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This paper introduces a modified Morris–Lecar neuron model that incorporates a memristor with a ReLU-based activation function. The impact of the memristor on the dynamics of the ML neuron model is analyzed using bifurcation diagrams and Lyapunov exponents. The findings reveal chaotic behavior within specific parameter ranges, while increased magnetic strength tends to maintain periodic dynamics. The emergence of various firing patterns, including periodic and chaotic spiking as well as square-wave and triangle-wave bursting is also evident. The modified model also demonstrates multistability across certain parameter ranges. Additionally, the dynamics of a network of these modified models are explored. This study shows that synchronization depends on the strength of the magnetic flux, with synchronization occurring at lower coupling strengths as the magnetic flux increases. The network patterns also reveal the formation of different chimera states, such as traveling and non-stationary chimera states. [ABSTRACT FROM AUTHOR]

A memristor is a two-terminal passive electronic device that exhibits memory of resistance. It is essentially a resistor with memory, hence the name "memristor". The unique property of memristors makes them useful in a wide range of applications, such as memory storage, neuromorphic computing, reconfigurable logic circuits, and especially chaotic systems. Fixed point-free maps or maps without fixed points, which are different from normal maps due to the absence of fixed points, have been explored recently. This work proposes an approach to build fixed point-free maps by connecting a cosine term and a memristor. Four new fixed point-free maps displaying chaos are reported to illustrate this approach. The dynamics of the proposed maps are verified by iterative plots, bifurcation diagram, and Lyapunov exponents. Because such chaotic maps are highly sensitive to the initial conditions and parameter variations, they are suitable for developing novel lightweight random number generators. [ABSTRACT FROM AUTHOR]

The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention in the past years. In this paper, we propose a new 2D fractional map with the simplest algebraic structure reported to date and with an infinite line of equilibrium. The conceived map possesses an interesting property not explored in literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of periodic, chaotic and hyper-chaotic attractors. Bifurcation diagrams, computation of the maximum Lyapunov exponents, phase plots and 0–1 test are reported, with the aim to analyse the dynamics of the 2D fractional map as well as to highlight the coexistence of initial-boosting chaotic and hyperchaotic attractors in commensurate and incommensurate order. Results show that the 2D fractional map has an infinite number of coexistence symmetrical chaotic and hyper-chaotic attractors. Finally, the complexity of the fractional-order map is investigated in detail via approximate entropy. [ABSTRACT FROM AUTHOR]

Synchronization in coupled oscillators is of high importance in secure communication and information processing. Due to this reason, a significant number of studies have been performed to investigate the synchronization state in coupled circuits. Diffusive coupling is the simplest connection between the oscillators, which can be implemented through a variable resistor between two variables of two circuits. The Chua's circuit is the most famous chaotic circuit whose dynamics have been investigated in many studies. However, Wang–Zhang–Bao (WZB) is another chaotic circuit that can exhibit exciting behaviors such as bistability. Thus, this study aims to investigate the cooperative dynamics of the WZB circuit in its elegant parameter values. To this issue, first, we explored the dynamic behavior of the elegant WZB circuit using the bifurcation diagrams, the Lyapunov exponents, and the basins of attraction. Based on the results, we found the range of the bifurcation parameter and the initial conditions wherein the system is bistable. Subsequently, setting the parameters in the monostable region, we studied the synchronization state of two diffusively coupled WZB circuits analytically and numerically. Consequently, we used master stability functions and temporally averaged synchronization error as the analytical and numerical tools to explore the synchronization state. Then we numerically examined the synchronization state in a network of 100 nonlocally coupled WZB oscillators. As a result, we found imperfect chimera and phase synchronization in the studied network before getting synchronized. [ABSTRACT FROM AUTHOR]

Recent works have focused the analysis of chaotic phenomena in fractional discrete memristor. However, most of the papers have been related to simulated results on the system dynamics rather than on their hardware implementations. This work reports the implementation of a new chaotic fractional memristor map with "hidden attractors". The fractional memristor map is developed based on a memristive map by using the Grunwald–Letnikov difference operator. The fractional memristor map has flexible fixed points depending on a system's parameters. We study system dynamics for different values of the fractional orders by using bifurcation diagrams, phase portraits, Lyapunov exponents, and the 0–1 test. We see that the fractional map generates rich dynamical behavior, including coexisting hidden dynamics and initial offset boosting. [ABSTRACT FROM AUTHOR]

