Vol. 7, No. 1 (2018): Regular Issue

In this article, we study the existence of solutions for a coupled system of higher order nonlinear fractional differential equations with non-local integralboundaryconditionbyusing Schaefer’sfixedpointtheoremandthe uniqueness result is proved by the contraction mapping principle. Finally, examples are provided to the applicability our main results.

Periodicity plays a significant role in the chaos theory from the beginning since the skeleton of chaos can consist of infinitely many unstable periodic motions. This is true for chaos in the sense of Devaney [1], Li-Yorke [2] and the one obtained through period-doubling cascade [3]. Countable number of periodic orbits exist in any neighborhood of a structurally stable Poincar´e homoclinic orbit, which can be considered as a criterion for the presence of complex dynamics [4–6]. It was certified by Shilnikov [7] and Seifert [8] that it is possible to replace periodic solutions by Poisson stable or almost periodic motions in a chaotic attractor. Despite the fact that the idea of replacing periodic solutions by other types of regular motions is attractive, very few results have been obtained on the subject. The present study contributes to the chaos theory in that direction. In this paper, we take into account chaos both through a cascade of almost periodic solutions and in the sense of Li-Yorke such that the original Li-Yorke definition is modified by replacing infinitely many periodic motions with almost periodic ones, which are separated from the motions of the scrambled set. The theoretical results are valid for systems with arbitrary high dimensions. Formation of the chaos is exemplified by means of unidirectionally coupled Duffing oscillators. The controllability of the extended chaos is demonstrated numerically by means of the Ott-Grebogi-Yorke [9] control technique. In particular, the stabilization of tori is illustrated.

In this paper we study a non-linear partial differential equation (PDE), proposed by Kudryashov [arXiv:1611.06813v1[nlin.SI]], using continuum limit approximation of mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models. This generalized semi-discrete equation can be considered as a model for the description of non-linear dislocation waves in crystal lattice and the corresponding continuous system can be called mixed generalized potential KdV and sine-Gordon equation. We obtain the Bäcklund transformation of this equation in Riccati form in inverse method. We further study the quasi-integrable deformation of this model.

This paper proposes a new comparison theorem and stability analysis of fractional order Cohen-Grossberg neural networks. Firstly, a new comparison theorem for fractional order systems is proved. Secondly, the stability of a class of fractional order Cohen-Grossberg neural networks with Caputo derivative is investigated on the basis of the above comparison theorem. Thirdly, sufficient conditions of stability of the neural networks are obtained utilizing the property of Mittag-Leffler functions, the generalized Gronwall-Bellman inequality and the method of the integral transform. Furthermore, a numerical simulation example is presented to illustrate the effectiveness of these results.

In this paper, a general class of stochastic fractional integrodifferential equations is investigated. The Picard-Lindel ¨of successive approximation scheme is used to establish the existence of solutions. The uniqueness of the solution is also studied under suitable conditions.

Some scaling properties describing the convergence for the steady state in a Hopf bifurcation are discussed. Two different procedures are considered in the investigation: (i) a phenomenological description obtained from time series coming from the numerical integration of the system, leading to a set of critical exponents and hence to scaling laws; (ii) a direct solution of the differential equations, which is possible only in the normal form. At the bifurcation, the convergence to the stationary state obeys a generalized and homogeneous function. For short time, the dynamics giving by the distance from the fixed point is mostly constant when a critical time is reached hence changing the dynamics to a convergence for the steady state given by a power law. Both the size of the constant plateau and the characteristic crossover time depend on the initial distance from the fixed point. Near the bifurcation, the convergence is described by an exponential decay with a relaxation time given by a power law.

Within the frame work of extended complex phase space characterized by x = x1 + ip4,y = x2 + ip5,z = x3 + ip6, px = p1 + ix4, py = p2 + ix5 and pz = p3 +ix6, we investigate the exact dynamical invariant for a coupled harmonic system in three dimensions. For this purpose Lie-algebraic method is employed and the invariant obtained in this work may play an important role in reducing the order of differential equations, solution of Cauchy system and to check the accuracy of a numerical simulation.

The aim of this paper is to establish and discuss the approximation of Caristi’s random fixed point theorems. Our theorem is used to determine a large numbers of nonlinear stochastic problems.

In this paper, we study fractional functional differential equations with three-point multi-term boundary conditions. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Existence and uniqueness results are obtained by using Schauder fixed point theorem and contraction principle. An illustrative example is also presented.