Articles

Problem Without Initial Conditions for a Countable System of Semilinear Hyperbolic Equations of the First Order

We derive sufficient conditions for the solvability of the problem without initial conditions for a countable hyperbolic system of semilinear equations of the first order and establish conditions for the classical solvability of the initial-boundary value problem for countable hyperbolic systems of semilinear equations of the first order in a semistrip.

Nonlocal Mixed-Value Problem for a Boussinesq-Type Integrodifferential Equation with Degenerate Kernel

We consider the problem of unique solvability of a mixed problem for a nonlinear Boussinesq-type fourthorder integrodifferential equation with degenerate kernel and integral conditions. A method of degenerate kernel is developed for the case of nonlinear Boussinesq-type fourth-order partial integrodifferential equation. The Fourier method of separation of variables is used. After redenoting, the integrodifferential equation is reduced to a system of countable systems of algebraic equations with nonlinear and complex right-hand sides. As a result of the solution of this system of countable systems of algebraic equations and substitution of the obtained solution in the previous formula, we get a countable system of nonlinear integral equations. To prove the theorem on unique solvability of the countable system of nonlinear integral equations, we use the method of successive approximations. Further, we establish the convergence of the Fourier series to the required function of the mixed problem. Our results can be regarded as a subsequent development of the theory of partial integrodifferential equations with degenerate kernels.

Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel

A mixed problem for a certain nonlinear third-order intregro-differential equation of the pseudoparabolic type with a degenerate kernel is considered. The method of degenerate kernel is essentially used and developed and the Fourier method of variable separation is employed for this equation. A system of countable systems of algebraic equations is first obtained; after it is solved, a countable system of nonlinear integral equations is derived. The method of sequential approximations is used to prove the theorem on the unique solvability of the mixed problem.

Countable hyperbolic systems in the theory of nonlinear oscillations

In this article a model example of a mixed problem for a fourth-order differential equation reduced to initial-boundary value problem for countable hyperbolic system of first order coherent differential equations.

The Borwein-Preiss variational principle for nonconvex countable systems of equilibrium problems

The Borwein-Preiss variational principle for nonconvex countable systems of equilibrium problems

Random countable iterated function systems with overlaps and applications

We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist many differences between our case and the finite deterministic case studied in [7], and we introduce new methods specific to the infinite and random case. We apply our results towards a problem related to a conjecture of Lyons about random continued fractions ([11]), and show that for all parameters λ>0, the invariant measure νλ is exact dimensional; and in addition, we give estimates for the pointwise and Hausdorff dimension of νλ, for λ in a certain interval. The finite IFS determining these continued fractions is not hyperbolic, but we can associate to it a random infinite IFS of contractions which have overlaps. We study then also other large classes of random countable iterated function systems with overlaps, namely: a) several types of random iterated function systems related to Kahane–Salem sets; and b) randomized infinite IFS in the plane which have uniformly bounded number of disc overlaps. For all the above classes, we prove dimensional exactness, and we find lower and upper estimates for the pointwise (and Hausdorff, packing) dimensions of the projection measures.

Random countable alphabet conformal iterated function systems satisfying the transversality condition

Dealing with with countable (finite and infinite alike) alphabet random conformal iterated function systems with overlaps, we formulate appropriate transversality conditions and then prove the relevant, in such a context, the Moran–Bowen formula which determines the Hausdorff dimension of random limit sets in dynamical terms. We also provide large classes of examples of such random systems satisfying the transversality condition.

Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems

Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems

Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions

We consider the classical solutions of mixed problems for infinite, countable systems of parabolic functional dierential equations. Dierence methods of two types are constructed and convergence theorems are proved. In the first type, we approximate the exact solutions by solutions of infinite dierence systems. Methods of second type are truncation of the infinite dierence system, so that the resulting dierence problem is finite and practically solvable. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for the given functions. The comparison system is infinite. Parabolic problems with deviated variables and integro-dierenti al problems can be obtained from the general model by specifying the given operators.

Invariant Measure Associated with a Generalized Countable Iterated Function System

Generalized countable iterated function systems (GCIFS) are an extension of countable iterated function systems by considering contractions from X × X into X instead of contractions on the compact metric space X into itself. For a GCIFS endowed with a system of probabilities we associate an invariant and normalized Borel measure whose support is just the attractor of the respective GCIFS, extending the classical Hutchinson’s construction.

Poisson representations of branching Markov and measure-valued branching processes

Representations of branching Markov processes and their measure-valued limits in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. Each particle has a location and a "level," but unlike earlier constructions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level $r$, or for the limiting models, hits infinity. For branching Markov processes, at each time $t$, conditioned on the state of the process, the levels are independent and uniformly distributed on $[0,r]$. For the limiting measure-valued process, at each time $t$, the joint distribution of locations and levels is conditionally Poisson distributed with mean measure $K(t)\times\varLambda$, where $\varLambda$ denotes Lebesgue measure, and $K$ is the desired measure-valued process. The representation simplifies or gives alternative proofs for a variety of calculations and results including conditioning on extinction or nonextinction, Harris's convergence theorem for supercritical branching processes, and diffusion approximations for processes in random environments.

Generalized countable iterated function systems

One of the most common and most general way to generate fractals is by using iterated function systems which consists of a finite or infinitely many maps. Generalized countable iterated function systems (GCIFS) are a generalization of countable iterated function systems by considering contractions from X ? X into X instead of contractions on the metric space X to itself, where (X, d) is a compact metric space. If all contractions of a GCIFS are Lipschitz with respect to a parameter and the supremum of the Lipschitz constants is finite, then the associated attractor depends continuously on the respective parameter.

