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This paper investigates the dynamical behaviors, experimental validation using a microcontroller technique, and control strategies for complex and coexisting behaviors in a nonsmooth-air-gap brushless direct current motor under the influence of temperature (NAGBDCMT). Steady-state analysis reveals three steady states with unstable branches in the three considered planes. The NAGBDCMT exhibits diverse dynamical structures, including bistable regular and chaotic domains, monostable attractors, periodic and continuous spiking, coexisting chaotic and regular spiking behaviors, various complex characteristics, and coexistence between continuous spiking oscillations and chaotic characteristics. These dynamical structures are corroborated by an ATMEGA2560 microcontroller realization scheme. Control of the complex characteristics and coexisting attractors in the NAGBDCMT is achieved using two configured single controllers and a linear augmentation control scheme, respectively. [ABSTRACT FROM AUTHOR]

Two types of neuron models are constructed in this paper, namely the single discrete memristive synaptic neuron model and the dual discrete memristive synaptic neuron model. Firstly, it is proved that both models have only one unstable equilibrium point. Then, the influence of the coupling strength parameters and neural membrane amplification coefficient of the corresponding system of the two models on the rich dynamical behavior of the systems is analyzed. Research has shown that when the number of discrete local active memristor used as simulation synapses in the system increases from one to two, the coupling strength parameter of the same memristor has significantly different effects on the dynamical behavior of the system within the same range, that is, from a state with periodicity, chaos, and periodicity window to a state with only chaos. In addition, under the influence of coupling strength parameters and neural membrane amplification coefficients, the complexity of the system weakens to varying degrees. Moreover, under the effect of two memristors, the system exhibits a rare and interesting phenomenon where the coupling strength parameter and the neural membrane amplification coefficient can mutually serve as control parameter, resulting in the generation of a remerging Feigenbaum tree. Finally, the pseudo-randomness of the chaotic systems corresponding to the two models are detected by NIST SP800-22, and relevant simulation results are verified on the DSP hardware experimental platform. The discrete memristive synaptic neuron models established in this article provide assistance in studying the relevant working principles of real neurons. [ABSTRACT FROM AUTHOR]

In this work, a four-dimensional fractional order hyperchaotic system with coexisting attractors is excogitated. The stability analysis of the equilibrium point is investigated, and the chaotic and hyperchaotic behavior of the system is well demonstrated numerically by the Lyapunov exponents. By applying delayed feedback control technique, we obtain the existence condition of Hopf bifurcation for the such system. Some numerical simulations are exhibited to substantiate the obtained results in the end, which also displays some interesting dynamical phenomena, such as quasi-periodic, and hyperchaotic. The work reveals that the fractional order and delays have a great significance to controlling the hyperchaotic system. [ABSTRACT FROM AUTHOR]

Copyright of Electronic Components & Materials is the property of Electronic Components & Materials and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we present a novel two-dimensional discrete memristor map that is based on a discrete memristor model and a sine–arcsine one-dimensional map. First, an analysis is conducted on the memristor model to understand its characteristics. Then, the model is coupled with the sine–arcsine one-dimensional map to achieve the two-dimensional discrete memristor map. Our investigation reveals the presence of coexisting attractors and hyperchaotic attractors as the bifurcation parameters vary. Numerical simulations show that the discrete memristors effectively enhance the complexity of chaos in the sine–arcsine map. Furthermore, a digital circuit is designed to experimentally verify the new chaotic system. The research results can enrich the theoretical analysis and circuit implementation of chaos. [ABSTRACT FROM AUTHOR]

We develop a nonlinear duopoly model in which the heuristic expectation formation and learning behavior of two boundedly rational firms may engender complex dynamics. Most importantly, we assume that the firms employ different forecasting models to predict the behavior of their opponent. Moreover, the firms learn by leaning more strongly on forecasting models that yield more precise predictions. An eight-dimensional nonlinear map drives the dynamics of our approach. We analytically derive the conditions under which its unique steady state is locally stable and numerically study its out-of-equilibrium behavior. In doing so, we detect multiple scenarios with coexisting attractors at which the firms' behavior yields distinctively different market outcomes. [ABSTRACT FROM AUTHOR]

