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The deadly outbreak of the second wave of Covid-19, especially in worst hit lower-middle-income countries like India, and the drastic rise of another growing epidemic of Mucormycosis, call for an efficient mathematical tool to model pandemics, analyse their course of outbreak and help in adopting quicker control strategies to converge to an infection-free equilibrium. This review paper on prominent pandemics reveals that their dispersion is chaotic in nature having long-range memory effects and features which the existing integer-order models fail to capture. This paper thus puts forward the use of fractional-order (FO) chaos theory that has memory capacity and hereditary properties, as a potential tool to model the pandemics with more accuracy and closeness to their real physical dynamics. We investigate eight FO models of Bombay plague, Cancer and Covid-19 pandemics through phase portraits, time series, Lyapunov exponents and bifurcation analysis. FO controllers (FOCs) on the concepts of fuzzy logic, adaptive sliding mode and active backstepping control are designed to stabilise chaos. Also, FOCs based on adaptive sliding mode and active backstepping synchronisation are designed to synchronise a chaotic epidemic with a non-chaotic one, to mitigate the unpredictability due to chaos during transmission. It is found that severity and complexity of the models increase as the memory fades, indicating that FO can be used as a crucial parameter to analyse the progression of a pandemic. To sum it up, this paper will help researchers to have an overview of using fractional calculus in modelling pandemics more precisely and also to approximate, choose, stabilise and synchronise the chaos control parameter that will eliminate the extreme sensitivity and irregularity of the models.

We explore an oscillator with nonlinear functions and equilibrium lines that displays chaos. The equilibrium stability and complexity of the oscillator have been analysed and investigated. The presence of multiple equilibrium lines sets it apart from previously reported oscillators. The synchronization of the oscillator is considered as an application for secure communications. An observer is designed by considering a transmitted signal as a state, in other words, by injecting a linear function satisfying Lipschitz's condition to the proposed oscillator. Moreover, the adaptive control of the new oscillator is obtained. [ABSTRACT FROM AUTHOR]

The use of the advancements in memristor technology to construct chaotic maps has garnered significant research attention in recent years. The combination of memristors and nonlinear terms provides an effective approach to proposing novel maps. In this study, we have leveraged memristors and sine terms to develop three-dimensional maps, capable of processing special fixed points. Additionally, we have conducted an in depth study of a specific example (TDMM 1 map) to demonstrate its dynamics, feasibility, and application for lightweight encryption. Notably, our general approach could be extended to develop higher-dimensional maps, including four- and five-dimensional ones, thereby opening up the possibility to create numerous higher-dimensional maps. [ABSTRACT FROM AUTHOR]

In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system (Jia, 2007) with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system (Jia, 2007) has the Lyapunov exponents L1 = 0.3684,L2 = 0.2174,L3 = 0 and L4 =−12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as DKY =3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L1 = 0.4195,L2 = 0.2430,L3 = 0.0145,L4 = 0 and L5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as DKY =4.0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE), which is greater than the maximal Lyapunov exponent (MLE) of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.

In this research work, a six-term 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel jerk system are obtained as L1 = 0.07765,L2 = 0, and L3 = −0.87912. The Kaplan-Yorke dimension of the novel jerk system is obtained as DKY = 2.08833. Next, an adaptive backstepping controller is designed to stabilize the novel jerk chaotic system with two unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve complete chaos synchronization of the identical novel jerk chaotic systems with two unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model

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]

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 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]

In the last decade, researchers, who work in the field of nonlinear circuits, have the "dream" to use a real memristor, which is the only nonlinear fundamental circuit element, in a new or other reported nonlinear circuit in literature, in order to experimentally investigate chaos. With this intention, for the first time, a well-known nonlinear circuit, in which its nonlinear element has been replaced with a commercially available memristor (KNOWM memristor), is presented in this work. Interesting phenomena concerning chaos theory, such as period-doubling route to chaos, coexisting attractors, one-scroll and double-scroll chaotic attractors are experimentally observed. [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]

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]

The memristor, a specialized circuit element, has found recent applications in various areas. The nonlinearity of memristors can be harnessed to construct memristive maps, capable of generating complex behavior. In this paper, we present a model of a memristive map that incorporates two memristors, two amplifiers, and a control component. The proposed model allows for adjustments not only in the number of fixed points but also in the symmetry of the memristor map. As a result, the map exhibits diverse features, including chaos, a plane of fixed points, absence of fixed points, as well as symmetry and asymmetry, which can be controlled through modifications to the control component. To facilitate quick implementation and testing of the introduced map, we employed a cost-effective microcontroller board. The experimental results demonstrate the feasibility of the map and its potential for utilization in chaos-based applications. Furthermore, the proposed model opens the door to the creation of novel memristive maps. • A simple approach to construct two-memristor based maps is proposed. • Complex dynamics of a two-memristor based map is discovered. • The map exhibits diverse features. • Hardware implementation is presented. [ABSTRACT FROM AUTHOR]

This paper deals with a new modified hyperchaotic van der Pol–Duffing (MVPD) snap oscillator. Various dynamical properties of the proposed system are investigated with the help of Lyapunov exponents, stability analysis of the equilibrium points and bifurcation plots. The existence of the Hopf bifurcation is established by analyzing the corresponding characteristic equation. It is also proved that the MVPD oscillator shows multistability with coexisting attractors. Various numerical simulations are conducted and presented to show the dynamical behavior of the MVPD system. To show that the system is hardware realizable, we derive the discrete model of the MVPD system using the Euler's method and using the hardware–software cosimulation, the proposed MVPD system is implemented in Field Programmable Gate Arrays. It is shown that the output of the digital implementations of the MVPD systems matches the numerical analysis. [ABSTRACT FROM AUTHOR]

