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This work proposes and systematically investigates the dynamics of a novel snap system with a single parameterized nonlinearity in the form φ k ( z ) = 0.5 ( exp ( k z ) − exp ( − z ) ) . The form of nonlinearity is physically interesting in the sense that the corresponding circuit realization involves only off-the shelf electronic components such as resistors, semiconductor diodes and operational amplifiers. Parameter k (i.e. a control resistor) serves to smoothly adjust the nonlinearity, and hence the symmetry of the system. In particular, for k = 1 , the nonlinearity is a hyperbolic sine, and thus the system is point symmetry about the origin. For k  ≠ 1, the system is non-symmetric. The fundamental dynamics of the system are investigated in terms of equilibria and stability, phase space trajectory plots, bifurcations diagrams, and graphs of Lyapunov exponents. When monitoring the system parameters, some striking phenomena are found including period doubling bifurcation, reverse bifurcations, merging crises, coexisting bifurcations, hysteresis and offset boosting. Several windows in the parameters space are depicted in which the novel snap system displays a plethora of coexisting attractors (i.e. two, three, four, five or six different attractors) depending solely on the choice of the initial conditions. The magnetization of the state space due to the presence of multiple competing solutions is illustrated by means of basins of attraction. Laboratory experimental results confirm the theoretical predictions. [ABSTRACT FROM AUTHOR]

Four types of scenarios occurring multistability are identified in a vibro-impact system. Besides the coexistence of high-periodic attractors and chaotic attractors, this system has some coexisting multi-frequency quasiperiodical attractors. We observe the coexistence of different two-dimensional tori T 2 , which the mechanism of torus formation involves phase-locked dynamics on three-dimensional tori. We also find that the tori T 3 and the tori T 2 (or period-3) coexist for a certain set of parameters. The mechanism leading to the multistability is investigated through interplay among the codimension two bifurcations, strong resonance bifurcations and global bifurcations. [ABSTRACT FROM AUTHOR]

This paper considers a large class of memristor circuits of arbitrary order, and containing an arbitrary number of flux- or charge-controlled memristors, for which a state equation (SE) description can be obtained. By means of the SEs, it is shown that the state space of each circuit can be decomposed in infinitely many manifolds, and that in the autonomous case, each manifold is positively invariant and is characterized by a different reduced-order dynamics and attractors. These results are the basis for extending the analysis to the non-autonomous case, where time-varying independent sources are present. In particular, the chief result obtained in this paper shows how to analytically design external pulses for programming memristor circuits, i.e., how invariant manifolds and attractors can be changed and controlled by applying suitable charge or flux sources via time-varying voltage and/or current pulses with finite time duration. The main results are obtained by relying on a recently introduced technique for the analysis of memristor circuits in the flux-charge domain. [ABSTRACT FROM AUTHOR]

The study of multi-scroll memristive chaotic system has attracted great interest due to their unique dynamics. In this paper, a novel five-dimensional chaotic system devoid of equilibria is constructed by introducing two memristors into a three-dimensional chaotic system, resulting in the emergence of a grid-scroll attractor. The evolution of the chaotic system is analyzed using techniques such as phase diagrams, Poincare maps, bifurcation diagrams, and Lyapunov exponents. The basin of attraction is delineated to investigate the coexisting attractors in the system. By manipulating the parameters, the partial amplitude of the system can be modulated. To experimentally confirm the existence of the proposed system, a circuit is meticulously designed and implemented. At last, an adaptive controller is designed, and it is observed from the simulation results that a synchronization state can be reached by the two chaotic systems within a finite time. [ABSTRACT FROM AUTHOR]