Frequency Locking in Countable Cellular Systems, Localization of (Asymptotic) Quasi-Periodic Solutions of Autonomous Differential Systems

We address the question of frequency locking in coupled differential systems and of the existence of some quasi-periodic solutions of a certain kind of differential systems. Those systems are named “cellular systems” quite generally as they deal with countable numbers of coupled systems in some general Banach spaces. Moreover, the inner dynamics of each subsystem does not have to be specified. We reach some general results about how the frequency locking phenomenon is related to the structure of the coupling map. Those results can be restated in terms of localization of a certain type of quasi-periodic solution of differential systems that may be seen as cellular systems. This paper gives some explanations about how and why synchronized behaviors naturally occur in a wide variety of complex systems.

Continuous and discrete models of neural systems in infinite-dimensional abstract spaces

Infinite countable or uncountable systems of nonlinear ordinary and partial differential equations, which are discrete and continuous models, respectively, of real-world phenomena and processes were analysed in physics, biology, neuroscience, in studying of neural systems or in pure mathematics. Discrete models and corresponding infinite countable systems have been investigated by numerous authors in Banach sequence spaces ℓ ∞ . But continuous model and corresponding infinite uncountable systems have been investigated by a few authors only. It is interesting that we can attack the above problems by using the same topological fixed-point theory tools as in the countable case. It is the study of infinite uncountable systems of parabolic reaction-diffusion-convection equations that is the subject matter of this paper. Motivation for considering such systems and examples thereof are presented. The paper is an introduction to the study of infinite uncountable systems of parabolic equations.

Stationary systems of Gaussian processes

We describe all countable particle systems on ℝ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson point process with intensity measure $\mathfrak{m}$ and moving independently of each other according to the law of some Gaussian process ξ. We classify all pairs $(\mathfrak{m},\xi)$ generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure $\mathfrak{m}$ is arbitrary, whereas the process ξ is stationary. In the second family, the measure $\mathfrak{m}$ is a multiple of the Lebesgue measure, and ξ is essentially a Gaussian stationary increment process with linear drift. In the third, most interesting family, the measure $\mathfrak{m}$ has a density of the form αe−λx, where α > 0, λ ∈ ℝ, whereas the process ξ is of the form ξ(t) = W(t) − λσ2(t) / 2 + c, where W is a zero-mean Gaussian process with stationary increments, σ2(t) = Var W(t), and c ∈ ℝ.

On the existence of infinite-dimensional invariant tori of nonlinear countable systems of difference-differential equations

In the space of bounded number sequences, we establish sufficient conditions for the existence of invariant tori for nonlinear countable systems of difference-differential equations defined on infinite-dimensional tori and containing an infinite set of constant deviations of a scalar argument.

On the existence of invariant tori of countable systems of difference-differential equations defined on infinite-dimensional tori

In the space of bounded number sequences, we establish sufficient conditions for the existence of invariant tori for linear and quasilinear countable systems of differential-difference equations defined on infinitedimensional tori and containing an infinite set of constant deviations of a scalar argument.

The renormalization transformation for two-type branching models

Cet article etudie des systemes denombrables de diffusions en interaction hierarchiques et lineaires vivant dans le quadrant positif. De tels systemes apparaissent dans la dynamique d’individus de deux types qui migrent tout en interagissant dans des colonies. Le comportement a grande echelle et temps long peut etre etudie en utilisant le programme de renormalisation. Ce programme, qui a permis de resoudre d’autres cas (principalement uni-dimensionnels) est base sur la construction et l’analyse d’une transformation de renormalisation non lineaire, agissant sur la fonction de diffusion des composants du systeme et connectant l’evolution de blocs moyennes sur le temps a differentes echelles. Nous identifions une classe generale de fonctions de diffusion dans le quadrant positif pour lequel la transformation de renormalisation est bien definie et qui, sous une conjecture de comportement aux bords, peut-etre iteree. A l’interieur de certaines sous-classes, nous identifiens les points fixes de la transformation et etudions leurs domaines d’attraction. Ces domaines d’attraction constitutent les classes d’universalite du systeme apres changement d’echelle dans le temps et l’espace.

Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems

The concepts of collective sensitivity and compact-type collective sensitivity are introduced as stronger conditions than the traditional sensitivity for dynamical systems and Hausdorff locally compact second countable (HLCSC) dynamical systems, respectively. It is proved that sensitivity of the induced hyperspace system defined on the space of non-empty compact subsets or non-empty finite subsets (Vietoris topology) is equivalent to the collective sensitivity of the original system; sensitivity of the induced hyperspace system defined on the space of all non-empty closed subsets (hit-or-miss topology) is equivalent to the compact-type collective sensitivity of the original HLCSC system. Moreover, relations between these two concepts and other dynamics concepts that describe chaos are investigated.

Non-linear Volterra IDE, infinite systems and normal forms of ODE

We study the Volterra integro-differential equation in R n (⋆ ) d x d t = X ( t , x , ∫ 0 t K ( t , s ) g ( x ( s ) d s ) ) . We establish a connection between system (⋆) with a kernel of the form (⋆ ⋆) K ( t , s ) = ∑ j = 1 ∞ C j F j ( t ) G j ( s ) and a countable system of ordinary differential equations. Such a reduction allows use of results obtained earlier for the countable systems of differential equations in the study of integro-differential equations. In this paper we discuss problems related to the stability of systems (⋆) and (⋆⋆), as well as applications of the method of normal forms to solving some problems in the qualitative theory of integro-differential equations. In particular, it can be employed for the study of critical cases of stability and bifurcation problems in integro-differential equations.