In this paper, a cyclic memristive neural network structure is proposed. There are counterclockwise connections between the neurons. The system generates a controllable number of multi-scroll chaos by means of memristors with multi-segment nonlinear functions, which can produce a controllable infinite coexistence of heterogeneous attractors with initial offsets and a large range of amplitude-modulation properties. Through numerical simulations, the phenomenon of multi-scroll chaos is demonstrated and the coexisting attractors are found to exhibit extreme multi-stability as well as parameter-dependent amplitude-modulation properties. In addition, the feasibility of the system is verified by the construction of the circuit platform, the results of the digital hardware experiments are given, and the PRNG is constructed by applying this circular memristor neural network system. [ABSTRACT FROM AUTHOR]

On the basis of the chaotic system proposed by Wang et al. in 2023, this paper constructs a 5D fractional-order memristive hyperchaotic system (FOMHS) with multiple coexisting attractors through coupling of magnetic control memristors and dimension expansion. Firstly, the divergence, Kaplan–Yorke dimension, and equilibrium stability of the chaotic model are studied. Subsequently, we explore the construction of the 5D FOMHS, introducing the definitions of the Caputo differential operator and the Riemann–Liouville integral operator and employing the Adomian resolving approach to decompose the linears, the nonlinears, and the constants of the system. The complex dynamic characteristics of the system are analyzed by phase diagrams, Lyapunov exponent spectra, time-domain diagrams, etc. Finally, the hardware circuit of the proposed 5D FOMHS is performed by FPGA, and its randomness is verified using the NIST tool. [ABSTRACT FROM AUTHOR]

Multiple delays and connection topology are the key parameters for the realistic modeling of networks. This paper discusses the influences of time delays and connection weight on multi-delay artificial neural models with inertial couplings. Firstly, sufficient conditions of some singularities involving static bifurcation, Hopf bifurcation, and pitchfork-Hopf bifurcation are presented by analyzing the transcendental characteristic equation. Secondly, taking self-connection weight and coupling delays as adjusting parameters and utilizing the parameter perturbation with the aid of the non-reduced order technique for the first time, rich dynamics near zero-Hopf interaction are obtained on the plane with self-connected weight and coupling delay as abscissa and ordinate. The multi-delay inertial neural system can exhibit coexisting attractors such as a pair of nontrivial equilibrium points and a periodic orbit with nontrivial equilibrium points. Self-connected weight can affect the number and dynamics of the system equilibrium points, while time delays can contribute to both trivial equilibrium and non-trivial equilibrium losing their stability and generating limit cycles. Simulation plots are displayed with computer software to support the established main results. Compared with the traditional reduced-order method, the used method here is simple and valid with less computation. The key research findings of this paper have significant theoretical guiding value in dominating and optimizing networks. [ABSTRACT FROM AUTHOR]

Purpose: Memristor has attracted extensive research interest and hot discussions in recent years. This paper reports new chaotic system with non-ideal memristor whose internal state is determined by extensible piecewise-linear function, and investigates the application of this system to image encryption. Methods: Its stabilities of equilibria and complex dynamics are studied. With the change of parameter, the coexisting double-scroll attractors will degenerate into coexisting periodic attractors via bifurcation and split into double number of coexisting single-scroll attractors. Results: The system can generate multiscroll attractors with any number of scrolls and are easily broken into any number of coexisting double-scroll attractors. This suggests that the introduction of the memristor can enable chaotic system to exhibit more complex and diverse dynamic behaviors. Conclusions: NIST test and circuit realization are given to verify its physical existence and engineering applicability. In addition, a comprehensive evaluation of the designed efficient chaos-based image encryption algorithm verifies its ability to effectively protect the confidentiality and privacy of image contents. [ABSTRACT FROM AUTHOR]