We consider a 2D piecewise-linear discontinuous map defined on three partitions that drives the dynamics of a stock market model. This model is a modification of our previous model associated with a map defined on two partitions. In the present paper, we add more realistic assumptions with respect to the behavior of sentiment traders. Sentiment traders optimistically buy (pessimistically sell) a certain amount of stocks when the stock market is sufficiently rising (falling); otherwise they are inactive. As a result, the action of the price adjustment is represented by a map defined by three different functions, on three different partitions. This leads, in particular, to families of attracting cycles which are new with respect to those associated with a map defined on two partitions. We illustrate how to detect analytically the periodicity regions of these cycles considering the simplest cases of rotation number 1 / n , n ≥ 3 , and obtaining in explicit form the bifurcation boundaries of the corresponding regions. We show that in the parameter space, these regions form two different overlapping period-adding structures that issue from the center bifurcation line. In particular, each point of this line, associated with a rational rotation number, is an issue point for two different periodicity regions related to attracting cycles with the same rotation number but with different symbolic sequences. Since these regions overlap with each other and with the domain of a locally stable fixed point, a characteristic feature of the map is multistability, which we describe by considering the corresponding basins of attraction. Our results contribute to the development of the bifurcation theory for discontinuous maps, as well as to the understanding of the excessively volatile boom-bust nature of stock markets. • A 2D discontinuous map on three partitions related to a stock market model is considered. • We detect analytically the periodicity regions of attracting cycles with rotation number 1/n, n > 2. • They form two overlapping period-adding structures issuing from the center bifurcation line. • We describe multistability by considering the corresponding basins of attraction. • We contribute to the theory of non-smooth maps and the understanding of stock market volatility. [ABSTRACT FROM AUTHOR]

The concept of self-similarity, widely employed in specialized fields of applied sciences, has been investigated through numerous works utilizing both analytical elements and experimental observations. As a significant component of both fractal and chaos theories, self-similarity can naturally manifest around us or be artificially simulated using mathematical arguments and algorithms. In this paper, we utilize analytical, numerical, and experimental circuit elements to study, analyze, and simulate a generalized chaotic system of attractors with multiple scrolls. This system presents various types of processes, including chaotic and hyperchaotic dynamics, as well as hidden attractors. We discuss the well-posedness of the system before examining its numerical solvability, error tolerance, and simulations for the model involving fractal and fractional operations. Simulations are conducted for various values of the model's parameters, demonstrating that the system displays fractal features combined with chaotic behavior. The system is implemented using a Field Programmable Gate Array (FPGA) board, generating self-similar results akin to those obtained numerically. • A chaotic model of attractors with multiple scrolls is analyzed. • The model presents processes that are chaotic, hyperchaotic & with hidden attractors. • Self-similarity is generated using analytical and numerical methods. • Circuit implementation is performed to recover and verify the self-similarity. [ABSTRACT FROM AUTHOR]

Memristors are widely used to simulate the effects of electromagnetic radiation on neurons or as synapses to simulate excitation and inhibition between neurons. This paper constructs discrete neural network models (DNNMs) in three scenarios: without discrete memristor, discrete memristor simulate electromagnetic radiation stimulation, and discrete local active memristor simulate synapses. Research reveals that the DNNM without fixed points under electromagnetic radiation stimulation, which in turn induces hidden hyperchaotic attractors and also observes the existence of infinite coexisting homogeneous attractors. Meanwhile, the DNNM can generate heterogeneous coexisting attractors after introducing a memristor to mimic synapses. Numerical results show that discrete memristors can induce the DNNM to develop more complex chaotic dynamics. In addition, a hardware platform is designed to validate the numerical results, and the DNNMs are applied to construct the pseudo-random number generator (PRNG). • A local active discrete memristor is proposed and applied to neural network. • Discrete neural network model for three different scenarios are proposed. • The coexisting attractors in discrete neural networks are found and analyzed. • Dynamic analysis, circuit realization and NIST test of the network are given. [ABSTRACT FROM AUTHOR]