We propose a modified Fitzhugh-Nagumo neuron (MFNN) model. Based on this model, an integer-order MFNN system (case A) and a fractional-order MFNN system (case B) were investigated. In the presence of electromagnetic induction and radiation, memductance and induction can show a variety of distributions. Fractional-order magnetic flux can then be considered. Indeed, a fractional-order setting can be acceptable for non-uniform diffusion. In the case of an MFNN system with integer-order discontinuous magnetic flux, the system has chaotic and non-chaotic attractors. Dynamical analysis of the system shows the birth and death of period doubling, which is a sign of antimonotonicity. Such a behavior has not been studied previously in the dynamics of neurons. In an MFNN system with fractional-order discontinuous magnetic flux, different attractors such as chaotic and periodic attractors can be observed. However, there is no sign of antimonotonicity. [ABSTRACT FROM AUTHOR]

Chaotic systems without equilibrium are of interest because they are the systems with hidden attractors. A nonequilibrium system with chaos is introduced in this work. Chaotic behavior of the system is verified by phase portraits, Lyapunov exponents, and entropy. We have implemented a real electronic circuit of the system and reported experimental results. By using this new chaotic system, we have constructed S-boxes which are applied to propose a novel image encryption algorithm. In the designed encryption algorithm, three S-boxes with strong cryptographic properties are used for the sub-byte operation. Particularly, the S-box for the sub-byte process is selected randomly. In addition, performance analyses of S-boxes and security analyses of the encryption processes have been presented. [ABSTRACT FROM AUTHOR]

Designing new chaotic system with specific features is an interesting field in nonlinear dynamics. In this paper, some new chaotic systems with cyclic symmetry are proposed. In order to understand the overall behavior of such systems, the dynamical analyses such as stability analysis, bifurcation and Lyapunov exponent analysis are done. The accurate examination of bifurcation plot represents that these systems are multistable which makes them more interesting. Also, the basin of attraction of these systems is investigated to detect the type of attractors of these systems which are self-excited. Finally, the circuit implementation is carried out to show their feasibility. [ABSTRACT FROM AUTHOR]

Systems with many equilibrium points have attracted considerable interest recently. A chaotic system with a line equilibrium has been studied in this work. The system has infinite equilibria and exhibits coexisting chaotic attractors. The system with an infinite number of equilibria has been realized by an electronic circuit, which confirms the feasibility of the system. Based on such a system, we have developed a new S-Box generation algorithm. With the developed algorithm, two new S-Boxes are produced. Performance tests of S-Boxes are performed. The tests have shown that proposed S-Boxes have good performance results. [ABSTRACT FROM AUTHOR]

This paper investigates the F – M synchronization between non-identical fractional-order systems characterized by different dimensions and different orders. F – M synchronization combines the inverse generalized synchronization with the matrix projective synchronization. In particular, the proposed approach enables the F – M synchronization to be achieved between an n-dimensional master system and an m-dimensional slave system. The developed approach is applied to chaotic and hyperchaotic fractional systems with the aim of illustrating its applicability and suitability. A multiple-input multiple-output (MIMO) secure communication system is also developed by using the F – M synchronization and verified through computer simulations. [ABSTRACT FROM AUTHOR]

Discovering chaotic systems with interesting features has been of interest in the recent years. One such important and interesting feature is the type and shape of equilibrium points. We introduce a class of chaotic systems which could show different types of infinite equilibrium points. The fundamental properties of the proposed systems like bifurcation diagram and Lyapunov exponents are investigated. An electronic circuit of the presented chaotic systems is implemented. In addition, a chaos-based communication application by the differential chaos shift keying method with the new chaotic system is designed and tested for engineering application. According to the design test results, the proposed chaosbased communication system is successful. Therefore, the new chaotic system can be used in chaos-based communication systems. [ABSTRACT FROM AUTHOR]

A novel 4-D chaotic hyperjerk system with four quadratic nonlinearities is presented in this work. It is interesting that the hyperjerk system has no equilibrium. A chaotic attractor is said to be a hidden attractor when its basin of attraction has no intersection with small neighborhoods of equilibrium points of the system. Thus, our new non-equilibrium hyperjerk system possesses a hidden attractor. Chaos in the system has been observed in phase portraits and verified by positive Lyapunov exponents. Adaptive backstepping controller is designed for the global chaos control of the non-equilibrium hyperjerk system with a hidden attractor. An electronic circuit for realizing the non-equilibrium hyperjerk system is also introduced, which validates the theoretical chaotic model of the hyperjerk system with a hidden chaotic attractor. [ABSTRACT FROM AUTHOR]

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium. [ABSTRACT FROM AUTHOR]

There has been an increasing interest in discovering no-equilibrium chaotic systems recently. In this paper, a novel three dimensional fractional-order chaotic system, which has no equilibrium, is introduced. Dynamics of the system has been studied. It is interesting that the system can exhibit coexisting chaotic attractors for the order as low as 2.7. The adjustable feature of a variable is studied by introducing a single controlled constant. Circuit implementation of the system is proposed to show its feasibility. In addition, we have designed the controllers to investigate coexisting synchronization types of such a new fractional-order system. Numerical examples have verified the proposed synchronization schemes. [ABSTRACT FROM AUTHOR]