Leslie matrix models with nonzero nonlinear fecundity elements are under consideration. It is proved that by use of the general Deriso–Schnute recruitment function the supercritical nature of bifurcations in 2- and 3-age class models and a thorough analysis of 1:2 and 1:3 resonance phenomena are also provided. A discussion of impact of coexisting attractors and structures of trapping regions is included as well. Results regarding stabilizing and destabilizing effects as well as dynamical outcomes found in general n-age class models are also presented. Suggestions with respect to where our models may apply are provided too. [ABSTRACT FROM AUTHOR]

Considering that the synaptic connection weight between two neurons in a neural network is time-dependent, a two-neuron model coupled with one synapse is proposed. Meanwhile, a type of memristor is introduced to reveal the electromagnetic induction effect. Complicated dynamics of the addressed system with system parameters changing are explored via phase portraits, time series, bifurcation diagram and Lyapunov exponents. The phenomenon of coexistence of various attractors, chaos transient as well as chaotic crisis is detected. In addition, the analog circuit implementation is used to simulate the dynamics of the addressed system. Some simulations of output voltages and time waveform are captured. The research results of numerical simulation and analog circuits have potential application for revealing the dynamical behaviors of neural network and designing integrated circuits. [ABSTRACT FROM AUTHOR]

This study investigated the Hopf bifurcation of the equilibrium point of chaotic systems with coexisting attractors under the time-delayed feedback control. First, the equilibrium point and Hopf bifurcation of chaotic systems with coexisting attractors were analyzed. Second, the chaotic systems were controlled by time-delayed feedback, the transversality condition of Hopf bifurcation at the equilibrium point was discussed, and the time-delayed value of Hopf bifurcation at the equilibrium point was obtained. Lastly, the correctness of the theoretical analysis was verified by using the numerical results.

Unlike the high-dimensional hyperchaotic system based on a continuous memristor, the low-dimensional map coupled by discrete memristor (DM) and traditional chaotic map can also generate hyperchaos. However, the hyperchaotic map constructed by two DMs has not attracted much attention. To this end, a generalized two-dimensional dual DM-coupled hyperchaotic mapping model is reported in this paper, and four specific maps are provided. The proposed maps have line invariant points, which can be interpreted as allowing arbitrary real values for the initial condition associated with the DM, and the stability is investigated in detail. Furthermore, the coupling strength-dependent and initial condition-dependent complex dynamics of four maps are studied by numerical simulations, and the dynamical performance is evaluated from the perspective of quantitative analysis. It is shown that the considered maps are capable of exhibiting the three characteristic fingerprints of memristors in arbitrary parameter spaces, and this characteristic has gained attention for the first time. In particular, the complete control of the considered maps by variable substitution is performed, which can generate arbitrary switched hyperchaotic behaviors. In addition, four pseudo-random number generators are designed based on the proposed maps, and the randomness is tested by using the NIST SP800-22 software. In general, the proposed maps can not only generate abundant dynamical behaviors, but also enrich the DM circuits and provide a reference for applications based on chaos. Finally, the developed digital hardware circuit implementation platform verifies the results of the numerical method. [ABSTRACT FROM AUTHOR]

In this article we study the behaviors of a piecewise linear map with initial condition in the second quadrant. There is a unique equilibrium point and two 4-cycles of the map. We found regions of initial condition that solutions become equilibrium point or 4-cycles. We divided the second quadrant into sub-regions and identify behaviors of solutions in each sub-region by direct calculations, and formulated inductive statements to explain the behaviors of the map without using stability theorems. [ABSTRACT FROM AUTHOR]

This paper investigates and analyzes the dynamics of the two-dimensional Duffing map. Multistability behavior has been observed from the system numerically. Such behavior, especially the coexistence of chaotic and periodic attractors, is undesirable in the applications of chaos-based cryptography. Therefore, we design and implement a Sine–Cosine chaotification technique to enhance chaos in the multi-stable regions. Furthermore, this paper proposes a new image encryption algorithm to examine the performance of the generalized Duffing map in cryptography applications. Simulation results and security analysis reveal that the proposed algorithm can effectively encrypt and decrypt several image types with a high level of security. [ABSTRACT FROM AUTHOR]