Chaotic maps have simple structures but can display complex behavior. In this paper, we apply discrete memristance and a nonlinear term in order to design new memristive maps. A general model for constructing memristive maps has been presented, in which a memristor is connected in serial with a nonlinear term. By using this general model, different memristive maps have been built. Such memristive maps process special fixed points (infinite and without fixed point). A typical memristive map has been studied as an example via fixed points, bifurcation diagram, symmetry, and coexisting iterative plots. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear dynamics of the biological system modeled by the fractional incommensurate order Van der Pol equations are investigated. The stability of the proposed fractional non-autonomous system is analyzed by varying both the fractional order derivative and system parameters. Moreover, very interesting phenomena such as symmetry, multi-stability and coexistence of attractors are discovered in the considered biological system. Numerical simulations are performed by considering the Caputo fractional derivative and results are reported by means of bifurcation diagrams, computation of the largest Lyapunov exponent, phase portraits in 2D and 3D projections. [ABSTRACT FROM AUTHOR]

It is almost more than a year that earth has faced a severe worldwide problem called COVID-19. In December 2019, the origin of the epidemic was found in China. After that, this contagious virus was reported almost all over the world with different variants. Besides all the healthcare system attempts, quarantine, and vaccination, it is needed to study the dynamical behavior of this disease specifically. One of the practical tools that may help scientists analyze the dynamical behavior of epidemic disease is mathematical models. Accordingly, here, a novel mathematical system is introduced. Also, the complex behavior of this model is investigated considering different dynamical analyses. The results represent that some range of parameters may lead the model to chaotic behavior. Moreover, comparing the two same bifurcation diagrams with different initial conditions reveals that the model has multi-stability. [ABSTRACT FROM AUTHOR]

In recent years, some efforts have been devoted to nonlinear dynamics of fractional discrete-time systems. A number of papers have so far discussed results related to the presence of chaos in fractional maps. However, less results have been published to date regarding the presence of hyperchaos in fractional discrete-time systems. This paper aims to bridge the gap by introducing a new three-dimensional fractional map that shows, for the first time, complex hyperchaotic behaviors. A detailed analysis of the map dynamics is conducted via computation of Lyapunov exponents, bifurcation diagrams, phase portraits, approximated entropy and C 0 complexity. Simulation results confirm the effectiveness of the approach illustrated herein. [ABSTRACT FROM AUTHOR]

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with "self-excited attractors." This paper makes a contribution to the topic of fractional-order discrete-time systems with "hidden attractors" by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system. [ABSTRACT FROM AUTHOR]

Investigating dynamic properties of discrete chaotic systems with fractional order has been receiving much attention recently. This paper provides a contribution to the topic by presenting a novel version of the fractional Grassi–Miller map, along with improved schemes for controlling and synchronizing its dynamics. By exploiting the Caputo h-difference operator, at first, the chaotic dynamics of the map are analyzed via bifurcation diagrams and phase plots. Then, a novel theorem is proved in order to stabilize the dynamics of the map at the origin by linear control laws. Additionally, two chaotic fractional Grassi–Miller maps are synchronized via linear controllers by utilizing a novel theorem based on a suitable Lyapunov function. Finally, simulation results are reported to show the effectiveness of the approach developed herein. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of two fractional-order chaotic maps based on two standard chaotic maps with sine terms. The dynamic behavior of this map is analyzed using numerical tools such as phase plots, bifurcation diagrams, Lyapunov exponents and 0–1 test. With the change of fractional-order, it is shown that the proposed fractional maps exhibit a range of different dynamical behaviors including coexisting attractors. The existence of coexistence attractors is depicted by plotting bifurcation diagram for two symmetrical initial conditions. In addition, three control schemes are introduced. The first two controllers stabilize the states of the proposed maps and ensure their convergence to zero asymptotically whereas the last synchronizes a pair of non-identical fractional maps. Numerical results are used to verify the findings. [ABSTRACT FROM AUTHOR]

