Symmetric Attractors Research Articles

Asymmetry-Induced Dynamics for a Class of Diode-Based Chaotic Circuits: A Case Study

We consider the modeling and asymmetry-induced dynamics for a class of chaotic circuits sharing the same feature of an antiparallel diodes pair as the nonlinear component. The simple autonomous jerk circuit of [J. Kengne, Z. T. Njitacke, A. N. Nguomkam, M. T. Fouodji and H. B. Fotsin, Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit, Int. J. Bifurcation Chaos Appl. Sci. Eng. 26 (2016) 1650081] is used as the prototype. In contrast to current approaches where the diodes are assumed to be identical (and thus a perfect symmetric circuit), we examine the more realistic situation where the diodes have different electrical properties in spite of unavoidable scattering of parameters. In this case, the nonlinear component formed by the diodes pair displays an asymmetric current–voltage characteristic which induces asymmetry of the whole circuit. The model is described by a continuous-time 3D autonomous system (ODEs) with exponential nonlinearities. We examine the chaos mechanism with respect to system parameters both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectory plots as the main indicators. Period doubling route to chaos, merging crisis, and multiple coexisting (i.e., two, four, or six) mutually symmetric attractors are reported in the symmetric mode of oscillation. In the asymmetric mode, several unusual nonlinear behaviors arise such as coexisting bifurcations, hysteresis, asymmetric double-band chaotic attractor, crisis, and coexisting multiple (i.e., two, three, four, or five) asymmetric attractors for some suitable ranges of parameters. Theoretical analyses and circuit experiments show a very good agreement. The results obtained in this work let us conjecture that chaotic circuits with antiparallel diodes pair are capable of much more complex dynamics than what is reported in the current literature and thus should be reconsidered accordingly in spite of the approach followed in this work.

Generating coexisting attractors from a new four-dimensional chaotic system

This paper introduces a new four-dimensional chaotic system with a unique unstable equilibrium and multiple coexisting attractors. The dynamic evolution analysis shows that the system concurrently generates two symmetric chaotic attractors for fixed parameter values. Based on this system, an effective method is established to construct an infinite number of coexisting chaotic attractors. It shows that the introduction of some non-linear functions with multiple zeros can increase the equilibria and inspire the generation of coexisting attractor of the system. Numerical simulations verify the availability of the method.

Symmetric Projection Attractor Reconstruction analysis of murine electrocardiograms: Retrospective prediction of Scn5a+/- genetic mutation attributable to Brugada syndrome.

BackgroundLife-threatening arrhythmias resulting from genetic mutations are often missed in current electrocardiogram (ECG) analysis. We combined a new method for ECG analysis that uses all the waveform data with machine learning to improve detection of such mutations from short ECG signals in a mouse model.ObjectiveWe sought to detect consequences of Na+ channel deficiencies known to compromise action potential conduction in comparisons of Scn5a+/- mutant and wild-type mice using short ECG signals, examining novel and standard features derived from lead I and II ECG recordings by machine learning algorithms.MethodsLead I and II ECG signals from anesthetized wild-type and Scn5a+/- mutant mice of length 130 seconds were analyzed by extracting various groups of features, which were used by machine learning to classify the mice as wild-type or mutant. The features used were standard ECG intervals and amplitudes, as well as features derived from attractors generated using the novel Symmetric Projection Attractor Reconstruction method, which reformulates the whole signal as a bounded, symmetric 2-dimensional attractor. All the features were also combined as a single feature group.ResultsClassification of genotype using the attractor features gave higher accuracy than using either the ECG intervals or the intervals and amplitudes. However, the highest accuracy (96%) was obtained using all the features. Accuracies for different subgroups of the data were obtained and compared.ConclusionDetection of the Scn5a+/- mutation from short mouse ECG signals with high accuracy is possible using our Symmetric Projection Attractor Reconstruction method.