In this work, a novel memristive Chua's oscillator circuit is presented. In the proposed circuit, a linear negative resistor, which is parallel coupled with a first-order memristive diode bridge, is used instead of the well-known Chua's diode. Following this, an extensive theoretical and dynamical analysis of the circuit is conducted. This involves numerical computations of the system's phase portraits, bifurcation diagrams, Lyapunov exponents, and continuation diagrams. A comprehensive comparison is made between the numerical simulations and the circuit's simulations performed in Multisim. The analysis reveals a range of intriguing phenomena, including the route to chaos through a period-doubling sequence, antimonotonicity, and coexisting attractors, all of which are corroborated by the circuit's simulation in Multisim. [ABSTRACT FROM AUTHOR]

Considering the dynamic characteristics of memristors, a new Jerk-like system without an equilibrium point is addressed based on a Jerk-like system, and the hidden dynamics are investigated. When changing system parameter b and fixing other parameters, the proposed system shows various hidden attractors, such as a hidden chaotic attractor (b = 5), a hidden period-1 attractor (b = 3.2), and a hidden period-2 attractor (b = 4). Furthermore, bifurcation analysis suggests that not only parameter b, but also the initial conditions of the system, have an effect on the hidden dynamics of the discussed system. The coexistence of various hidden attractors is explored and different coexistences of hidden attractors can be found for suitable system parameters. Offset boosting of different hidden attractors is discussed. It is observed that offset boosting can occur for hidden chaotic attractor, period-1 attractor, and period-2 attractor, but not for period-3 attractor and period-4 attractor. The antimonotonicity of the proposed system is debated and a full Feigenbaum remerging tree can be detected when system parameters a or b change within a certain range. On account of the complicated dynamics of the proposed system, an image encryption scheme is designed, and its encryption effectiveness is analyzed via simulation and comparison. [ABSTRACT FROM AUTHOR]

Grid multi-scroll attractors are widely studied in the traditional chaotic systems but are rarely appeared in the neural network systems. This paper proposes a novel method for generating the grid multi-scroll attractors based on a memristive Hopfield neural network (HNN). Firstly, the mathematical model of a simple memristive HNN is developed with an original memristor as a connecting synapse, and its equilibrium points and dynamic behaviors are analyzed. Then, a pulse-controlled memristive HNN is constructed when an external multi-level-logic pulse current is applied to one neuron. Theoretical analysis and numerical simulations reveal that an appropriate external pulse current stimulation can stabilize the chaotic HNN by inducing a dynamic transition from chaotic to weakly chaotic and then to periodic behavior. Additionally, by introducing a multi-piecewise memristor into the pulse-controlled memristive HNN, this study demonstrates that the various complex grid multi-scroll attractors can be generated. By setting the different series of multi-level-logic pulse currents and multi-piecewise memristor control parameters, the structure of the grid multi-scroll attractors can be controlled, including multi-double-scroll, multi-three-scroll and multi-four-scroll attractors. Finally, a physical circuit implementing the grid multi-scroll attractors is presented using the basic commercial electronic components. The proposed approach has the potential to be applied in the treatment of neurological diseases. [ABSTRACT FROM AUTHOR]

Purpose: The construction of memristor-based chaotic system with complex dynamics has been a research hotspot in recent years. This paper proposes a new memristive chaotic system characterized by the abundant coexisting attractors. The new system which is established by inserting a memristor is dissipative, symmetric, chaotic and has two line equilibria. Methods: The evolution of chaos and the existence of coexisting attractors are investigated by using bifurcation diagrams and phase portraits with respect to parameters and initial conditions. Moreover, the system can realize partial amplitude control by adjusting parameters. Results: The new system can produce infinitely many coexisting attractors, including symmetric periodic attractors and chaotic attractors. This shows that the multistability of chaotic systems can be achieved by adding memristors. Conclusion: The analog circuit and hardware circuit are used to illustrate the existence of the proposed system. In addition, a pseudo-random number generator (PRNG) is designed based on this system and its NIST test is given. It can work reliably in the engineering environment. [ABSTRACT FROM AUTHOR]