In this paper, we are particularly interested in the fractional form of the Hénon-Lozi type map. Using discrete fractional calculus, we show that the general behavior of the proposed fractional order map depends on the fractional order. The dynamical properties of the new generalized map are investigated by applying numerical tools such as: phase portrait, bifurcation diagram, largest Lyapunov exponent, and 0-1 test. It shows that the fractional order Hénon-Lozi map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. Furthermore, a one-dimensional control law is proposed to stabilize the states of the fractional order map. Numerical results are presented to illustrate the findings. [ABSTRACT FROM AUTHOR]

A three-dimensional (3D) Hénon map of fractional order is proposed in this paper. The dynamics of the suggested map are numerically illustrated for different fractional orders using phase plots and bifurcation diagrams. Lorenz-like attractors for the considered map are realized. Then, using the linear fractional-order systems stability criterion, a controller is proposed to globally stabilize the fractional-order Hénon map. Furthermore, synchronization control scheme has been designed to exhibit a synchronization behavior between a given 2D fractional-order chaotic map and the 3D fractional-order Hénon map. Numerical simulations are also performed to verify the main results of the study. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics and control of a Caputo fractional difference form of the Duffing map. We use phase plots, bifurcation diagrams, and Lyapunov exponents to establish the existence of chaos over a wide range of fractional orders and examine the nature of the dynamics. Also, we present the 0–1 test to detect chaos and C 0 complexity, which is an alternative nonlinear statistical measure that can quantify the regularity of a time series. In addition, we measure the approximate entropy to see the performance of our numerical results. Through phase plots and bifurcation diagrams, it is shown that the proposed fractional map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. A one-dimensional feedback stabilization controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Simulation results have been carried out to illustrate the findings of the study. [ABSTRACT FROM AUTHOR]

Chaotic systems with no equilibrium are a very important topic in nonlinear dynamics. In this paper, a new fractional order discrete-time system with no equilibrium is proposed, and the complex dynamical behaviors of such a system are discussed numerically by means of a bifurcation diagram, the largest Lyapunov exponents, a phase portrait, and a 0–1 test. In addition, a one-dimensional controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Next, a synchronization control scheme for two different fractional order discrete-time systems with hidden attractors is reported, where the master system is a two-dimensional system that has been reported in the literature. Numerical results are presented to confirm the results. [ABSTRACT FROM AUTHOR]

Investigating new chaotic flows has been a hot topic for many years. Studying the chaotic attractors of systems with various properties illuminates a lamp to reveal the vague of the generation of chaotic attractors. A new chaotic system in the spherical coordinates is proposed in this paper. The system's solution is inside a predefined sphere, and its attractor cannot cross the sphere. Investigation of equilibrium points of the system shows that the system has eight equilibria, and all of them are saddle. Bifurcation analysis of the system depicts the period-doubling route to chaos with changing the bifurcation parameter. Also, Lyapunov exponents in the studied interval of the bifurcation parameter are discussed. The basin of attraction of the system is investigated to show the sensitivity of the system to initial conditions. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of a new fractional-order map with no fixed points. Through phase plots, bifurcation diagrams, largest Lyapunov exponent, it is shown that the proposed fractional map exhibit chaotic and periodic behavior. New Hidden chaotic attractors are observed, and transient state is found to exist. Complexity of the new map is also analyzed by employing approximate entropy. Results, show that the fractional map without fixed point have high complexity for certain fractional order. In addition, a control scheme is introduced. The controllers stabilize the states of the fractional map and ensure their convergence to zero asymptotically. Numerical results are used to verify the findings. [ABSTRACT FROM AUTHOR]

Multistability is an essential topic in nonlinear dynamics. Recently, two critical subsets of multistable systems have been introduced: systems with extreme multistability and systems with megastability. In this paper, based on a newly introduced megastable system, a megastable forced oscillator is introduced. The effect of adding a forcing term and its parameters on the dynamical behavior of the designed system is investigated. By the help of bifurcation diagram and Lyapunov exponents, it is shown that the modified oscillator can show a variety of dynamical solutions including limit cycle, torus, and strange attractor. [ABSTRACT FROM AUTHOR]

This paper proposes a new chaotic system with a specific attractor which is bounded in a sphere. The system is offered in the spherical coordinate. Dynamical properties of the system are investigated in this paper. The system shows multistability, and all of its attractors are inside or on the surface of the specific sphere. Bifurcation diagram of the system displays an inverse period-doubling route to chaos. Lyapunov exponents of the system are studied to show its chaotic attractors and predict its bifurcation points. [ABSTRACT FROM AUTHOR]

