In the present framework, the coupled mathematical model of the atmosphere-ocean system called El Nino-Southern Oscillation (ENSO) is analyzed with the aid Adams-Bashforth numerical scheme. The fundamental aim of the present work is to demonstrate the chaotic behaviour of the coupled fractional-order system. The existence and uniqueness are demonstrated within the frame of the fixed-point hypothesis with the Caputo--Fabrizio fractional operator. Moreover, we captured the chaotic behaviour for the attained results with diverse order. The effect of the perturbation parameter and others associated with the model is captured. The obtained results elucidate that, the present study helps to understand the importance of fractional order and also initial conditions for the nonlinear models to analyze and capture the corresponding consequence of the fractional-order dynamical systems.

In this study, an alternative method has been applied to obtain the new wave solution of mathematical equations used in physics, engineering, and many applied sciences. We argue that this method can be used for some special nonlinear partial differential equations (NPDEs) in which the balancing methods are not integer. A number of new complex hyperbolic trigonometric traveling wave solutions have been successfully generated in the Eckhaus equation (EE) and nonlinear Klein-Gordon (nKG) equation models associated with the Schrödinger equation. The graphs representing the stationary wave are presented by giving specific values to the parameters contained in these solutions. Finally, some discussions about new complex solutions are given. It is discussed by giving physical meaning to the constants in traveling wave solutions, which are physically important as well as mathematically. These discussions are supported by three-dimensional simulation. In order to eliminate the complexity of the process and to save time, computer package programs have been utilized.

In the present paper, we implement a novel numerical method for solving differential equations with fractional variable-order in the Caputo sense to research the dynamics of a circulant Halvorsen system. Control laws are derived analytically to make synchronization of two identical commensurate Halvorsen systems with fractional variable-order time derivatives. The chaotic dynamics of the Halvorsen system with variable-order fractional derivatives are investigated and the identical synchronization between two systems is achieved. Moreover, graph simulations are provided to validate the theoretical analysis.

In the present paper, we implement a novel numerical method for solving differential equations with fractional variable-order in the Caputo sense to research the dynamics of a circulant Halvorsen system. Control laws are derived analytically to make synchronization of two identical commensurate Halvorsen systems with fractional variable-order time derivatives. The chaotic dynamics of the Halvorsen system with variable-order fractional derivatives are investigated and the identical synchronization between two systems is achieved. Moreover, graph simulations are provided to validate the theoretical analysis.