A Simple Method for Generating Mirror Symmetry Composite Multiscroll Chaotic Attractors

For further increasing the complexity of chaotic attractors, a new method for generating Mirror Symmetry Composite Multiscroll Chaotic Attractors (MSCMCA) is proposed. We take the Lorenz system as an example to explain the mechanism of the method. Firstly, by varying the signs and magnitudes of the nonlinear terms, the Lorenz system generates symmetrical attractors and different-magnitude attractors, respectively. Secondly, a modified Lorenz system is constructed by imposing several unified multilevel-logic pulse signals to the Lorenz system. The new system generates a novel chaotic attractor consisting of two pairs of different-magnitude symmetrical attractors. By adjusting the parameters of the pulse signals, the modified Lorenz system can also be controlled to generate novel grid multiscroll chaotic attractors, namely MSCMCA. Several dynamical behaviors of the new system are shown by equilibria analysis and Lyapunov exponent spectrum. Moreover, the method can be applied to other chaotic systems. Finally, a circuit of the modified Lorenz system is designed by Multisim software, and the simulation result proves the effectiveness of the method.

Extremely rich dynamics in a memristor-based chaotic system

By introducing a memristor to the jerk system, a four-dimensional (4D) chaotic system is proposed. This system has five line equilibrium points with different stability. The extremely rich dynamic behaviors are studied by numerical simulation and theoretical analysis. Specifically, transient dynamics of coexisting chaotic attractors with different amplitudes at different time scales are found. And extreme multistability of mirror symmetric chaotic attractors, different scroll or amplitude of chaotic attractors, and different periodic attractors can be observed by selecting appropriate parameters and initial conditions. It is also found that the initial condition of the memristor can be used as the controller to realize the offset boost of the chaotic attractor. Then, the HSVII method, which includes state increment integral transformation and linear transformation, is used to reduce the dimension of the system and realize the transformation of correlation dynamics of initial condition into correlation dynamics of system parameter. It is proved numerically that the HSVII method is effective for the multistability analysis of the memristive chaotic system.

Extreme Multistability and Complex Dynamics of a Memristor-Based Chaotic System

A memristor with coexisting pinched hysteresis loops and twin local activity domains is presented and analyzed, with an emulator being designed and applied to the classic Chua’s circuit to replace the diode. The memristive system is modeled with four coupled first-order autonomous differential equations, which has three equilibria determined by three static equilibria of the memristor but not controlled by the system parameters. The complex dynamics of the system are analyzed by using compound coexisting bifurcation diagrams, Lyapunov exponent spectra and phase portraits, including point attractors, limit cycles, symmetrical chaotic attractors and their blasting, extreme multistability, state-switching without parameter, and transient chaos. Of particular surprise is that the extreme multistability of the system is hidden and symmetrically distributed. It is found that the existence of transient chaos in the specified parameter domain is determined by using bifurcation diagrams within different time durations and Lyapunov exponents with chaotic sequences. Finally, the symmetrical chaotic attractor and the system blasting are verified by digital signal processing experiments, which are consistent with the numerical analysis.

Complex dynamics of a non-volatile memcapacitor-aided hyperchaotic oscillator

To explore memcapacitors and their characteristics in chaotic oscillators, this paper proposes a logarithmic charge-controlled memcapacitor model. Using power-off plot analysis, we show that the memcapacitor possesses continuous non-volatile characteristic. Also, its dynamic route map shows the memcapacitor can rapidly switch from one memcapacitance to another by applying a single voltage pulse. Based on the memcapacitor model, we design a chaotic oscillator, which can exhibit some complex dynamic characteristics, such as chaos, hyperchaos and various coexisting attractors. The multistable coexisting oscillation of the system is further analyzed by using phase portraits, basins of attraction and double-bifurcation diagrams. Symmetric coexistence attractors with infinite homogeneity and heterogeneity are also found, which can evolve into hyperchaos under certain initial conditions. Finally, the chaotic oscillator is verified by numerical simulations and digital signal processor experiments.