A new fractional-order cellular neural network (CNN) system is solved using the Adomian decomposition method (ADM) with the hyperbolic tangent activation function in this paper. The equilibrium point is analyzed in this CNN system. The dynamical behaviors are studied as well, using a phase diagram, bifurcation diagram, Lyapunov Exponent spectrum (LEs), and spectral entropy (SE) complexity algorithm. Changing the template parameters and the order values has an impact on the dynamical behaviors. The results indicate that rich dynamical properties exist in the system, such as hyperchaotic attractors, chaotic attractors, asymptotic periodic loops, complex coexisting attractors, and interesting state transition phenomena. In addition, the digital circuit implementation of this fractional-order CNN system is completed on a digital signal processing (DSP) platform, which proves the accuracy of ADM and the physical feasibility of the CNN system. The study in this paper offers a fundamental theory for the fractional-order CNN system as it applies to secure communication and image encryption. [ABSTRACT FROM AUTHOR]

A chaotic oscillator has been designed based on a Sallen Key-type high-pass filter (HPF). The HPF has been converted to a chaotic oscillator using a parallel combination of a PN junction diode as a nonlinear element and an inductor as an energy storage element. The dynamics of the proposed system has been simulated numerically using fourth-order Runge–Kutta method. The circuit exhibits period-doubling route to chaos as well as period-adding route to chaos depending on the choice of system parameters. Striking features like antimonotonicity and coexistence of attractors are also observed. Bifurcation diagram, phase plane plots and spectrum of Lyapunov exponents have been employed to describe the chaotic behavior of the system. A hardware experiment has been carried out to verify the same in the laboratory using off-the-shelf components. [ABSTRACT FROM AUTHOR]

A soft impacting oscillator with symmetric constraints is considered with a focus on grazing-induced bifurcations and chaotic crises. PF-type grazing bifurcation is defined and described schematically. (ω, g)-parameter transition regions lying between adjacent SP(1–m–m)S (m ≥ 0) attractors are of two types, hysteresis zones HZm and beat motion zones BMZm. The global dynamics in two types of transition regions is discussed by applying numerical simulation, continuation shooting method and cell mapping method. The UP(1–(m + 1)–(m + 1))S attractor exists in the whole HZm and BMZm. The difference is that it arises and then ends up by a SN-type grazing bifurcation in the HZm, while it arises and then ends up by a PF-type grazing bifurcation in the BMZm. In soft impacting oscillators, a grazing bifurcation of unstable attractor is bound to happen when an unstable branch coalesces with two branches of stable periodic attractors with different number of impacts. Merging crisis occurs when two chaotic attractors simultaneously collide with an unstable symmetric periodic attractor lying on the border between the basins of attraction, with which two coexisting chaotic attractors merge into a larger chaotic attractor. Then, the interior crisis brings the chaotic attractor to grow up suddenly. And lastly, the chaotic attractor ends up at the boundary crisis. Subcritical characteristic of period-doubling bifurcation is also revealed in the soft impacting oscillator. [ABSTRACT FROM AUTHOR]

A new tri-cellular neural network(CNN) system based on double memristors is constructed which used a hyperbolic tangent function instead of the conventional segmentation function in this paper. The multiple equilibrium points existing in the CNN system are analyzed. Through Lyapunov exponential spectrum, bifurcation diagram, phase diagram, SE complexity and digital circuit implementation, the rich and complex dynamical characteristics of the double-MR-based CNN system are presented. Interestingly, changing different parameters and initial values, the system has multiple coexisting attractors which include periodic-periodic attractors, periodic-chaotic attractors, and chaotic-chaotic attractors. Finally, a hardware circuit of the memristive cellular neural network is designed and built on the basis of a DSP platform to verify the implementability of the network model. The improved double-MR-based Cellular neural network system provides a theoretical foundation in other fields of application, especially for secure communications. [ABSTRACT FROM AUTHOR]

