ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

January 2018 , Volume 23 , Issue 1

Special issue on dynamical systems and applications

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*+*[Abstract](1041)

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**Abstract:**

We consider discrete dynamical systems coming from the models of evolution of populations playing *rock*-*paper*-*scissors game*. Asymptotic behaviour of trajectories of these systems is described, occurrence of the Neimark-Sacker bifurcation and nonexistence of time averages are proved.

*+*[Abstract](1056)

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**Abstract:**

We study the large time behaviour of the mass (size) of particles described by the fragmentation equation with homogeneous breakup kernel. We give necessary and sufficient conditions for the convergence of solutions to the unique self-similar solution.

*+*[Abstract](960)

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**Abstract:**

In the paper we consider a boundary value problem involving a differential equation with the fractional Laplacian

*+*[Abstract](628)

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**Abstract:**

The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate value of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.

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**Abstract:**

The article is devoted to semilinear Schrödinger equations in bounded domains. A unified semigroup approach is applied following a concept of Trotter-Kato approximations.Critical exponents are exhibited and global solutions are constructed for nonlinearities satisfying even a certain critical growth condition in

*+*[Abstract](970)

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**Abstract:**

We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that medical treatment is not immediate we add a new compartment,

*+*[Abstract](607)

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**Abstract:**

The paper considers the Dickman equation

for

*+*[Abstract](702)

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**Abstract:**

In the case of first-order linear systems with single constant delay and with constant matrix, the application of the well-known "step by step" method (when ordinary differential equations with delay are solved) has recently been formalized using a special type matrix, called delayed matrix exponential. This matrix function is defined on the intervals

*+*[Abstract](1255)

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**Abstract:**

This paper is concerned with analysis of two anticancer therapy models focused on sensitivity of therapy outcome with respect to model structure and parameters. Realistic periodic therapies are considered, combining cytotoxic and antiangiogenic agents, defined on a fixed time horizon. Tumor size at the end of therapy and average tumor size calculated over therapy horizon are chosen to represent therapy outcome. Sensitivity analysis has been performed numerically, concentrating on model parameters and structure at one hand, and on treatment protocol parameters, on the other. The results show that sensitivity of the therapy outcome highly depends on the model structure, helping to discern a good model. Moreover, it is possible to use this analysis to find a good protocol in case of heterogeneous tumors.

*+*[Abstract](923)

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**Abstract:**

Epilepsy is among the most common serious disabling disorders of the brain, and the global burden of epilepsy exerts a tremendous cost to society. Most people with epilepsy have acquired forms, and the development of antiepileptogenic interventions could potentially prevent or cure these epilepsies [

Using scalp and depth EEG recordings for six patients, the goal of our analysis is to find a way to quantitatively detect features in the EEG that could potentially help predict seizure onset post trauma. Unsupervised Diffusion Component Analysis [

Some new results that demonstrate how this algorithm is used to detect spikes in the EEG data as well as other changes over time are shown. This nonlinear and local network approach has been used to determine if the early occurrences of specific electrical features of epileptogenesis, such as interictal epileptiform activity and morphologic changes in spikes and seizures, during the initial week after TBI predicts the development of PTE.

*+*[Abstract](533)

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**Abstract:**

In this paper, we consider a certain theorem on the global invertibility of a

*+*[Abstract](698)

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**Abstract:**

In the paper we consider mathematical model proposed by Gottman, Murray and collaborators to describe marital dissolution. This model is described in the framework of discrete dynamical system reflecting emotional states of wife and husband during consecutive rounds of talks between spouses. The model is, however, non-symmetric. To make it symmetric, one need to assume that the husband reacts with delay. Following this idea we consider the influence of time delays in the reaction terms of wife or/and husband. The delay means that one or both of spouses split their attention between present and previous rounds of talks. We study possibility of the change of stability with increasing delay. Surprisingly, it occurs that the delay has no impact on the stability, that is the condition of stability proposed by Murray remains unchanged.

*+*[Abstract](800)

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**Abstract:**

The present paper describes the general structure of free boundary problems for systems of PDEs modeling biological processes. It then proceeds to review two recent examples of the evolution of a plaque in the artery, and of a granuloma in the lung. Simplified versions of these models are formulated, and rigorous mathematical results and open questions are stated.

*+*[Abstract](654)

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**Abstract:**

In this article we continue the study of discrete anisotropic equations and we will provide a new multiplicity results of the solutions for a discrete anisotropic equation. The procedure viewed here is according to variational methods and critical point theory. In fact, using a consequence of the local minimum theorem due Bonanno and mountain pass theorem we look into the existence results for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by mingling two algebraic conditions on the nonlinear term employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

*+*[Abstract](590)

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**Abstract:**

We study periodic problems for nonlinear evolution inclusions defined in the framework of an evolution triple

*+*[Abstract](611)

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**Abstract:**

In the paper the general higher order difference equation

with several deviating arguments is considered. According to the kind of the deviations *property A* and *B* are established.

*+*[Abstract](612)

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**Abstract:**

A class of a higher-order nonlinear difference system with delayed arguments where the first equation of the system is of a neutral type is considered. A classification of non-oscillatory solutions is given and results on their boundedness or unboundedness are derived. The obtained results are illustrated by examples.

