Bifurcation analysis of predator-prey systems with constant rate harvesting using non-standard discretization

We formulate and apply non-standard discretization methods that enable us to study the saddle, elliptic and parabolic cases of the predator-prey system with constant rate harvesting as difference dynamical systems. Our models have the same qualitative features as their corresponding continuous models. By choosing appropriate bifurcation parameters, we combine analytical and numerical investigations to produce interesting global bifurcation diagrams, including saddle-node, Hopf and Bogdanov-Takens bifurcations.     

On analytical justification of phase synchronization in different chaotic systems

In analytical or numerical synchronizations studies of coupled chaotic systems the phase synchronizations have less considered in the leading literatures. This article is an attempt to find a sufficient analytical condition for stability of phase synchronization in some coupled chaotic systems. The method of nonlinear feedback function and the scheme of matrix measure have been used to justify this analytical stability, and tested numerically for the existence of the phase synchronization in some coupled chaotic systems.     

Dynamical analysis of public health education on HIV/AIDS transmission

An Human Immunodeficiency Virus/Acquired Immuno‐Deficiency Syndrome (HIV/AIDS) epidemic model for sexual transmission with asymptomatic and symptomatic phase is proposed as a system of differential equations. The threshold and steady state for the model are determined and stabilities of disease free steady state is investigated. We use the model and study the effect of public health education on the spread of HIV/AIDS as a single‐strategy in HIV prevention. The education, including basic reproduction number for the model with public health education, is compared with the basic reproduction number for the HIV/AIDS in the absence of public health education. By comparing these two values, influence of public health education appears. According to property of , threshold proportion of educated adolescents, education rate for susceptible individuals and education efficacy is obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015     

Numerical stability of chaotic synchronization using a nonlinear coupling function

Nonlinear coupling has been used to synchronize some chaotic systems. The difference evolutional equation between coupled systems, determined via the linear approximation, can be used to analyze the stability of the synchronization between drive and response systems. According to the stability criteria the coupled chaotic systems are asymptotically synchronized, if all eigenvalues of the matrix found in this linear approximation have negative real parts. There is no synchronization, if at least one of these eigenvalues has positive real part. Nevertheless, in this paper we have considered some cases on which there is at least one zero eigenvalue for the matrix in the linear approximation. Such cases demonstrate synchronization-like behavior between coupled chaotic systems if all other eigenvalues have negative real parts.     

Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations

The aim of this article is to present an analytical approximation solution for linear and nonlinear multi-order fractional differential equations (FDEs) by extending the application of the shifted Chebyshev operational matrix. For this purpose, we convert FDE into a counterpart system and then using proposed method to solve the resultant system. Our results in solving four different linear and nonlinear FDE, confirm the accuracy of proposed method.     

Phase synchronization in fractional differential chaotic systems

We propose an analytical justification for phase synchronization of fractional differential equations. This justification is based upon a linear stability criterion for fractional differential equations. We then investigate the existence of phase synchronization in chaotic forced Duffing and Sprott-L fractional differential systems of equations. Our numerical results agree with those analytical justifications.