Megastable and extreme multistable systems comprise two major new branches of multistable systems. So far, they have been studied separately in various chaotic systems. Nevertheless, to the best of our knowledge, no chaotic system has so far been reported that possesses both types of multistability. This paper introduces the first three-dimensional non-autonomous chaotic system that displays megastability and extreme multistability, jointly called mega-extreme multistability. Our model shows extreme multistability for a variation of an initial condition associated with one system variable and megastability concerning another variable. The different types of coexisting attractors are characterized by the corresponding phase portraits and first return maps, as well as by constructing the appropriate bifurcation diagrams, calculating the Lyapunov spectra, the Kaplan-Yorke dimension and the connecting curves, and by determining the corresponding basins of attraction. The system is explicitly shown to be dissipative, with the dissipation being state-dependent. We demonstrate the feasibility and applicability of our model by designing and simulating an appropriate analog circuit. • Recently, an entirely new realm of dissipative systems that feature infinitely many coexisting attractors was discovered • So far, such systems were deemed to belong to two distinct classes: extreme multistable or megastable systems • We introduce the first system that exhibits both mega and extreme multistability features • Model comprises a non-autonomous three-dimensional system that can present chaotic, torus, and periodic attractors • The model's applicability is demonstrated by designing and simulating an analog circuit with mega-extreme multistability [ABSTRACT FROM AUTHOR]

This letter proposes a new 4D autonomous chaotic system characterized by the abundant coexisting attractors and a simple mathematical description. The new system which is constructed from the Sprott B system is dissipative, symmetric, chaotic and has two unstable equilibria. For a given set of parameters, butterfly attractors are emerged from the system. These butterfly attractors will be broken into a pair of symmetric strange attractors with the variation of the parameters. A variety of coexisting attractors are spotted in the system including six periodic attractors, four periodic attractors with two chaotic attractors, two periodic attractors with three chaotic attractors, two periodic attractors with two chaotic attractors, four periodic attractors, etc. Finally, the system is established via an electronic circuit which can physically confirm the complex dynamics of the system. [ABSTRACT FROM AUTHOR]

This paper presents a new chaotic system that exhibits a two wing (2W) chaotic attractor in its integer order dynamics, three-wing (3W) and four-wing (4W) chaotic attractors in its fractional-order (FO) dynamics, and an eight-wing (8W) attractor in its nonautonomous fractional dynamics. An interesting feature of the proposed system is that two distinct periodic orbits coexist with a strange attractor that gradually evolves into a 4W attractor. The asymmetry, dissimilarity and topological structure of this proposed system with respect to those available in literature, manifest increased irregularity, which in turn indicate more chaos. Besides, the authors have drawn its comparison with various well-known fractional-order chaotic systems (FOCS)s to prove its enhanced features in terms of higher Lyapunov Exponent, fractional order orbital velocities, bandwidth, density, range of dynamical behaviour, etc. A control scheme is proposed to enable switching hybrid synchronisation between the 8W nonautonomous FOCS and the 4W autonomous FOCS, using the former as master and the latter as slave. This work throws light on the potential practical applicability of the proposed system by designing a circuit using minimum circuit components possible, thus signifying the objectives of the paper are finally achieved. [ABSTRACT FROM AUTHOR]

A special type of basins of attraction for high-period coexisting attractors is investigated, which basin boundaries possess multiple nested structures in a driven oscillator. We analyze the global organization of basins and discuss the mechanism for the appearance of layered structures. The unstable periodic orbits and unstable limit cycle are also detected in the oscillator. The basin organization is governed by the ordering of regular saddles and the regular saddle connections are the interrupted by the unstable limit cycle. Wada basin boundary with different Wada number is discovered. Wada basin boundaries for the hidden and rare attractors are also verified. [ABSTRACT FROM AUTHOR]

