A 2-D Discrete Cubic Chaotic Mapping with Symmetry

In the theoretical research of chaotic dynamical system, the different type of bifurcations is a very interesting powerful tool for analyzing the qualitative behavior of chaotic dynamical system; this short paper is devoted to analysis of a simple 2-D symmetry discrete chaotic map with quadratic and cubic nonlinearities. The dynamical behaviors of the map are investigated by mathematical analysis and simulated numerically using package of Matlab . We compute numerically the bifurcation diagram and largest Lyapunov exponent and phase portraits. The research results indicate that there are interesting nonlinear physical phenomena in this simple 2-D symmetry discrete cubic map, such as symmetry bifurcation, Hopf bifurcation, symmetry breaking bifurcation and identical symmetric attractors. The important nonlinear physical phenomena obtained in this paper would benefit the study of the cubic chaotic map and the development of the theory of chaotic discrete dynamical systems. En la investigación teórica de los sistemas dinámicos caóticos, los diferentes tipos de bifurcaciones son una herramienta poderosa muy interesante para analizar el comportamiento cualitativo de los sistemas dinámicos caóticos; este breve artículo está dedicado al análisis de un mapa caótico discreto de simetría bidimensional simple con no linealidades cuadráticas y cúbicas. Los comportamientos dinámicos del mapa se investigan mediante análisis matemático y se simulan numéricamente utilizando el paquete de Matlab . Calculamos numéricamente el diagrama de bifurcación y el mayor exponente de Lyapunov y los retratos de fase. Los resultados de la investigación indican que existen interesantes fenómenos físicos no lineales en este sencillo mapa cúbico discreto de simetría 2-D, como la bifurcación de simetría, la bifurcación de Hopf, la bifurcación de ruptura de simetría y los atractores simétricos idénticos. Los importantes fenómenos físicos no lineales obtenidos en este trabajo beneficiarían el estudio del mapa cúbico caótico y el desarrollo de la teoría de los sistemas dinámicos discretos caóticos.

Multiple explicit solutions of the 2D variable coefficients Chafee–Infante model via a generalized expansion method

In this work, we do apply a generalized expansion method to the realistic two-dimensional (2D) Chafee–Infante model with time-variable coefficients which is encountered in physical sciences.The key ideas of this method consist in: (i) to choose a nonlinear wave variable in respect to time-variable into the general finite series solution of the governing model; (ii) to take a full advantage from the general elliptic equation introduced as an auxiliary equation which can degenerate into sub-equations such as Riccati equation, the Jacobian elliptic equations, the generalized Riccati equation. Based upon this powerful technique, we successfully construct for the first time several types of non-autonomous solitary waves as well as some non-autonomous triangular solutions, rational or doubly periodic type ones. We investigate the propagation of non-autonomous solitons and we emphasize as well upon the influence of the variable coefficients. We are providing and analyzing a few graphical representations of some specific solutions. The results of this paper will be valuable in the study of nonlinear physical phenomena. The above- mentioned method could be employed to solve other partial differential equations with variable coefficients which describe various complicated natural phenomena.

An Analytical Technique Implemented in the Fractional Clannish Random Walker’s Parabolic Equation with Nonlinear Physical Phenomena

In this paper, the adapted (G’/G)-expansion scheme is executed to obtain exact solutions to the fractional Clannish Random Walker’s Parabolic (FCRWP) equation. Some innovative results of the FCRWP equation are gained via the scheme. A diverse variety of exact outcomes are obtained. The proposed procedure could also be used to acquire exact solutions for other nonlinear fractional mathematical models (NLFMMs).

Analysis of optical solitons solutions of two nonlinear models using analytical technique

<abstract><p>Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.</p></abstract>

Closed-form wave structures of the space-time fractional Hirota–Satsuma coupled KdV equation with nonlinear physical phenomena

The present paper applies the variation of (G^{\prime} /G)-expansion method on the space-time fractional Hirota–Satsuma coupled KdV equation with applications in physics. We employ the new approach to receive some closed form wave solutions for any nonlinear fractional ordinary differential equations. First, the fractional derivatives in this research are manifested in terms of Riemann–Liouville derivative. A complex fractional transformation is applied to transform the fractional-order ordinary and partial differential equation into the integer order ordinary differential equation. The reduced equations are then solved by the method. Some novel and more comprehensive solutions of these equations are successfully constructed. Besides, the intended approach is simplistic, conventional, and able to significantly reduce the size of computational work associated with other existing methods.

An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

Numerical solutions of the generalized Rosenau–Kawahara-RLW equation arising in fluid mechanics via B-spline collocation method

Present study reports the solution of generalized Rosenau–Kawahara-RLW equation. It includes motion of single solitary wave, interaction of two solitary waves along with the calculated invariants and error norms. Gaussian and undular bore initial conditions are studied to show evolution of solitons. Developed train of solitons and conservation of invariants are shown via figures and tables in the respective sections. Various case studies are presented to demonstrate the efficiency of the proposed numerical scheme. Solutions so produced may be helpful for explaining various nonlinear physical phenomena in nonlinear dynamical systems.