Recently, chaotic systems with hidden attractors and multistability have been of great interest in the field of chaos and nonlinear dynamics. Two special categories of systems with multistability are systems with extreme multistability and systems with megastability. In this paper, the simplest (yet) megastable chaotic oscillator is designed and introduced. Dynamical properties of this new system are completely investigated through tools like bifurcation diagram, Lyapunov exponents, and basin of attraction. It is shown that between its countable infinite coexisting attractors, only one is self-excited and the rest are hidden. [ABSTRACT FROM AUTHOR]

In this paper, an autonomous Toda jerk oscillator is proposed and analysed. The autonomous Toda jerk oscillator is obtained by converting an autonomous two-dimensional Toda oscillator with an exponential nonlinear term to a jerk oscillator. The existence of Hopf bifurcation is established during the stability analysis of the unique equilibrium point. For a suitable choice of the parameters, the proposed autonomous Toda jerk oscillator can generate antimonotonicity, periodic oscillations, chaotic oscillations and bubbles. By introducing two additional parameters in the proposed autonomous Toda jerk oscillator, it is possible to control partially or totally the amplitude of its signals. In addition, electronic circuit realization of the proposed Toda jerk oscillator is carried out to confirm results found during numerical simulations. The commensurate fractional-order version of the proposed autonomous chaotic Toda jerk oscillator is studied using the stability theorem of fractional-order oscillators and numerical simulations. It is found that periodic oscillations and chaos exist in the fractional-order form of the proposed Toda jerk oscillator with order less than three. Finally, combination synchronization of two fractional-order proposed autonomous chaotic Toda jerk oscillators with another fractional-order proposed autonomous chaotic Toda jerk oscillator is analysed using the nonlinear feedback control method. [ABSTRACT FROM AUTHOR]

In this paper, by modifying a known two-dimensional oscillator, we obtain an interesting new oscillator with coexisting limit cycles and point attractors. Then by changing this new system to its forced version and choosing a proper set of parameters, we introduce a chaotic system with some very interesting features. In this system, not only can we see the coexistence of different types of attractors, but also a fascinating phenomenon: some initial conditions can escape from the gravity of nearby attractors and travel far away before being trapped in an attractor beyond the usual access. [ABSTRACT FROM AUTHOR]

For the past 4 years, there has been a rapid rise in the study of chaotic systems with curves of equilibria which are categorized as systems with hidden attractors. There is still significant controversy surrounding the shapes of equilibrium points. This paper presents a new three-dimensional autonomous chaotic system with hyperbolic sine equilibrium. Fundamental dynamical properties and complex dynamics of the system have been discovered by using equilibrium analysis, phase portrait, Poincaré map, bifurcation diagram and Lyapunov spectrum. It is crucial to note that there are bistable hidden chaotic attractors in the introduced system. Furthermore, in order to show the feasibility of the new system with hyperbolic sine equilibrium, its electronic circuit has been implemented. [ABSTRACT FROM AUTHOR]

This paper investigates a three-dimensional autonomous chaotic flow without linear terms. Dynamical behaviour of the proposed system is investigated through eigenvalue structures, phase portraits, bifurcation diagram, Lyapunov exponents and basin of attraction. For a suitable choice of the parameters, the proposed system can exhibit anti-monotonicity, periodic oscillations and doublescroll chaotic attractor. Basin of attraction of the proposed system shows that the chaotic attractor is self-excited. Furthermore, feasibility of double-scroll chaotic attractor in the real word is investigated by using the OrCAD-PSpice software via an electronic implementation of the proposed system. A good qualitative agreement is illustrated between the numerical simulations and the OrCAD-PSpice results. Finally, a finite-time control method based on dynamic sliding surface for the synchronisation of master and slave chaotic systems in the presence of external disturbances is performed. Using the suggested control technique, the superior master–slave synchronisation is attained. Illustrative simulation results on the studied chaotic system are presented to indicate the effectiveness of the suggested scheme. [ABSTRACT FROM AUTHOR]

