A Distributed-Order Fractional Hyperchaotic Detuned Laser Model: Dynamics, Multistability and Dual Combination Synchronization

The aim of this article is to introduce the distributed-order hyperchaotic detuned (DOHD) laser model. Its dissipative dynamics, invariance, fixed points (FPs) and their stability are investigated. Numerical solutions of the DOHD laser model are computed using the modified Predictor-Corrector approach. Its viscoelasticity is described by the so-called DO derivative, allowing for the study of different technical systems and materials, and the model is found to have a whole circle of FPs as a hyperchaotic attractor. We discuss the coexistence of more attractors under various initial conditions and the same sets of parameters for our model (multistability). We also introduce the notion of dual combination synchronization (DCS), using four integer-order drive models and two DO response models. A theorem is stated and proved to obtain an analytical control function that ensures DCS for our models. Numerical simulations are presented to support these analytical results. Regarding the use of the well--known Caputo derivative, the results are very similar to those of DO, except when the Caputo order, 0< σ1, is very close to $1$, where the dynamics shows a "spiralling behavior" towards a fixed point. In all other cases, both Caputo and DO exhibit a very similar behavior.