This paper proposes a simple ring memristive neural network (MNN) with self-connection, bidirectional connection and a single memristive synapse. Compared with some existing MNNs, the most distinctive feature of the proposed MNN is that it can generate heterogeneous coexisting attractors and large-scale amplitude control. Various kinds of heterogeneous coexisting attractors are numerically found in the MNN, including chaos with a stable point, chaos with a limit cycle, a limit cycle with a stable point. By increasing the parameter values, the chaotic variables of the MNN can be accordingly increased and their corresponding areas are extremely wide, yielding parameter-dependent large-scale amplitude control. A circuit implementation platform is established and the obtained results demonstrate its validity and reliability. [ABSTRACT FROM AUTHOR]

Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of the incommensurate fractional-order Hopfield neural network (FOHNN) system is solved. The equilibrium point of the system is discussed, and the dissipative characteristics are verified and discussed. By varying the order values of the proposed system, different dynamical behaviors of the incommensurate FOHNN system are explored and discussed via bifurcation diagrams, the Lyapunov exponent spectrum, complexity, etc. Finally, using the DSP platform to implement the system, the results are in good agreement with those of the simulation. The actual results indicate that the system shows many complex and interesting phenomena, such as attractor coexistence and an inversion property, with dynamic changes of the order of q0, q1, and q2. These phenomena provide important insights for simulating complex neural system states in pathological conditions and provide the theoretical basis for the later study of incommensurate fractional-order neural network systems. [ABSTRACT FROM AUTHOR]

Previous results show that some oscillators possess finite number of Wada basins. Here we find that a nonlinear oscillator can possess a countable infinity of Wada basins and these Wada basins are connected. Infinite number of coexisting attractors and their Wada basins are investigated by the basin cell theorem and generalized basin cell theorem. Infinite number of Wada basins are systematic, which identical basins structure can be identified in each periodic X-axis coordinate interval. This type of Wada basin boundary can lead to a high level of indeterminacy and an extreme sensitive dependence on initial condition. [ABSTRACT FROM AUTHOR]

The membrane current I m of the Hodgkin-Huxley (HH) cardiac Purkinje fiber (CPF) model is usually calculated as direct current (DC). In this paper, a conventional alternating current (AC) is used, namely I AC = Asin (2 π f t) , as the impressed injection. Dynamic characters of the model with different AC parameters and various initial conditions are observed through phase plane orbits, waveforms, bifurcation diagrams, and Lyapunov exponent spectra, which reveal multiple coexisting membrane action potential patterns, corresponding to the coexisting attractors of the same or different periods in the phase diagram. Meanwhile, the model undergoes period, quasi-period and local forward or inverse period-doubling bifurcations with changes in amplitude A or frequency f, which further proves the complex nonlinear property of the AC-injected model. In addition, by changing the external current I m , the sodium-ion and potassium-ion equilibrium potentials, i.e. E Na and E K , respectively, the regularity of the CPF heartbeat frequency is observed. The state transformations of CPF are found between normal, abnormal and sudden cardiac arrest, and the method adjusting from the dangerous state to the normal heartbeat frequency range is investigated. This study may provide a reference for exploring the evolution of nonlinear dynamics in HH CPF model and protecting the health of life and heart. [ABSTRACT FROM AUTHOR]

In this work, a new cubic-like smooth nonlinearity is generated by modifying Chua's piecewise-linear segmental nonlinear function using logarithmic cos-hyperbolic function implementation. A logarithmic cos-hyperbolic function possessing smooth symmetric nonlinear characteristics is implemented through CMOS-based circuit design using the current mode approach. The nonlinear design is then incorporated in a new third-order chaotic oscillator configuration to produce chaotic oscillations. This chaotic circuit is tuned to develop different attractors through the bifurcation parameter. Moreover, the dynamics of chaos such as antimonotonicity and coexistence of attractors are also depicted in circuit simulation by tuning various controlling parameters. Additionally, some numerical analyses are performed on this dynamic system to justify the existence of chaoticity and attractors' development. This design has been optimized for low-voltage and moderately high dominant frequency of oscillations. Simulations are done using 180-nm CMOS technology in Cadence Virtuoso. Experimental results are presented to verify the workability of the proposed chaotic system. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic analysis and cryptographic applications of hyperbolic memristor-coupled neurons are reported. A memristor model is proposed, and its locally active property is verified by DC V-I diagram. A 5D hyperbolic memristor-coupled neuron is constructed. The boundedness and Hamiltonian energy of this 5D hyperbolic memristor-coupled neuron are analyzed. The nonlinear behavior of the 5D hyperbolic memristor-coupled neuron such as coexistence bifurcation mode and coexistence attractor is revealed by the bifurcation diagrams and phase diagrams. Furthermore, state switching without parameters is also explored. In order to enhance the security of image transmission, a full-process DNA encryption algorithm based on the proposed 5D hyperbolic memristor-coupled neuron combining with DNA sequence is presented. The histogram and correlation analysis of the encrypted image show that the image encryption algorithm has strong anti-attack ability. [ABSTRACT FROM AUTHOR]

