Multistability Emergence through Fractional-Order-Derivatives in a PWL Multi-Scroll System

Abstract

In this paper, the emergence of multistable behavior through the use of fractional-order-derivatives in a Piece-Wise Linear (PWL) multi-scroll generator is presented. Using the integration-order as a bifurcation parameter, the stability in the system is modified in such a form that produces a basin of attraction segmentation, creating many stable states as scrolls are generated in the integer-order system. The results here presented reproduce the same phenomenon reported in systems with integer-order derivatives, where the multistable regimen is obtained through a parameter variation. The multistable behavior reported is also validated through electronic simulation. The presented results are not only applicable in engineering fields, but they also enrich the analysis and the understanding of the implications of using fractional integration orders, boosting the development of further and better studies.

Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems

“…To attain the complete range of the signal's linear transformations, the offset boosting can be set together with the so-called amplitude control. It appeared that a novel boosting controller, which was introduced by [20], can destroy the symmetry of the variableboostable system [36,37]. In this section, we introduce three additional controlled constants η, ω, and in accordance with the variables x, y, and z, respectively.…”

Active Realization of Fractional‐Order Integrators and Their Application in Multiscroll Chaotic Systems

“…In this section, we design electronic circuits based on the building blocks of Figure 2 and the fractional integrators of Figure 3 to realize commensurate fractional-order Lü chaotic system, i.e., system with equal fractional order for integrators. A commensurate realization is acceptable to test the functionality and benefits of using the active fractionalorder integrators of Figure 3, but incommensurate fractional-order oscillators, as those reported in [45][46][47] or [48] can also be realized with the proposed approach without increasing hardware requirements. In fact, the proposed approach is an attractive alternative for the design of incommensurate fractional-order oscillators because it is necessary to calculate the parameter A using (17) instead of designing ladder networks.…”

“…In particular, the multiscroll chaotic attractors present plenty of complex topological structures contrary to single-or double-scroll attractors. One of the most typical applications is secure communications, e.g., in random number generators, cryptosystems in wireless networks, and image encryption [21][22][23].…”

Predicting Tipping Points in Chaotic Maps with Period‐Doubling Bifurcations

“…Various dynamics of the Bogdanov map were investigated in [24]. Multistability is an interesting behavior of dynamical systems [15,25,26]. Multistability is a condition in which the system's attractor is dependent on the initial values [27].…”