Constructing multiwing attractors from a robust chaotic system with non-hyperbolic equilibrium points

We investigate a three-dimensional (3D) robust chaotic system which only holds two nonhyperbolic equilibrium points, and finds the complex dynamical behaviour of position modulation beyond amplitude modulation. To extend the application of this chaotic system, we initiate a novel methodology to construct multiwing chaotic attractors by modifying the position and amplitude parameters. Moreover, the signal amplitude, range and distance of the generated multiwings can be easily adjusted by using the control parameters, which enable us to enhance the potential application in chaotic cryptography and secure communication. The effectiveness of the theoretical analyses is confirmed by numerical simulations. Particularly, the multiwing attractor is physically realized by using DSP (digital signal processor) chip

Un attracteur étrange 3D versatile à six ailes

To create additional wings to a given strange attractor, several methods based on the heteroclinic loop or switching controls for example are applied, but complicate the approach and require the extension of the system to one or more other dimensions of the phase space. This deflects us from the objectives of research on low-dimensional chaotic systems. Remaining in this narrow area of 3D phase spaces to invent multi-wing attractors constitutes the main scope of the present paper. Indeed, we present a rapid investigation of a very simple autonomous 3D system of firts-order differential equations with a rich variety of phase portraits. This new intentionally constructed model exhibits double, four-or even six-wing strange attractors. We point out that under the influence of the scalar parameters, such versatile chaotic attractors are obtained. A similar sequence was likewise observed for the periodic behaviors. Besides, both chaotic or regular featured trajectories are found to be in bilateral agreement even when the morphology of the portrait changes. Obviously, we present the basic attributes of the system and its bifurcation diagram. Eventually, we emphasize that the study of the relationship between the written differential equations and the observed characteristics of attractors remains undervalued.En vue de susciter l"apparition d"ailes supplémentaires pour un attracteur étrange donné, plusieurs méthodes basées sur la boucle hétéroclinique ou une commande de commutation, par exemple, sont appliquées. Toutefois, elles compliquent l'approche et nécessitent l'extension du système vers une ou plusieurs autres dimensions de l'espace des phases. Cela nous détourne des objectifs même de la recherche sur les systèmes chaotiques à faible dimension. C'est ainsi que rester dans cette zone étroite d'espaces de phase 3D pour inventer des attracteurs à ailes multiples constitue le principal sujet du présent article. En effet, nous présentons une étude rapide d’un système 3D très simple et autonome d’équations différentielles du premier ordre avec une grande variété de portraits de phase. Ce nouveau modèle construit intentionnellement présente des attracteurs étranges à deux, quatre voire même six ailes. Nous soulignons que sous l’influence des seuls paramètres scalaires, de tels attracteurs chaotiques versatiles sont obtenus. Une séquence similaire a également été observée pour les comportements périodiques. En outre, les trajectoires chaotiques ou régulières sont en accord bilatéral même lorsque la morphologie du portrait de phase change. Nous présentons évidemment les attributs de base du système et son diagramme de bifurcation. Finalement, nous soulignons que l’étude de la relation entre la formulation des équations différentielles et les caractéristiques observées des attracteurs demeure marginale dans la littérature scientifique

Grid multi-wing butterfly chaotic attractors generated from a new 3-D quadratic autonomous system

Due to the dynamic characteristics of the Lorenz system, multi-wing chaotic systems are still confined in the positive half-space and fail to break the threshold limit. In this paper, a new approach for generating complex grid multi-wing attractors that can break the threshold limit via a novel nonlinear modulating function is proposed from the firstly proposed double-wing chaotic system. The proposed method is different from that of classical multi-scroll chaotic attractors generated by odd-symmetric multi-segment linear functions from Chua system. The new system is autonomous and can generate various grid multi-wing butterfly chaotic attractors without requiring any external forcing, it also can produce grid multi-wing both on the xz-plane and yz-plane. Basic properties of the new system such as dissipation property, equilibrium, stability, the Lyapunov exponent spectrum and bifurcation diagram are introduced by numerical simulation, theoretical analysis and circuit experiment, which confirm that the multi-wing attractors chaotic system has more rich and complicated chaotic dynamics. Finally, a novel module-based unified circuit is designed which provides some principles and guidelines for future circuitry design and engineering application. The circuit experimental results are consistent with the numerical simulation results.&nbsp

Analysis of a class of complex system without equilibria via switched control law

"In this article we introduce a new class of complex system without equilibria which exhibits a chaotic multiscroll attractor in each node. The number of scrolls in the attractor is determined by a switched control law to allow the operation of different linear affine systems. Thus, the system is composed of many subsystems which interact with each other to generate a multiscroll attractor. This new class of piecewise linear (PWL) system presents no positive real part in the eigenvalues of the Jacobian matrix as opposed to the reported systems with multiscrolls. The scrolls present a complex behavior since these don’t unwrap and don’t appear close to an unstable manifold of a saddle-focus equilibrium point. A particular case is taken as case study and simulation plots of the attractor are provided.