A broken symmetry approach for the modeling and analysis of antiparallel diodes-based chaotic circuits: a case study

This paper focuses on the modeling and symmetry breaking analysis of the large class of chaotic electronic circuits utilizing an antiparallel diodes pair as nonlinear device necessary for chaotic oscillations. A new and relatively simple autonomous jerk circuit is used as a paradigm. Unlike current approaches assuming identical diodes (and thus a perfect symmetric circuit), we consider the more realistic situation where both antiparallel diodes present different electrical properties in spite of unavoidable scattering of parameters. Hence, the nonlinear component synthesized by the diodes pair exhibits an asymmetric current–voltage characteristic which engenders the explicit symmetry break of the whole electronic circuit. The mathematical model of the new circuit consists of a continuous time 3D autonomous system with exponential nonlinear terms. We investigate the chaos mechanism with respect to model parameters and initial conditions as well, both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectories plots as main indicators. We report period doubling route to chaos, spontaneous symmetry breaking, merging crisis and the coexistence of two, four, or six mutually symmetric attractors in the symmetric regime of operation. More intriguing and complex nonlinear behaviors are revealed in the asymmetric system such as coexisting asymmetric bubbles of bifurcation (a new kind of phenomenon discovered in this work), hysteretic dynamics, critical phenomena, and coexisting multiple (i.e. two, three, four, or five) asymmetric attractors for some appropriately selected values of parameters. Laboratory experimental tests are conducted to support the theoretical analysis. The results obtained in this work clearly indicate that chaotic circuits with back to back diodes can demonstrate much more complex dynamics than what is reported in the relevant literature and thus should be reconsidered accordingly following the method described in this paper.

Extracting new information from old waveforms: Symmetric projection attractor reconstruction: Where maths meets medicine.

What is the topic of this review? Symmetric Projection Attractor Reconstruction (SPAR) is a relatively new mathematical method that can extract additional information pertaining to the morphology and variability of physiological waveforms, such as arterial pulse pressure. Herein, we describe the potential utility of the method for more sensitive quantification of cardiovascular changes. What advances does it highlight? We use a simple example of a human tilt table to illustrate these concepts. SPAR can be used on any approximately periodic waveform and may add value to experimental and clinical settings, where such signals are collected routinely. Periodic physiological waveform data, such as blood pressure, pulse oximetry and ECG, are routinely sampled between 100 and 1000Hz in preclinical research and in the clinical setting from a wide variety of implantable, bedside and wearable monitoring devices. Despite the underlying numerical waveform data being captured at such high fidelity, conventional analysis tends to reside in reporting only averages of minimum, maximum, amplitude and rate, as single point averages. Although these averages are undoubtedly of value, simplification of the data in this way means that most of the available numerical data are discarded. In turn, this may lead to subtle physiological changes being missed when investigating the cardiovascular system over time. We have developed a mathematical method (symmetric projection attractor reconstruction) that uses all the numerical data, replotting and revisualizing them in a manner that allows unique quantification of multiple changes in waveform morphology and variability. We propose that the additional quantification of these features will allow the complex behaviour of the cardiovascular system to be mapped more sensitively in different physiological and pathophysiological settings.

Circuit Realization of a 3D Multistability Chaotic System and Its Synchronization via Linear Resistor and Linear Capacitor in Parallel Coupling

In this paper, a 3D multistability chaotic system with two coexisting conditional symmetric attractors is studied by using a circuit block diagram and realized by using an electronic circuit. The simulation results show that two coexisting conditional symmetric attractors are emerged in this electronic circuit. Furthermore, synchronization of this 3D multistability chaotic system and its electronic circuit is studied. It shows that linear resistor and linear capacitor in parallel coupling can achieve synchronization in this chaotic electronic circuit. That is, the output voltage of chaotic electronic circuit is coupled via one linear resistor and one linear capacitor in parallel coupling. The simulation results verify that synchronization of the chaotic electronic circuit can be achieved.