In this work a memristive circuit consisting of a first-order memristive diode bridge is presented. The proposed circuit is the simplest memristive circuit containing the specific circuitry realization of the memristor to be so far presented in the literature. Characterization of the proposed circuit confirms its complex dynamic behavior, which is studied by using well-known numerical tools of nonlinear theory, such as bifurcation diagram, Lyapunov exponents and phase portraits. Various dynamical phenomena concerning chaos theory, such as antimonotonicity, which is observed for the first time in this type of memristive circuits, crisis phenomenon and multiple attractors, have been observed. An electronic circuit to reproduce the proposed memristive circuit was designed, and experiments were conducted to verify the results obtained from the numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear dynamics of permanent magnet (PM) DC motor drive with proportional (P) controller have been investigated. The drive system shows different dynamical behaviors; periodic, quasi-periodic, and chaotic behaviors, and those are characterized by using bifurcation diagram, phase portrait, and time series. The stability analysis of period-1 behavior is studied by using Filippov's method, the analytic results show good agreement with simulation ones. Then, the stabilization of chaos to fixed point is carried out by using the sliding mode control (SMC). In addition, experimentally the nonlinear dynamics and the proposed stabilization method to PM DC motor drive system have been achieved by using a microcontroller. For the first time, it is noted that when the system is in chaotic dynamics, the vibration of the motor is increased approximately 400% compared with the system in periodic dynamical behavior. [ABSTRACT FROM AUTHOR]

For the past five years there has been much attention to systems with uncountable equilibria. This work introduces a new system with an infinite number of equilibrium points. It is worth noting that equilibrium points, which located on a heart-shaped curve, lead the presence of hidden attractors in the system. By using phase portraits, bifurcation diagram, maximal Lyapunov exponents, dynamics of the system have been investigated. Moreover, an electronic implementation is proposed to illustrate the feasibility of the theoretical system. [ABSTRACT FROM AUTHOR]

In this note, hidden attractors in chaotic maps are investigated. Although there are many new researches on hidden attractors in chaotic flows, no investigation has been done on hidden attractors in maps based on our knowledge. In addition, a new interesting chaotic map with a bifurcation diagram starting from any desired period and then continuing with period doubling is introduced in this paper. [ABSTRACT FROM AUTHOR]

Discovering dynamical systems with hidden attractors has become an attractive topic recently. A novel four-dimensional continuous-time autonomous system with a cubic nonlinear term is introduced in this work. It is worth noting that this no-equilibrium system can generate hidden hyperchaotic attractors. The fundamental properties of such systems are investigated by means of equilibrium points, phase portrait, bifurcation diagram and Lyapunov exponents. In addition, an adaptive scheme has been presented to synchronize two such identical hyperchaotic systems. Moreover, an electronic circuit is also designed and implemented to verify the feasibility of the theoretical model. [ABSTRACT FROM AUTHOR]

A novel four-dimensional continuous-time autonomous hyperchaotic system which has no equilibrium is proposed in this paper. By starting from a third-order chaotic system and introducing a further variable performing state feedback, a four-dimensional system exhibiting hyperchaos is obtained. The basic dynamical properties of this system are investigated, such as equilibria and stability, Lyapunov exponent spectrum, and bifurcation diagrams. Furthermore, synchronization via diffusive coupling or control has been addressed. In the latter, parameter identification and synchronization are performed simultaneously. The circuit realization and experimental results are also presented. [ABSTRACT FROM AUTHOR]

This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincaré map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology. [ABSTRACT FROM AUTHOR]

Oscillations and oscillators appear in various fields and find applications in numerous areas. We present an oscillator with infinite equilibria in this work. The oscillator includes only nonlinear elements (quadratic, absolute, and cubic ones). It is different from common oscillators, in which there are linear elements. Special features of the oscillator are suitable for secure applications. The oscillator's dynamics have been discovered via simulations and an electronic circuit. Chaotic attractors, bifurcation diagrams, Lyapunov exponents, and the boosting feature are presented while measurements of the implemented oscillator are reported by using an oscilloscope. We introduce a random number generator using such an oscillator, which is applied in biomedical image encryption. Moreover, the security and performance analysis are considered to confirm the correctness of encryption and decryption processes. [ABSTRACT FROM AUTHOR]

Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. Results show that the dynamics of the proposed fractional maps are sensitive to both initial conditions and fractional order. There are coexisting attractors which have been displayed in terms of bifurcation diagrams, phase portraits and a 0-1 test. Furthermore, control schemes are introduced to stabilize the chaotic trajectories of the two novel systems. [ABSTRACT FROM AUTHOR]

Chaotic systems have attracted considerable attention and been applied in various applications. Investigating simple systems and counterexamples with chaotic behaviors is still an important topic. The purpose of this work was to study a simple symmetrical system including only five nonlinear terms. We discovered the system's rich behavior such as chaos through phase portraits, bifurcation diagrams, Lyapunov exponents, and entropy. Interestingly, multi-stability was observed when changing system's initial conditions. Chaos of such a system was predicted by applying a machine learning approach based on a neural network. [ABSTRACT FROM AUTHOR]

Fractional calculus in discrete-time systems is a recent research topic. The fractional maps introduced in the literature often display chaotic attractors belonging to the class of "self-excited attractors". The field of fractional map with "hidden attractors" is completely unexplored. Based on these considerations, this paper presents the first example of fractional map without equilibria showing a number of hidden attractors for different values of the fractional order. The presence of the chaotic hidden attractors is validated via the computation of bifurcation diagrams, maximum Lyapunov exponent, 0–1 test, phase diagrams, complexity, and entropy. Finally, an active controller with the aim for stabilizing the proposed fractional map is successfully designed. [ABSTRACT FROM AUTHOR]

In this work a novel chaotic system with a line equilibrium is presented. First, a dynamical analysis on the system is performed, by computing its bifurcation diagram, continuation diagram, phase portraits and Lyapunov exponents. Then, the system is applied to the problem of secure communication. We assume that the transmitted signal is an additional state. For this reason, the nonlinear system is rewritten in a rectangular descriptor form and then an observer is constructed for achieving synchronization and input reconstruction. If we assume some rank conditions (on the nonlinearities and the solvability of a linear matrix inequality (LMI)) on the system matrices then the observer synchronization can be feasible. We evaluate and demonstrate our approach with specific numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose three fractional chaotic maps based on the well known 3D Stefanski, Rössler, and Wang maps. The dynamics of the proposed fractional maps are investigated experimentally by means of phase portraits, bifurcation diagrams, and Lyapunov exponents. In addition, three control laws are introduced for these fractional maps and the convergence of the controlled states towards zero is guaranteed by means of the stability theory of linear fractional discrete systems. Furthermore, a combined synchronization scheme is introduced whereby the fractional Rössler map is considered as a drive system with the response system being a combination of the remaining two maps. Numerical results are presented throughout the paper to illustrate the findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional form of a new three-dimensional generalized Hénon map and study the existence of chaos and its control. Using bifurcation diagrams, phase portraits and Lyapunov exponents, we show that the general behavior of the proposed fractional map depends on the fractional order. We also present two control schemes for the proposed map, one that adaptively stabilizes the fractional map, and another to achieve the synchronization of the proposed fractional map. [ABSTRACT FROM AUTHOR]

A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents' diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map. [ABSTRACT FROM AUTHOR]

Today, four-dimensional chaotic systems are attracting considerable attention because of their special characteristics. This paper presents a non-equilibrium four-dimensional chaotic system with hidden attractors and investigates its dynamical behavior using a bifurcation diagram, as well as three well-known entropy measures, such as approximate entropy, sample entropy, and Fuzzy entropy. In order to stabilize the proposed chaotic system, an adaptive radial-basis function neural network (RBF-NN)–based control method is proposed to represent the model of the uncertain nonlinear dynamics of the system. The Lyapunov direct method-based stability analysis of the proposed approach guarantees that all of the closed-loop signals are semi-globally uniformly ultimately bounded. Also, adaptive learning laws are proposed to tune the weight coefficients of the RBF-NN. The proposed adaptive control approach requires neither the prior information about the uncertain dynamics nor the parameters value of the considered system. Results of simulation validate the performance of the proposed control method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method. [ABSTRACT FROM AUTHOR]

