Vol. 7, No. 2 (2018): Regular Issue

This paper is concerned with trajectory controllability of a class of fractional-order systems of order α ∈ (1,2] with delay in state variable and with a nonlinear control term. Firstly, the existence and uniqueness of the system is proved under suitable conditions on the nonlinear term involving state variable. Then the trajectory controllability of this class of systems is studied using Mittag-Leffler functions and Gronwall-Bellman inequality. Finally, examples are given to illustrate the proposed theory.

The localization of the last invariant spanning curve – also known as the last invariant tori – in a family of generalized standard mappings is discussed. The position of the curve dictates the size of the chaotic sea hence influencing the scaling properties observed for such region. The mapping is area preserving and is constructed such its dynamical variables are the action, J, and the angle θ . The action is controlled by a parameter ε , controlling the intensity of a generic nonlinear function, which defines a transition from integrable for ε = 0 to non integrable for ε≠ 0. The angle is dependent on a parameter γ. If γ > 0, the angle has the property that it diverges in the limit of vanishingly action and is added, by a finite function dependent on a free parameter γ , when the action is larger than zero. The case γ = −1 reproduces the expression of the angle for the traditional standard mapping. The phase space is mixed and shows, for certain ranges of control parameters, a set of periodic islands, chaotic seas and invariant spanning curves. Statistical properties for an ensemble of noninteracting particles starting in the chaotic sea with very low action is considered and we show: (i) the saturation of chaotic orbits grows with εα ; (ii) the regime of growth scales with nβ ; and (iii) the regime that marks the changeover from the diffusive dynamics to the stationary state scales with ε z. The exponents α and z depend on γ and are independent of the nonlinear function f while β is universal. To illustrate the theory here proposed, we obtain an estimation for the critical parameter Kc for a generalized standard mapping considering three different periodic functions. We also find α, β and z for different nonlinear functions.

This paper portrays and interpret the fuzzy delay differential equations using the generalized differentiability concept by applying the Generalized Characterization Theorem. Subsequently we also investigate the problem of finding a numerical approximation of solutions. Moreover, the Runge-Kutta approximation methods is implemented and its error analysis are also discussed. The applicability of the theoretical results are illustrated with some examples.

In this paper, we have investigated the existence of positive solutions for system of nonlinear itegro-differential equations with multi(m)-point boundary conditions on time scales. Existence of positive solutions are established via Guo-Krasnosel’skii fixed point theorem for operators on a cone in a Banach space. An example is given to illustrate the effectiveness of our proposed result.

The present investigation deals with a diffusive predator-prey model in order to study the dynamic response of a reaction-diffusion model with linear prey harvesting. The governing equations of the proposed model system subject to the homogeneous Neumann boundary condition provide some qualitative interpretations of solutions to the reaction-diffusion system. The conditions of diffusion-driven instability and the Turing bifurcation region in two parameter space are explored. From the outcome of the present mathematical analysis carried out followed by the numerical simulations based on the model parameters, it reveals that for unequal diffusive coefficients, prey harvesting may induce that diffusion-driven instability resulting in stationary Turing patterns. The choice of parameter values is important to study the effect of prey harvesting and diffusion, while it depends more on the non-linearity of the model system. Moreover, the model dynamics exhibits the influence of both prey harvesting and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, labyrinthine, stripes-spots mixture and spots replication. All these features illustrate that the dynamics of the proposed model with the control of prey harvesting is not straightforward, but rich and complex in nature.

In this paper, we study the existence, uniqueness and stability of solutions for fractional pantograph equations with complex order. The Krasnoselkii’s fixed point theorem and Banach contraction principle are used to obtain the desired results.

The variational iteration method (VIM) is a analysis tool efficient for approximate non-linear fractional differential equations. Recently differents investigators are used this method in your works and we study the Lagrange multipliers of the variational iteration method for the time fractional Burgers equation and apply those in differents particular cases. In this conference we present approximations of the solutions for a particular case of the time fractional Burgers equation (BF), with the use of the variational iteration method, the Caputo derivate for 0 <α≤ 1, after make an comparation with the Adomian descomposition method (ADM).

We illustrate that, like the truncation error, the round-off error has a significant influence on the reliability of numerical simulations of chaotic dynamic systems. Due to the butterfly-effect, all numerical approaches in double precision cannot give a reliable simulation of chaotic dynamic systems. So, in order to avoid man-made uncertainty of numerical simulations of chaos, we had to greatly decrease both of the truncation and round-off error to a small enough level, plus a verification of solution reliability by means of an additional computation using even smaller truncation and round-off errors.

The dynamics and spectrum of vibrations of non-uniform Euler-Bernoulli beam were investigated. The beam cross section varies linearly in direction of its length. Spacial and temporal solutions were investigated, spacial by Differential transform Method and temporal by four order Runge Kutta. A comparison between temporal numerical solutions and generalized Landau analytical approximations was conducted. There is a significant region, in parameter spaces, that the generalized Landau solution works well, and that there is a region, in parameter space, where the system behaves as it was periodic for practical reasons, thus we name it as almost stable attractor. It was shown that the solutions are strongly sensible to parameters that are related to geometrical and physical proprieties of the beam, it means that a small deviation on parameters can change the dynamics from convergent to divergent, and between those regimes, there is a region on parameter space that the model imitates a periodic system.