A universal variable extension method for designing multi-scroll/wing chaotic systems

© 2023 IEEE. This is the accepted manuscript version of an article which has been published in final form at https://doi.org/10.1109/TIE.2023.3299020Developing a universal design method to construct different multiscroll/wing chaotic systems (MS/WCSs) has been challenging. This article proposes a general design method for MS $/$ WCSs called the universal variable extension method (UVEM). It is a simple but effective approach that generates one-direction (1-D) and 2-D multiscroll/wing chaotic attractors. Using any double-scroll/wing chaotic system as the basic system, the UVEM is able to construct different MS/WCSs. Employing Chua's chaotic system and Lorenz chaotic system as two examples, we construct two MSCSs (including 1-D and 2-D) and two MWCSs (including 1-D and 2-D), respectively. Theoretical analysis and numerical simulation show that the constructed MS/WCSs not only can generate 1-D and 2-D multiscroll/wing chaotic attractors but also have 1-D and 2-D initial boosting behaviors. This means that the MS/WCSs designed by the UVEM are very sensitive to their initial states, and have better unpredictability and more complex chaotic behaviors. To show the simplicity of UVEM in hardware implementation, we develop a field-programmable gate array-based digital hardware platform to implement the designed MS $/$ WCSs. Finally, a new pseudorandom number generator is proposed to investigate the application of the MS/WCSs. All P-values obtained by the NIST SP800-22 test are larger than 0.01, which indicates that the MS/WCSs designed by UVEM have high randomness.Peer reviewe

Grid Multiscroll Hyperchaotic Attractors Based on Colpitts Oscillator Mode with Controllable Grid Gradient and Scroll Numbers

AbstractThrough introducing two piecewise-linear triangular wave functions in a three-dimensional spiral chaotic Colpitts oscillator model, a four-dimensional grid multiscroll hyperchaotic system is constructed. Interestingly, by adjusting a build-in parameter in a variable of one triangle wave function, the control of the gradient of the multiscroll grid is achieved. Whereas by deploying the zero points of the two triangular wave functions to extend the saddle-focus equilibrium points with index-2 in phase space the scroll numbers do not only increase along with the number of turning points, but they can also generate arbitrary multiples of products. The basic dynamical behaviors of the proposed four-dimensional multiscroll hyperchaotic system are analyzed. Finally, the hardware experimental circuit is designed and the interrelated circuit implementation is realized. The experimental results are in agreement with both theoretical analyses and numerical simulations, which verify the feasibility of the design methods

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

Diversified Butterfly Attractors of Memristive HNN With Two Memristive Systems and Application in IoMT for Privacy Protection

© 2024, IEEE. This is an open access accepted manuscript distributed under the terms of the Creative Commons Attribution License (CC BY), https://creativecommons.org/licenses/by/4.0/Memristors are often used to emulate neural synapses or to describe electromagnetic induction effects in neural networks. However, when these two things occur in one neuron concurrently, what dynamical behaviors could be generated in the neural network? Up to now, it has not been comprehensively studied in the literature. To this end, this paper constructs a new memristive Hopfield neural network (HNN) by simultaneously introducing two memristors into one Hopfield-type neuron, in which one memristor is employed to mimic an autapse of the neuron and the other memristor is utilized to describe the electromagnetic induction effect. Dynamical behaviors related to the two memristive systems are investigated. Research results show that the constructed memristive HNN can generate Lorenz-like double-wing and four-wing butterfly attractors by changing the parameters of the first memristive system. Under the simultaneous influence of the two memristive systems, the memristive HNN can generate complex multi-butterfly chaotic attractors including multi-double-wing-butterfly attractors and multi-four-wing-butterfly attractors, and the number of butterflies contained in an attractor can be freely controlled by adjusting the control parameter of the second memristive system. Moreover, by switching the initial state of the second memristive system, the multi-butterfly memristive HNN exhibits initial-boosted coexisting double-wing and four-wing butterfly attractors. Undoubtedly, such diversified butterfly attractors make the proposed memristive HNN more suitable for chaos-based engineering applications. Finally, based on the multi-butterfly memristive HNN, a novel privacy protection scheme in the IoMT is designed. Its effectiveness is demonstrated through encryption tests and hardware experiments.Peer reviewe