Most of the papers published so far on fractional discrete systems are related to theoretical results on their chaotic behaviors or to applications based on the mathematical modeling of the chaotic phenomena. No paper has been published to date regarding the hardware implementation of hyperchaotic fractional maps. This manuscript makes a contribution to the topic by presenting the first example of hardware implementation of hyperchaotic fractional maps. In particular, by exploiting the Grunwald–Letnikov difference operator, the paper introduces a new version of the fractional Grassi–Miller map. The system dynamics are analyzed via bifurcation diagrams and Lyapunov exponents, showing that the conceived map is hyperchaotic when the fractional order μ belongs to the interval [ 0. 966 , 1 ]. Finally, a hardware implementation of the fractional map is illustrated, with the aim to concretely highlight the presence of hyperchaos in physical systems described by fractional difference equations. Arduino, an open-source platform, has been used to illustrate the simplicity as well as the feasibility of the implementation. • A new version of the fractional Grassi–Miller map is introduced. • We discover system's dynamics by using phase portraits, bifurcation diagrams, and Lyapunov exponents. • Hyperchaotic behaviors are investigated. • The hardware implementation of the fractional map is reported. [ABSTRACT FROM AUTHOR]

[Display omitted] • This article proposes a new robust nonlinear controller that stabilizes a chaotic finance system in a finite-time without cancellation of the spacecraft's nonlinear terms, it improves the efficiency of the closed-loop. • It accomplishes an oscillation-free faster convergence of the perturbed state variables to the desired steady-state. • The proposed controller is insensitive to the parameter uncertainties of the nonlinear terms and exogenous disturbances. • The paper performs a comparative study to verify the performance and efficiency of the proposed controller. Robust, stable financial systems significantly improve the growth of an economic system. The stabilization of financial systems poses the following challenges. The state variables' trajectories (i) lie outside the basin of attraction, (ii) have high oscillations, and (iii) converge to the equilibrium state slowly. This paper aims to design a controller that develops a robust, stable financial closed-loop system to address the challenges above by (i) attracting all state variables to the origin, (ii) reducing the oscillations, and (iii) increasing the gradient of the convergence. This paper proposes a detailed mathematical analysis of the steady-state stability, dissipative characteristics, the Lyapunov exponents, bifurcation phenomena, and Poincare maps of chaotic financial dynamic systems. The proposed controller does not cancel the nonlinear terms appearing in the closed-loop. This structure is robust to the smoothly varying system parameters and improves closed-loop efficiency. Further, the controller eradicates the effects of inevitable exogenous disturbances and accomplishes a faster, oscillation-free convergence of the perturbed state variables to the desired steady-state within a finite time. The Lyapunov stability analysis proves the closed-loop global stability. The paper also discusses finite-time stability analysis and describes the controller parameters' effects on the convergence rates. Computer-based simulations endorse the theoretical findings, and the comparative study highlights the benefits. Theoretical analysis proofs and computer simulation results verify that the proposed controller compels the state trajectories, including trajectories outside the basin of attraction, to the origin within finite time without oscillations while being faster than the other controllers discussed in the comparative study section. This article proposes a novel robust, nonlinear finite-time controller for the robust stabilization of the chaotic finance model. It provides an in-depth analysis based on the Lyapunov stability theory and computer simulation results to verify the robust convergence of the state variables to the origin. [ABSTRACT FROM AUTHOR]

• In this paper, we present a new fractional-order hyperchaotic memristor oscillator. • The proposed system has been investigated through numerical simulations. • A novel robust adaptive controller is implemented to synchronize the system. • As an application, the system has been implemented for voice encryption. The present study proposes a new fractional-order hyperchaotic memristor oscillator. The proposed system is studied through numerical simulations and analyses, such as the Lyapunov exponents, bifurcation diagrams, and phase portraits. Then, using the sliding mode concept, a robust adaptive control scheme is designed to synchronize the proposed system. The adaptation mechanism is implemented to estimate the unknown parameters of the slave system. Then, the output of the proposed adaptation mechanism is used for the control scheme. The stability of the closed-loop system is proven via a fractional version of the Lyapunov stability theorem and Barbalat's lemma. Numerical simulations of synchronization are shown to investigate the performance of the developed control technique on the uncertain fractional-order hyperchaotic memristor oscillator. Finally, as an engineering application, the proposed fractional-order system is implemented for voice encryption. In this regard, to show the appropriate performance of the proposed system for voice encryption, statistical characteristic of the encryption and decryption processes are performed through different methods including correlation, entropy, root mean square, and root sum of squares. [ABSTRACT FROM AUTHOR]