In a global dynamic analysis, the coexisting attractors and their basins are the main tools to understand the system behavior and safety. However, both basins and attractors can be drastically influenced by uncertainties. The aim of this work is to illustrate a methodology for the global dynamic analysis of nondeterministic dynamical systems with competing attractors. Accordingly, analytical and numerical tools for calculation of nondeterministic global structures, namely attractors and basins, are proposed. First, based on the definition of the Perron-Frobenius, Koopman and Foias linear operators, a global dynamic description through phase-space operators is presented for both deterministic and nondeterministic cases. In this context, the stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. Then, numerical implementation of these concepts is accomplished via an adaptative phase-space discretization strategy based on the classical Ulam method. Sample results of the methodology are presented for a canonical dynamical system.

A hyperchaotic map with various patterns of coexisting attractors is found by introducing trigonometric functions. The periodicity of trigonometric functions, as a key factor of coexisting attractors, brings various possibilities for attractor self-producing. By introducing orthorhombic feedback of sinusoidal and cosine functions, the newly constructed multistability can be flexibly controlled, and consequently layered coexisting attractors, externally wrapped coexisting attractors, and scissor-type coexisting attractors are produced. As a result, the offset boosting of initial conditions may draw out chaotic signals with desired amplitudes and polarities. Furthermore, such coexisting attractors may connect and grow in the phase space. This new finding is further verified based on the platform STM32. [ABSTRACT FROM AUTHOR]

The memristor-based chaotic systems are more and more popular with academia because of their abundant dynamics. This paper constructs a novel 4D chaotic system by introducing a flux-controlled memristor into the existing Sprott-J system. Dynamic behaviors of the system are studied by theoretical analysis and numerical simulations. It surprisingly shows that the system can yield infinite coexisting attractors via changing the initial values. And the amplitudes of all signals can be controlled by adjusting a certain parameter. The circuit and microcontroller realization is given as well; corresponding results agree well with numeral simulations. [ABSTRACT FROM AUTHOR]

This paper explores nonlinear analysis of a novel small-scale grid (SSG) and studies the impact of electromagnetic power, random wind power, periodic load and additive white Gaussian noise. Different behaviours such as period doubling bifurcation, chaos, chaos breaking and multistability are investigated and stability issues are revealed in the proposed small-scale grid. Qualitative and quantitative tools such as phase portraits, bifurcation diagrams, Lyapunov exponents, Lyapunov spectrum and basin of attraction are utilised to verify different dynamic behaviours. Erosion of basin region with varying parameter and external noise is reported. Further, the study of unlike behaviours, multistability and coexistence of attractors may be of capital importance in the dynamic evolution of SSG behaviour since serious impediment may occur even after the required safeguards. The present study is expected to be potentially useful in a variety of modern or future power systems, microgrids etc. Numerical simulation is achieved and presented in the MATLAB environment. [ABSTRACT FROM AUTHOR]

