Compound-Combination Synchronization for Fractional Hyperchaotic Models with Different Orders

Mahmoud, Gamal M.; Themairi, A. Al; Abed‐Elhameed, Tarek M.; Farghaly, Ahmed Sabry

doi:10.3390/sym15020279

Cited by 3 publications

(2 citation statements)

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“…For different dimensions, the combination synchronization among one integer and two fractional orders models was investigated [34]. The compound-combination synchronization between three master and two slave fractional orders models with different orders was presented by Mahmoud et al [35]. We extended the pervious work to introduce the compound synchronization among three integer, commensurate and non-commensurate orders MNSEs as the master models and distributed MNSE model as a slave one.…”

Section: Introductionmentioning

confidence: 99%

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

Abed-Elhameed,

Otefy,

Mahmoud

2024

Phys. Scr.

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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.

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Section: Introductionmentioning

confidence: 99%

“…There have only been a few researches on the synchronization of different orders in literature (e.g. [33][34][35]). Yang et al [33] studied the complete synchronization between one integer and one fractional orders models.…”

Section: Introductionmentioning

confidence: 99%