*+*[Abstract](652)

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**Abstract:**

This paper is concerned with the optimal control problem of the vibrations of a viscoelastic beam, which is governed by a nonlinear partial differential equation. We discuss the initial-boundary problem for the cases when the ends of the beam are clamped or hinged. We define the weak solution of this initial-boundary problem. Our control problem is formulated by minimization of a functional where the state of a system is the solution of viscoelastic beam equation. We use the Galerkin method to approximate the solution of our control problem with respect to a spatial variable. Based on the finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation points of the optimal solutions of the discrete family control problems exist and each of these points is the solution of the original optimal control problem.

*+*[Abstract](764)

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**Abstract:**

We study the nonlinear boundary value problem consisting of a system of second order differential equations and boundary conditions involving a Riemann-Stieltjes integrals. Our proofs are based on the generalized Miranda Theorem.

*+*[Abstract](648)

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**Abstract:**

In the paper, we generalize the Arzelà-Ascoli's theorem in the setting of uniform spaces. At first, we recall the Arzelà-Ascoli theorem for functions with locally compact domains and images in uniform spaces, coming from monographs of Kelley and Willard. The main part of the paper introduces the notion of the extension property which, similarly as equicontinuity, equates different topologies on

*+*[Abstract](644)

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**Abstract:**

We consider a constrained semilinear evolution inclusion of parabolic type involving an

*+*[Abstract](762)

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**Abstract:**

We generalize a previously-studied model for chronic myeloid leu-kemia (CML) [

*+*[Abstract](798)

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**Abstract:**

We obtain necessary optimality conditions for the discrete-time fractional-order Cucker-Smale optimal control problem. By using fractional order differences on the left side of nonlinear system we introduce memory effects to the considered problem.

*+*[Abstract](612)

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**Abstract:**

The 2-dimensional system of neutral type nonlinear difference equations with delays in the following form

is considered. In this paper we use Schauder's fixed point theorem to study the existence of periodic solutions of the above system.

*+*[Abstract](537)

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**Abstract:**

This work is devoted to the study of the existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with *p*-Laplacian.

*+*[Abstract](746)

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**Abstract:**

A new theorem on asymptotic stability of stochastic semigroups is given. This theorem is applied to a stochastic semigroup corresponding to Stein's neuronal model. Asymptotic properties of models with and without the refractory period are compared.

*+*[Abstract](606)

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**Abstract:**

The paper deals with the formulation and the finite element approximation of a quasi-static thermoviscoelastic problem which describes frictional contact between a deformable body and a rigid foundation. The contact is modeled by normal damped response condition whereas the friction is described by the Coulomb law of dry friction. The weak formulation of the model consists of a coupled system of the variational inequality for the displacement and the parabolic equation for the temperature. The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model.

*+*[Abstract](1018)

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**Abstract:**

In this paper we make an attempt to study the influence of optimism and pessimism into our social life. We base on the model considered earlier by Rinaldi and Gragnani (1998) and Rinaldi *et al.* (2010) in the context of romantic relationships. Liebovitch *et al.* (2008) used the same model to describe competition between communicating people or groups of people. Considered system of non-linear differential equations assumes that the emotional state of an actor at any time is affected by the state of each actor alone, rate of return to that state, second actor's emotional state and mutual sympathy. Using this model we describe the change of emotions of both actors as a result of a single meeting. We try to explain who wants to meet whom and why. Interpreting the results, we focus on the analysis of the impact of a person's attitude to life (optimism or pessimism) on establishing emotional relations. It occurs that our conclusions are not always obvious from the psychological point of view. Moreover, using this model, we are able to explain such strange behavior as so-called Stockholm syndrom.

*+*[Abstract](585)

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**Abstract:**

Oncolytic viruses are genetically altered replication-competent vi-ruses which upon death of a cancer cell produce many new viruses that then infect neighboring tumor cells. A mathematical model for virotherapy of glioma is analyzed as a dynamical system for the case of constant viral infusions and TNF-*α* inhibitors. Aside from a tumor free equilibrium point, the system also has positive equilibrium point solutions. We investigate the number of equilibrium point solutions depending on the burst number, i.e., depending on the number of new viruses that are released from a dead cancer cell and then infect neighboring tumor cells. After a transcritical bifurcation with a positive equilibrium point solution, the tumor free equilibrium point becomes asymptotically stable and if the average viral load in the system lies above a threshold value related to the transcritical bifurcation parameter, the tumor size shrinks to zero exponentially. Other bifurcation events such as saddle-node and Hopf bifurcations are explored numerically.

*+*[Abstract](950)

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**Abstract:**

We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function representing the efficiency of reverse transcriptase inhibitors and consider the pharmacological delay associated to the control. Finally, we propose and analyze an optimal control problem with state and control delays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs.

*+*[Abstract](864)

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**Abstract:**

We consider a model of phenotypic evolution in populations with assortative mating of individuals. The model is given by a nonlinear operator acting on the space of probability measures and describes the relation between parental and offspring trait distributions. We study long-time behavior of trait distribution and show that it converges to a combination of Dirac measures. This result means that assortative mating can lead to a polymorphic population and sympatric speciation.

*+*[Abstract](650)

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**Abstract:**

The paper investigates the navigation problem of following a moving target, using a mathematical model described by a system of differential equations with random parameters. The differential equations, which employ controls for following the target, are solved by a new approach using moment equations. Simulations are presented to test effectiveness of the approach.

*+*[Abstract](731)

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**Abstract:**

In this paper, we obtain conditions under which the difference equation

has infinitely many homoclinic solutions. A variant of the fountain theorem is utilized in the proof of our theorem. Some known results in the literature are extended and complemented.

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