Stable Node-foci Research Articles

Revealing More Hidden Attractors from a New Sub-Quadratic Lorenz-Like System of Degree 6 5

In the sense that the descending powers of some certain variables may widen the range of parameters of self-excited and hidden attractors, this technical note proposes a new three-dimensional Lorenz-like system of degree [Formula: see text]. In contrast to the previously studied one of degree [Formula: see text], the newly reported one creates more hidden Lorenz-like attractors coexisting with the unstable origin and a pair of stable node-foci in a broader range of parameters, which confirms the generalization of the second part of the celebrated Hilbert’s 16th problem once more. In addition, some other dynamics, i.e. Hopf bifurcation, the generic and degenerate pitchfork bifurcation, invariant algebraic surface, first integral, singularly degenerate heteroclinic cycle with nearby chaotic attractor, ultimate bounded set and existence of a pair of heteroclinic orbits, are discussed.

Generating Multiwing Hidden Chaotic Attractors With Only Stable Node-Foci: Analysis, Implementation, and Application

Based on Shil'nikov criterion, i.e., extending the number of unstable saddle-focus -type equilibria of chaotic system, a rich and varied multiwing chaos with complicated topology and remarkable pseudorandomness has been coined. However, it is a challenging task to construct multiwing chaos with only stable node-focus -type equilibria as the Shil'nikov criterion is not applicable in the circumstances. To eliminate this difficulty and enrich the multiwing chaos, we construct a simple three-dimensional autonomous system with only two symmetric stable equilibria, and further introduce a sawtooth wave modulation function to configure more equilibria. The highlight is that multiwing chaotic attractors can be generated from system with two or multiple stable node-foci. This characteristic indicates that the attractors can be hidden oscillating and are difficult to be located in the small neighborhoods of equilibria, which makes the system has good concealment performance for security applications. The design mechanism and system properties are theoretically analyzed and numerically investigated using phase portraits, finite-time Lyapunov exponents, and basins of attraction. Hardware experiments based on field-programmable gate array verify the feasibility of the system. Finally, based on the generated multiwing hidden chaotic attractors, an image encryption scheme with high security performance is designed to enhance its application.

A New Chaotic System with Two Stable Node-Foci Equilibria and an Unstable Saddle-Focus Equilibrium: Bifurcation and Multistability Analysis, Circuit Design, Voice Cryptosystem Application, and FPGA Implementation

A New Chaotic System with Two Stable Node-Foci Equilibria and an Unstable Saddle-Focus Equilibrium: Bifurcation and Multistability Analysis, Circuit Design, Voice Cryptosystem Application, and FPGA Implementation

Generating multi-folded hidden Chua’s attractors: Two-case study

In recent decades, Chua’s attractor has been taken as the classical paradigm for chaos demonstration, but it still presents some new results one after another. Hidden Chua’s attractor is a new category of attractor with basin of attraction that does not intersect with any system equilibrium point. The existing hidden Chua’s attractors have the simple single-hollow structure, and can only be generated from Chua’s system with both unstable and stable equilibria. In this paper, we report the generation of new hidden Chua’s attractors via the operation of complex number. Two cases of multi-folded hidden Chua’s attractors with only stable node-foci or with only unstable saddles are respectively generated from two seed Chua’s systems with double-scroll self-excited attractor or with single-hollow hidden attractor. The detailed system construction, equilibrium calculation, and numerical simulation are presented. Besides, the local basins of attraction of these new multi-folded hidden Chua’s attractors are visualized to show the evolutions under the operation of complex number. Finally, the physical experiment using field programmable gate array implemented these hidden attractors.

Extreme multistability and phase synchronization in a heterogeneous bi-neuron Rulkov network with memristive electromagnetic induction.