In 1971, Chua suggested that there should be a fourth electrical component in addition to resistor, capacitor and inductor. This new component was proposed to be called "memristor". Due to lack of commercially available memristors many emulators are mainly used in nonlinear circuits. Also, the memristance functions are used in many dynamical systems in order to enhance their complexity. Furthermore, the use of memristors in discrete chaotic maps is an interesting research topic, due to the applications of such systems. In this work, a memristor-based Ikeda mapping model is presented by coupling a discrete memristance function with Ikeda map. To investigate system's dynamical behavior a host of nonlinear tools has been used, such as bifurcation and continuation diagrams, maximal Lyapunov exponent diagrams, and Kaplan–Yorke conjecture. Interesting phenomena related to chaos has been observed. More specifically, regular (periodic and quasiperiodic) and chaotic orbits, route to chaos through the mechanism of period doubling and crisis phenomena, have been found. Moreover, higher value of the internal state of the memristor, revealed chaotic behavior in a bigger area. Also, from the comparison of the bifurcation diagrams with the respective continuation diagrams coexisting attractors have been found. Finally, two-parameter bifurcation-like diagrams revealed the system's rich dynamical behavior. • Ikeda map presented chaos through the period doubling and crisis phenomena. • The change in initial conditions revealed the existence of coexisting attractors. • Higher value of the internal state of the memristor, causes chaotic behavior in a bigger area. • Two-parameter bifurcation-like diagrams showed system's dynamical behavior. [ABSTRACT FROM AUTHOR]

This paper deals with the effects of an amplitude modulated (AM) excitation on the nonlinear dynamics of reactions between four molecules. The computation of the fixed points of the autonomous nonlinear chemical system has been made in detail using Cardan's method. Routes to chaos have been investigated through bifurcations structures, Lyapunov exponent and phase portraits. The effects of the control force on chaotic motions have been strongly analyzed and the control efficiency is found in the cases g = 0 (unmodulated case), g = 0 with Ω = nω; n a natural number and ... p and q are simple positive integers. Vibrational Resonance (VR), hysteresis and coexistence of several attractors have been studied in details based on the relationship between the frequencies of the AM force. Results of analytical investigations are validated and complemented by numerical simulations. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Dalian Polytechnic University is the property of Journal of Dalian Polytechnic University Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we enhance the study of the tâtonnement model with cautious price adjustment introduced by Weddepohl in 1995. Specifically, we investigate the discrete time dynamics of the model when the original assumption of equal values for the maximum rate of increase and decrease of the price is relaxed. As a result, its analytic definition is expressed by a bimodal one-dimensional continuous piecewise smooth map which depends on three parameters. As it happens in general for bimodal maps, it is possible to describe the bifurcation structure in some regions of the parameter space of the model using the skew tent map scenario as a normal form. Nonetheless, we show that some border collision bifurcations which play a fundamental role for the asymptotic behavior of the map essentially pertain to its bimodal shape. Among them we highlight the ones that lead, for some specific parameter values, to the coexistence of two chaotic attractors. Moreover, we identify degenerate border collision bifurcations responsible for the peculiar shapes of the chaotic attractors that distinguish the model. [ABSTRACT FROM AUTHOR]

In this paper, we first present a simple seven-term 4D hyperchaotic system based on the classical Sprott-C 3D chaotic system. This novel system is inspired by the simple 4D hyperchaotic system based on Sprott-B proposed by A. T. Sheet (2022). We discuss the phenomenon of premature divergence brought about by the improper choice of coupling parameters in that paper and describe the basic properties of the new system with phase diagrams, Lyapunov exponential spectra and bifurcation diagrams. Then, we find that the dynamical behaviors of the system suffer from the limitation of the control parameters and cannot represent the process of motion in detail. To improve the system, we expand the dimensionality and add the control parameters and memristors. A 5D memristive hyperchaotic system with hidden attractors is proposed, and the basic dynamical properties of the system, such as its dissipation, equilibrium point, stability, Lyapunov exponential spectra and bifurcation diagram, are analyzed. Finally, the hardware circuits of the 4D Sprott-C system and the 5D memristive hyperchaotic system were realized by a field programmable gate array (FPGA) and verified by an experiment. The experimental results are consistent with the numerical simulation results obtained in MATLAB, which demonstrates the feasibility and potential of the system. [ABSTRACT FROM AUTHOR]