Dynamics of chaotic and hyperchaotic modified nonlinear Schrödinger equations and their compound synchronization

Abstract

We present in this paper four versions of chaotic and hyperchaotic modified nonlinear Schr"{o}dinger equations (MNSEs). These versions are hyperchaotic integer order, hyperchaotic commensurate fractional order, chaotic non-commensurate fractional order, and chaotic distributed order MNSEs. These models are regarded as extensions of previous models found in literature. We also studied their dynamics which include symmetry, stability, chaotic and hyperchaotic solutions. The sufficient condition is stated as a theorem to study the existence and uniqueness of the solutions of hyperchaotic integer order MNSE. We state and prove another theorem to test the dependence of the solution of hyperchaotic integer order MNSE on initial conditions. By similar way, we can introduce the previous two theorems for the other versions of MNSEs. The Runge–Kutta of the order 4, the Predictor-Corrector and the modified spectral numerical methods are used to evaluate the numerical solutions for integer, fractional and distributed orders MNSEs, respectively. We calculate numerically using the Lyapunov exponents the intervals of parameters of the purposed models at which hyperchaotic, chaotic and stable solutions are exist. The MNSEs have an important role in many fields of science and technology, such as nonlinear optics, electromagnetic theory, superconductivity, chemical and biological dynamics, lasers and plasmas. The compound synchronization for these chaotic and hyperchaotic models is investigated. We state its scheme using the tracking control technique among three integer commensurate and non-commensurate orders as the derive models and one distributed order as a slave model. We presented and proved a theorem that provides us with the analytical formula for the control functions which are required to achieve compound synchronization. The analytical results are supported by numerical calculations and agreement is found.

The design, circuit realization and applications of chaotic system with offset-boosting and multistability

“…Since 1963, the research branches of chaotic systems have increased significantly, such as multi-vortex systems [1], conservative systems [2], fractional-order systems [3], Jerk systems [4], and neural networks [5], etc. At the same time, chaotic systems have been widely applied in areas such as image encryption [6], weak signal detection [7], and chaotic synchronization secure communication [8].…”