Memristive electromagnetic induction effect has been widely explored in bi-neuron network with homogeneous neurons, but rarely in bi-neuron network with heterogeneous ones. This paper builds a bi-neuron network by coupling heterogeneous Rulkov neurons with memristor and investigates the memristive electromagnetic induction effect. Theoretical analysis discloses that the bi-neuron network possesses a line equilibrium state and its stability depends on the memristor coupling strength and initial condition. That is, the stability of the line equilibrium state has a transition between unstable saddle-focus and stable node-focus via Hopf bifurcation. By employing parameters located in the stable node-focus region, dynamical behaviors related to the memristor coupling strength and initial conditions are revealed by Julia- and MATLAB-based multiple numerical tools. Numerical results demonstrate that the proposed heterogeneous bi-neuron Rulkov network can generate point attractor, period, chaos, chaos crisis, and period-doubling bifurcation. Note that extreme multistability are disclosed with respect to initial conditions of memristor and gated ion concentration. Coexisting infinitely multiple firing patterns of periodic firing patterns with different periodicities and chaotic firing patterns for different memristor initial conditions are demonstrated by phase portrait and time-domain waveform. Besides, the phase synchronization related to the memristor coupling strength and its initial condition is explored, which suggests that the two heterogeneous neurons become phase synchronization with large memristor coupling strength and initial condition. This also reflects that the plasticity of memristor synapse enables adaptive regulation in keeping energy balance between the neurons. What's more, MCU-based hardware experiments are executed to further confirm the numerical simulations.

Hidden multiwing chaotic attractors with multiple stable equilibrium points

PurposeThe purpose of this paper is to construct a multiwing chaotic system that has hidden attractors with multiple stable equilibrium points. Because the multiwing hidden attractors chaotic systems are safer and have more dynamic behaviors, it is necessary to construct such a system to meet the needs of developing engineering.Design/methodology/approachBy introducing a multilevel pulse function into a three-dimensional chaotic system with two stable node–foci equilibrium points, a hidden multiwing attractor with multiple stable equilibrium points can be generated. The switching behavior of a hidden four-wing attractor is studied by phase portraits and time series. The dynamical properties of the multiwing attractor are analyzed via the Poincaré map, Lyapunov exponent spectrum and bifurcation diagram. Furthermore, the hardware experiment of the proposed four-wing hidden attractors was carried out.FindingsNot only unstable equilibrium points can produce multiwing attractors but stable node–foci equilibrium points can also produce multiwing attractors. And this system can obtain 2N + 2-wing attractors as the stage pulse of the multilevel pulse function is N. Moreover, the hardware experiment matches the simulation results well.Originality/valueThis paper constructs a new multiwing chaotic system by enlarging the number of stable node–foci equilibrium points. In addition, it is a nonautonomous system that is more suitable for practical projects. And the hardware experiment is also given in this article which has not been seen before. So, this paper promotes the development of hidden multiwing chaotic attractors in nonautonomous systems and makes sense for applications.

Dynamics, Periodic Orbit Analysis, and Circuit Implementation of a New Chaotic System with Hidden Attractor

Hidden attractors are associated with multistability phenomena, which have considerable application prospects in engineering. By modifying a simple three-dimensional continuous quadratic dynamical system, this paper reports a new autonomous chaotic system with two stable node-foci that can generate double-wing hidden chaotic attractors. We discuss the rich dynamics of the proposed system, which have some interesting characteristics for different parameters and initial conditions, through the use of dynamic analysis tools such as the phase portrait, Lyapunov exponent spectrum, and bifurcation diagram. The topological classification of the periodic orbits of the system is investigated by a recently devised variational method. Symbolic dynamics of four and six letters are successfully established under two sets of system parameters, including hidden and self-excited chaotic attractors. The system is implemented by a corresponding analog electronic circuit to verify its realizability.

Existence of periodic orbits and chaos in a class of three-dimensional piecewise linear systems with two virtual stable node-foci

In this paper, we consider a new class of piecewise linear (PWL) systems with two virtual stable node-foci (the meaning of “virtual” is from Bernardo et al. (2008)) which exhibits periodic orbits and chaos. This fact that PWL systems have no unstable equilibria but has chaos will unavoidably make the exploration of this chaos more complicated. Particular values for bifurcation diagram are provided. Based on mathematical analysis and Poincaré map, periodic orbits of this kind of system without unstable equilibrium points are derived, the corresponding existence theorems are given, and the obtained results are applied to specific examples.

