Vol. 9, No. 1 (2020): Regular Issue

Of concern in the present theoretical study is to carry out the complex dynamics of a reaction-diffusion predator-prey model incorporating constant proportion of prey refuge and nonlinear prey harvesting with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the feasible interior equilibrium point for nonspatial model was established. The conditions of diffusion-driven instability were obtained and the Turing space in the parameters space was given as well. Consequently, we present the evolutionary procedure that occupies organism distribution and their interaction of spatially distributed species with diffusion and locate that the model dynamics reveals a diffusion-controlled formation growth to hole patterns or labyrinthine patterns or hole-stripe patterns replication over the whole spatial domain. The analytical results are then authenticated with the help of numerical simulations. Our results points out that the diffusion has an immense impact on the prey refuge as well as prey harvesting and extend well the findings of spatiotemporal dynamics in the reaction-diffusion model.

We analyze the uniqueness and Ulam stability results for implicit impulsive fractional differential equations connecting nonlocal form of the Hadamard derivative of fractional order ϑ. The main results are studied by using the Banach contraction principle and Ulam stability. The finding of the result evoluted by the example.

In the present work, we present a new method for assessing causality in uni-directionally coupled chaotic oscillators with small frequency mismatch. This method is based on the correlation between changes in the phase dynamics of the slave oscillator and the dynamics of the phase difference between the oscillators. Application of the proposed approach to master-slave R¨ossler systems showed that the new algorithm is well-suited for assessing the presence and direction of coupling, especially in the case of weak coupling.

This paper is concerned with the dynamics of the steady state twodelayed models when the involved functions describe the saturation effect whereby the prey density increases, the predation rate becomes less dependent on the population of the prey and only ependent on the predator population. The stability of the steady state together with its dependence on the magnitude of time delays is examined and by means Hopf-bifurcation theory it is showed that a generator of selfsustaining cycles appear. The normal form is applied to determine the direction of the bifurcation and the stability of the periodic orbits bifurcating from a steady state. This paper is dedicated to Albino and Conceição.

The aim of this work is to study the effect of the fractional-time derivative acting on a fractional version of the Boussinesq system that reads 0 = Dαt u(x, t)+Hx(x, t)+u(x, t)ux(x, t), 0 = Dαt H(x, t)+(u(x, t)H(x, t))x+uxxx(x, t), subject to the initial conditions f (x) = u(x,0), g(x) = H(x,0). Dαt is the Caputo fractional operator with α ∈ (0,1] and f (x), g(x) ∈C∞[ℜ]. To achieve our goal, we solved analytically the proposed model using a new technique called modified residual power series method (RPS). The reliability of RPS technique has been verified using tabular and graphical analysis which reveal the fact when the time-memory index “time-fractional order” is close to zero “full memory”, the solution bifurcate and produce a wave-like pattern, whereas the pattern vanishes when the memory is close to 1 “no memory”.

In this work a four-variable dimensionless model of a quantum dot light emitting diode (QDLED) under optical feedback effect is studied. The bifurcations of these dynamics is fully determined by the increasing the optical feedback strength OFBS. Our results show that small OFBS leads to a selection of QDLED dynamics. Increasing of the τ leads to increase both in region of the double period and the chaos. Otherwise, bias dominates the course of photons. Adding the grating mirror as a special technique stabilize the oscillation of a QDLED. It is worth mentioning that we did not get a perfect behavior except using this technique. Delayed feedback and turn-on dynamics are studied. Results show that there is no change QDLED dynamics because of controlling the bias current on the behavior of photons and the effect of delay forces the system to enter the state of chaos while the turn-on dynamics of the QDLED structure show damping of the relaxation oscillations and the increase of phase shift with increasing both of OFBS and delay time. Dependence the outset of chaos on the linewidth enhancement factor is examined.

The transient proper orthogonal decomposition (TPOD) method is generalized to high-dimensional rotor system models with common faults and the efficiency of the reduced models is discussed in this paper. The method to confirm the optimal reduced rotor model for order reduction is proposed based on the physical significance of the TPOD method. The physical significance of the TPOD method can be provided by the proper orthogonalmode (POM). Three rotor models with faults are established by the Newton’s second law: the first is crack fault, the second is looseness fault and the third is the model with coupling faults. The model with coupling faults contains more complex characteristics than the other two models with single fault (looseness, crack). The TPOD method is applied to obtain the relatively optimal reduced model based on the POM energy. The efficiency of order reduction method is verified via the energy curves of POM and many other dynamical behaviors (the bifurcation diagrams, the amplitude-frequency curves, the phase curves, etc.). The optimal reduced models of the rotor systems can be obtained via applying the TPOD method on the basis of the POM energy.

In this article, the stability analysis and chaos control of Simple chaotic system have been discussed. A new lemma based on Caputo derivative is used during the stability analysis of the fractional order Simple chaotic system through Lyapunov stability theory. During hybrid projective synchronization the Simple chaotic system is considered as drive system and Lu chaotic system is taken as response system. Nonlinear control method has been used to analyse the hybrid projective synchronization of fractional systems. For numerical simulations Adams-Bashforth-Moulton method has been used and results obtained are presented graphically.

Symmetry analysis is performed on a coupled Jaulent-Miodek system, which arises in many branches of physics such as particle physics and fluid dynamics. The similarity reductions and new exact solutions are constructed. Subsequently, conservation laws are derived using the multiplier approach.

The structural response of a slender beam subjected to impact by several rigid particles is investigated. A higher-order shear deformation beam model and the von-Karman geometric measure are employed. The equations of motion, continuity conditions at the points of impact by particles, and geometric and natural boundary conditions for the beam are derived by means of Hamilton’s principle. The generalized differential quadrature and Newmark time integration methods are utilized to carry out the numerical solution of the system of nonlinear partial differential equations comprised of the equations of motion and stress resultants. The durations of contact, displacements, and stress resultants are obtained for beams with various types of boundary conditions.

In this paper, we present a geometric approach to form the governing equations of motion of dynamic systems. A geometric form of the d’Alembert-Lagrange equation on the configuration manifold is first developed and extended to a principal bundle. We can then obtain the explicit form of equations of motion by explicating the geometric form in a coordinate neighborhood on the principal bundle. This approach conveniently permits the choice of quantities to be used which best describe configurations, motions or constraints, and it yields equations of motion in concise forms. Examples are presented to illustrate the use and effectiveness of the approach. The objective of this paper is to provide a general geometric perspective of the governing equations of motion, and explain its suitability for studying complex dynamic systems subject to non-holonomic constraints.

This article deals with the non-commensurate fractional systems represented by a state-space model. It presents the advantages and disadvantages of using this type of model. It focuses on the disadvantages of fractional systems, especially the increase of computing time due to the accumulation of history. A method is proposed to minimize the calculation time, it consists in limiting the use of history of the state variables to a reduced number and taking into account the uncertainty of model in the predictive control. This method will be compared to the classical method in term of performance and calculation time.