Multistability Control of Space Magnetization in Hyperjerk Oscillator: A Case Study

Abstract In this paper, multistability control of a 5D autonomous hyperjerk oscillator through linear augmentation scheme is investigated. The space magnetization is characterized by the coexistence of five different stable states including an asymmetric pair of chaotic attractors, an asymmetric pair of period-3 cycle, and a symmetric chaotic attractor for a given/fixed set of parameters. The linear augmentation method is applied here to control, for the first time, five coexisting attractors. Standard Lyapunov exponents, bifurcation diagrams, basins of attraction, and 3D phase portraits are presented as methods to conduct the efficaciousness of the control scheme. The results of the applied methods reveal that the monostable chaotic attractor is obtained through three important crises when varying the coupling strength. In particular, below the first critical value of the coupling strength, five distinct attractors are coexisting. Above that critical value, three and then two chaotic attractors are now coexisting, respectively. While for higher values of the coupling strength, only the symmetric chaotic attractor is viewed in the controlled system. The process of annihilation of coexisting multiple attractors to monostable one is confirmed experimentally. The important results of the controlled hyperjerk system with its unique survived chaotic attractor are suited in applications like secure communications.

Extremely rich dynamics from hyperchaotic Hopfield neural network: Hysteretic dynamics, parallel bifurcation branches, coexistence of multiple stable states and its analog circuit implementation

In this work, we investigate the dynamics of a model of 4-neurons based hyperchaotic Hopfield neural network (HHNN) with a unique unstable node as a fixed point. The basic properties of the model including symmetry, dissipation, and condition of the existence of an attractor are explored. Our numerical simulations highlight several complex phenomena such as periodic orbits, quasi-periodic orbits, and chaotic and hyperchaotic orbits. More interestingly, it has been revealed several sets of synaptic weights matrix for which the HHNN studied display multiple coexisting attractors including two, three and four symmetric and disconnected attractors. Both hysteretic dynamics and parallel bifurcation branches justify the presence of these various coexisting attractors. Basins of attraction with the riddle structure of some of the coexisting attractors have been computed showing different regions in which each solution can be captured. Finally, PSpice simulations are used to further support the results of our previous analyses.

Dynamics, control and symmetry breaking aspects of a modified van der Pol–Duffing oscillator, and its analog circuit implementation

This article focuses on the dynamics of a modified van der Pol–Duffing circuit (MVDPD hereafter) (Fotsin and Woafo in Chaos Solitons and Fractals 24(5):1363–1371, 2005) whose symmetry is explicitly broken with the presence an offset term. When ignoring offset terms, the system displays an exact symmetry which is reflected in the location of the equilibrium points, the attractor topologies and the attraction basins shapes as well. In this mode of operation, the system displays typical behaviors such as period doubling sequences; spontaneous symmetry breaking, symmetry recovering, and multistability involving several pairs of mutually symmetric attractors. In the presence of offset terms, the MVDPD circuit is non-symmetric and more complex nonlinear phenomena arise such as parallel bifurcation branches, coexisting multiple (i.e. two, three, four or five) asymmetric attractors, and crises. It should be noted that for each case of multistability discussed in this work, a hidden attractor (period-1 limit cycle) coexists with self-excited others. To the best of our knowledge, the coexistence of five attractors (symmetrical or asymmetrical), one of which is hidden has not yet been reported in the MVDPD circuit and thus deserves dissemination. PSpice simulation investigations based on the implementation of the MVDPD confirm the theoretical predictions.

Dynamics, control and symmetry breaking aspects of a single Opamp-based autonomous LC oscillator

In this contribution, we consider the dynamics of the single opamp-based autonomous LC oscillator with diode pair nonlinearity previously introduced by Kengne et al. (2018). Unlike current approaches assuming identical diodes (and thus a perfect symmetric circuit), we consider the more realistic situation where both antiparallel diodes present different electrical properties in spite of unavoidable scattering of parameters (and so a non-symmetric circuit). We report period doubling route to chaos, spontaneous symmetry breaking, merging crisis and the coexistence of two and four mutually symmetric attractors in the symmetric regime of operation. More intriguing and complex nonlinear behaviors are revealed in the asymmetric system such as coexisting asymmetric bubbles of bifurcation, critical phenomena, and coexisting multiple asymmetric attractors for some appropriately selected values of parameters. The experimental and PSpice simulation results are consistent with the numerical predictions. We present a simple control strategy based on linear augmentation scheme that enables to drive the system from a multistable state to a monostable state. To the best of the authors’ knowledge, the results obtained in this work have not yet been reported in the literature and thus deserve dissemination.