Complex Dynamics of a Four-Dimensional Circuit System

Combining qualitative analysis and numerical technique, the present work revisits a four-dimensional circuit system in [Ma et al., 2016] and mainly reveals some of its rich dynamics not yet investigated: pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, globally exponentially attractive set, invariant algebraic surface and heteroclinic orbit. The main contributions of the work are summarized as follows: Firstly, it is proved that there exists a globally exponentially attractive set with three different exponential rates by constructing a suitable Lyapunov function. Secondly, the existence of a pair of heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, numerical simulations not only are consistent with theoretical results, but also illustrate potential existence of hidden attractors in its Lorenz-type subsystem, singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small [Formula: see text], i.e. hyperchaotic attractors and nearby pseudo singularly degenerate heteroclinic cycles, i.e. a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity, or the true ones consisting of normally hyperbolic saddle-foci (or saddle-nodes) and stable node-foci, giving some kind of forming mechanism of hyperchaos.

Generating multi-wing hidden attractors with only stable node-foci via non-autonomous approach

Hidden attractor is associated with the multistability phenomenon, which has important application prospects in security communication. Recently, the study of multi-wing hidden chaotic attractor with only stable equilibria is a particular interest. By modifying simple 3D quadratic systems, this paper first proposes two systems with two symmetric equilibria. In particular, in a wide parameter region, the two systems have stable node-foci but can generate structural stable double-wing chaotic attractors, which belongs to the category of hidden attractor. The rotation matrix is applied to implement the attractor rotation. In combination with multi-level pulse sequences, multi-wing chaotic attractors with switchable stable node-foci are generated. The multi-wing hidden attractors are experimentally validated by microcontroller-based hardware experiments.

Dynamical Analysis and Periodic Solution of a Chaotic System with Coexisting Attractors

Chaotic attractors with no equilibria, with an unstable node, and with stable node-focus are presented in this paper. The conservative solutions are investigated by the semianalytical and seminumerical method. Furthermore, multiple coexisting attractors are investigated, and circuit is carried out. To study the system’s global structure, dynamics at infinity for this new chaotic system are studied using Poincaré compactification of polynomial vector fields in R 3 . Meanwhile, the dynamics near the infinity of the singularities are obtained by reducing the system’s dimensions on a Poincaré ball. The averaging theory analyzes the periodic solution’s stability or instability that bifurcates from Hopf-zero bifurcation.

Design of multi-wing 3D chaotic systems with only stable equilibria or no equilibrium point using rotation symmetry

Some dissipative dynamical systems only have stable equilibria or have no equilibrium point but are able to generate chaos of non-Sil’nikov type. This category of chaotic attractors belong to hidden attractor, which is difficult to be located in the phase space. In this work, we aim at designing multi-wing 3D chaotic systems with hidden attractors. We first proposes a quadratic system with only two stable node-foci, which can generate double-wing chaotic hidden attractor. Using rotation symmetry, we construct multi-fold cover of the quadratic system and obtain multiple symmetric stable equilibria. Two examples of 2-fold and 3-fold covers of the chaotic system are presented. Furthermore, the rotation symmetry is applied to no-equilibrium system to generate multi-wing chaotic hidden attractors. By means of bifurcation diagram, finite-time Lyapunov exponents, phase portraits, and attraction basins, it is shown that the multi-wing chaotic attractors always coexist with stable point attractors. The local initial condition areas and complex boundaries for different coexisting attractors are addressed. Finally, the designed systems are implemented by analog circuit and microcontroller to verify the multi-wing hidden attractors.