When both an absolute value function and a cosine function are introduced for chaotic sequence generation, coexisting attractors with different polarities and locations could be extracted by the offset boosting of the initial condition, and thus a hyperchaotic map with distance-increasing coexisting attractors is constructed. It is found that this map has two controllers for rescaling the oscillation totally and partially. The digital circuit implementation shows good consistency with numerical simulation. Moreover, the parameter adaptation in the basic Pelican Optimization Algorithm (POA) is improved by the above-mentioned hyperchaotic map. Correspondingly, the Hyperchaotic Pelican Optimization Algorithm (HCPOA) is constructed, where the hyperchaotic sequences are employed as the sampling pool when random numbers are required in the basic POA. Furthermore, the impact of chaos regulation on the performance of HCPOA is investigated, and the indicators of HCPOA are analyzed by employing five benchmark functions, where the single attractor shows better performance than the double-cavity attractor. The methods of HCPOA proposed in this paper bring the chance to improve the quality of solutions and the ability to escape local optima. • The initially-controlled offset boosting gives birth to various regimes of double-cavity hyperchaotic attractors. And two control knobs are extracted for amplitude rescaling, including total and partial controllers. • An improved HyperChaotic Pelican Optimization Algorithm (HCPOA) is firstly developed by employing the hyperchaotic map, and the performance influenced by the chaotic property is exhaustively explored. • The performance criteria of HCPOA with various chaotic sequences associated with five benchmark functions are compared, indicating that the solution performance of the proposed algorithm can be modified flexibly by different initial conditions and amplitude. [ABSTRACT FROM AUTHOR]

The construction of multi-scroll chaotic systems has been a hot research topic in recent years. In this paper, a four-dimensional multi-scroll chaotic system is established on the basis of a simple three-dimensional chaotic system by adding a flux-controlled non-volatile memristor. After that, a novel system is extended to generate grid-scroll chaotic attractors by replacing the linear term in the multi-scroll chaotic system with a triangular wave function. The evolution of chaos is studied by means of bifurcation diagrams and Lyapunov exponent diagrams. The existence of coexisting attractors is observed, and the attraction basin of coexisting attractors is drawn. In addition, the partial amplitude of the proposed system can be controlled by adjusting the parameters. The results of circuit implementation are consistent with the numerical simulation, verifying the physical existence of the system. Finally, according to the proposed grid-scroll memristive chaotic system, a new image encryption algorithm is designed. It adopts a novel confusion-diffusion structure to scramble and diffuse all pixels of the plain-image. And the security analysis experiments are performed to verify the good encryption performance of the proposed scheme such as information entropy analysis, correlation coefficient analysis, differential attack analysis and robustness analysis. • New grid-scroll chaotic system with non-volatile memristor is constructed. • The generation of grid-scroll chaos and extreme multistability are studied. • New encryption algorithm is given to verify the application values of the system. [ABSTRACT FROM AUTHOR]

In this paper, a novel three-dimensional modified van der Pol-Duffing circuit with a quintic nonlinear resistor is proposed, and the multiple stable attractors are observed. The complex bursting patterns are investigated, various patterns of bursting oscillations are obtained under slowly changing frequency, and the transition mechanism of which can be revealed via the modified slow–fast analysis as well as the transformed phase portrait. It can be found that for the system with mono-stability, only one bursting pattern exists, while for the case with multi-stability, the trajectory can enter into different attraction areas in the transition region, leading to complex forms of bursting oscillations. Furthermore, the multi-valued characteristic of the system are observed by the domain of attraction. • The multiple stable attractors of the modified van der Pol-Duffing circuit are observed. • Various patterns of bursting oscillations are obtained under slowly changing frequency. • The transition mechanism is revealed via the modified slow–fast analysis. • The multi-valued characteristic of the system are observed by the domain of attraction. [ABSTRACT FROM AUTHOR]

The investigation of discrete memristive chaotic systems with complex dynamics has been an interesting and meaningful research work. This paper is devoted to constructing a hyperchaotic system with no fixed point and infinitely many coexisting attractors. The memristive Gaussian map (MGM) is coupled by a sinusoidal discrete memristor with the Gaussian map and induces extreme multistability associated with the initial state due to the import of sinusoidal nonlinearity. Creatively, the iterative number is imported into the system as a variable, which effectively boosts the complexity of its export. The most remarkable feature of the system is a large range of chaos and hyperchaos captured under multiple sets of parameters, and with various coexisting attractors. In addition, digital experiments are designed to verify the feasibility of the hardware implementation, an image encryption algorithm and PRNG are proposed based on the MGM chaotic sequence. • New memristive system with infinite hidden coexisting hyperchaos is presented. • Complex dynamics and digital circuit implementation of the system is given. • New chaos-based image encryption algorithm with high security is designed. [ABSTRACT FROM AUTHOR]