Asymmetric memristive Chua’s chaotic circuits

ABSTRACT Asymmetric property of the pinched hysteresis loop usually appears in a real memristor. To imitate this property, an asymmetric memristor emulator is constructed by an analog circuit consisting of a non-balance diode-bridge and a resistor-capacitor (RC) filter. By coupling this emulator to the classical Chua’s circuit in parallel in both forward and reverse ways, two asymmetric memristive Chua’s circuits are proposed and their mathematical models are established. Because of the symmetry between the forward and reverse circuits, the forward memristive Chua’s (FAMC) circuit is focused on modelling the normalised system and investigating the equilibrium stability. With the FAMC system, several numerical plots are employed to discuss the parameter-dependent complex dynamics and initial-dependent coexisting behaviours. The numerical results manifest that there are three equilibrium points in the FAMC system, among which two non-zero equilibrium points are asymmetric, resulting in the emergence of asymmetric attractors. With the change of system parameters, the stability of one non-zero equilibrium point converts from unstable saddle-focus into stable node-focus, leading to the coexistence of multiple attractors. In addition, hardware experiments on a breadboard level are executed and the captured results well verify the numerical plots.

Riddled Attraction Basin and Multistability in Three-Element-Based Memristive Circuit

By coupling a diode bridge-based second-order memristor and an active voltage-controlled memristor with a capacitor, a three-element-based memristive circuit is synthesized and its system model is then built. The boundedness of the three-element-based memristive circuit is theoretically proved by employing the contraction mapping principle. Besides, the stability distributions of equilibrium points are theoretically and numerically expounded in a 2D parameter plane. The results imply the memristive circuit has a zero unstable saddle focus and a pair of nonzero stable node-foci or unstable saddle-foci depending on the considered parameters. The dynamical behaviors include point attractor, period, chaos, coexisting bifurcation mode, period-doubling bifurcation route, and crisis scenarios, which are explored using some common dynamical methods. Of particular concern, riddled attraction basins and multistability are uncovered under two sets of specified model parameters nearing the tiny neighborhood of crisis scenarios by local attraction basins and phase plane plots. The riddled attraction basins with island-like structure demonstrate that their dynamical behaviors are extremely sensitive to the initial conditions, resulting in the coexistence of limit cycles with period-2 and period-6, as well as the coexistence of period-1 limit cycles and single-scroll chaotic attractors. Moreover, a feasible on-breadboard hardware circuit is manually made and the experimental measurements are executed, upon which phase plane trajectories for some discrete model parameters are captured to further confirm the numerically simulated ones.

Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

<p style='text-indent:20px;'>The present work is devoted to giving new insights into a chaotic system with two stable node-foci, which is named Yang-Chen system. Firstly, based on the global view of the influence of equilibrium point on the complexity of the system, the dynamic behavior of the system at infinity is analyzed. Secondly, the Jacobi stability of the trajectories for the system is discussed from the viewpoint of Kosambi-Cartan-Chern theory (KCC-theory). The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The obtained results show that in the sense of Jacobi stability, all equilibrium points of the system, including those of the two linear stable node-foci, are Jacobi unstable. These studies show that one might witness chaotic behavior of the system trajectories before they enter in a neighborhood of equilibrium point or periodic orbit. There exists a sort of stability artifact that cannot be found without using the powerful method of Jacobi stability analysis.

Remerging Feigenbaum Trees, Coexisting Behaviors and Bursting Oscillations in a Novel 3D Generalized Hopfield Neural Network

This paper, a novel 3D generalized Hopfield neural network is proposed and investigated in order to generate and highlight some unknown behaviors related to such type of neural network. This generalized model is constructed by exploiting the effect of an external stimulus on the dynamics of a simplest 3D autonomous Hopfield neural network reported to date. The stability of the model around its ac-equilibrium point is studied. We note that the model has three types of stability depending on the value of the external stimulus. The stable node-focus, the unstable saddle focus, and the stable node characterize the stability of the equilibrium point of the model. Traditional nonlinear analyses tools are used to highlight and support several complex phenomena such as remerging Feigenbaum trees, the coexistence of up to two, four and six disconnected stable states when the amplitude of the external stimulus is set to zero. Furthermore, for some values of the frequency and amplitudes of the external stimulus, bursting oscillations occur. This latter behavior is characterized by the fact that oscillations switch between quiescent states and spiking states, repetitively. The transformed phase portraits are used to support the bursting mode of oscillations found. Finally, PSpice simulations enable to support the results of the theoretical studies.

Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system

In this note, by using the theory of bifurcation and Lyapunov function, one performs a qualitative analysis on a novel four-dimensional unified hyperchaotic Lorenz-type system (UHLTS), including stability, pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, ultimate bound estimation, global exponential attractive set, heteroclinic orbit and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small \begin{document}$ b > 0 $\end{document} , i.e. conjugate hyperchaotic Lorenz-type attractors (CHCLTA) and nearby a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity or singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci and stable node-foci, etc. In particular, by a linear scaling, a possibly new forming mechanism behind the creation of well-known hyperchaotic attractor with \begin{document}$ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-12, 12, 23,-1, -1, 1, 2.1, -6, -0.2) $\end{document} , consisting of occurrence of degenerate pitchfork bifurcation at \begin{document}$ S_{z} $\end{document} , the change in the stability index of the saddle at the origin as \begin{document}$ b $\end{document} crosses the null value, explosion of normally hyperbolic stable node-foci, collapse of singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci or saddle-nodes, and stable node-foci, is revealed. The findings and results of this paper may provide theoretical support in some future applications, since they improve and complement the known ones.

A Simple Nonautonomous Hidden Chaotic System with a Switchable Stable Node-Focus

This paper presents a simple two-dimensional nonautonomous system, which possesses piecewise linearity constructed by a simple absolute value function. The nonautonomous system has only one switchable equilibrium state with a stable node-focus in the considered control parameter region but can generate periodic, chaotic and coexisting attractors. Therefore, the presented simple two-dimensional nonautonomous system always operates with hidden oscillations, which is not similar to any example reported in the literature. Furthermore, specific hidden dynamical behaviors are numerically disclosed by employing one-dimensional and two-dimensional bifurcation plots, phase plane plots, Poincaré mappings, local attraction basins, and complexity plots. In addition, by utilizing the circuit module of the absolute value function, a multiplierless analog circuit is designed, based on which breadboard experiments are performed to validate the numerically simulated phase plane plots of coexisting attractors.

Non-ideal memristor synapse-coupled bi-neuron Hopfield neural network: Numerical simulations and breadboard experiments

This paper presents a third-order non-ideal memristor synapse-coupled bi-neuron Hopfield neural network (HNN), in which a non-ideal memristor synapse is employed to generate an electromagnetic induction current caused by the potential difference between two neurons. The memristive HNN model possesses an odd number of equilibrium points including one (or three, dependent on the memristor coupling strength) unstable saddle point, two unstable index-2 saddle-foci, and two (or four) stable node-foci. Bifurcation plots, local attraction basins, and phase plane plots show that the memristive HNN model behaves bi-stability of coexisting chaotic and stable point attractors, which is perfectly validated by breadboard experiments based on discrete components.

Hidden Hyperchaotic Attractors in a New 5D System Based on Chaotic System with Two Stable Node-Foci

This paper reports some hidden hyperchaotic attractors and complex dynamics in a new five-dimensional (5D) system with only two nonlinear terms. The system is generated by adding two linear controllers to an unusual 3D autonomous quadratic chaotic system with two stable node-foci. In particular, the hyperchaotic system without equilibrium or with only one stable equilibrium can generate two kinds of hidden hyperchaotic attractors with three positive Lyapunov exponents. Numerical methods not only verify the existence of such attractors and hyperchaotic attractors, but also show the dynamical evolution of this system. The 5D system has self-excited attractors and two types of hidden attractors with the change of its parameter. The parameter switching algorithm is further utilized to numerically approximate the attractor. Specifically, the hidden hyperchaotic attractor can be approximated by switching between two self-excited chaotic attractors. Finally, the circuit realization results are consistent with the numerical results.