Crisis and hyperchaos in a simplified model of magnetoconvection

In this work, a numerical study of a reduced model of magnetoconvection is presented. By considering a background magnetic field in a conducting fluid in a plane layer, a generalized Lorenz model was originally derived by Macek and Strumik (2010) and is used here to investigate transition to hyperchaos as a function of the background magnetic field. In this work a merging crisis involving the collision of two symmetric chaotic attractors with a hyperchaotic saddle is shown to give rise to a hyperchaotic attractor. A new numerical technique for showing a collision between a hyperchaotic set and a chaotic attractor in a high-dimensional phase space is introduced.

Hyperchaotic Circuit Based on Memristor Feedback with Multistability and Symmetries

A novel hyperchaotic circuit is proposed by introducing a memristor feedback in a simple Lorenz-like chaotic system. Dynamic analysis shows that it has infinite equilibrium points and multistability. Additionally, the symmetrical coexistent attractors are investigated. Further, the hyperchaotic system is implemented by analogue circuits. Corresponding experimental results are completely consistent with the theoretical analysis.

Sequence of Routes to Chaos in a Lorenz-Type System

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

A Unique Jerk System with Abundant Dynamics: Symmetric or Asymmetric Bistability, Tristability, and Coexisting Bubbles

Finding and revealing new features and behaviors of simple chaotic systems have always been an important and attractive research topic. This paper aims to introduce a new 3D autonomous jerk chaotic system and explore its rich dynamical behaviors including period-doubling bifurcation and reverse period-doubling bifurcation routes to chaos, crisis and internal crisis, multiple symmetric coexisting attractors, and antimonotonicity. Especially, the phenomena of asymmetric bistability (e.g., coexistence of a point attractor and chaotic attractor or coexistence of a point attractor and period-5 limit cycle), tristability (e.g., coexistence of a point attractor and a pair of symmetric chaotic or periodic attractors), and coexisting bubbles are found, which have been rarely reported before. By using standard nonlinear analysis methods such as bifurcation diagrams, the largest Lyapunov exponent, phase portraits, Poincare sections, 0–1 test chart and the basin of attraction for an attractor, and the complex dynamical behaviors of the system are investigated in detail. Furthermore, a corresponding hardware electronic circuit is designed to verify the numerical simulations.

A chaos-based image encryption algorithm based on multiresolution singular value decomposition and a symmetric attractor

ABSTRACTThis paper proposes a new image cryptosystem using multiple chaotic maps and multiresolution singular value decomposition (MR-SVD). The encryption process starts with implementing the MR-SVD to decompose the original image into the four fundamental sub-bands, i.e., Approximation (A), Vertical (V), Horizontal (H), Diagonal (D) sub-bands respectively. Since the approximation part gives the most information about the image, this sub-band is selected to perform permutation and diffusion. The permutation of all the four sub-bands is done by deploying the Baker map. The diffusion of pixels in the permuted approximation part is done by our neighbourhood diffusion scheme that uses the numerical solution of the chaotic Thomas' cyclically symmetric attractor. The final cipher is obtained by combining all the four (A, H, V, D) partial ciphers by performing inverse MR-SVD. The experimental results of our proposed scheme on various benchmarks tests indicate that the algorithm is highly secure and can withstand various attacks.

Modeling Systems Based on the Assessment of Weak Symmetry Breaking in Reconstructed Attractors

The paper proposes to use the research method for chaotic processes based on the identification of weak symmetry breaking of the restored attractor. It is shown how the calculations results can be used to identify systems. An algorithm for creating finite-difference models has been developed, including: calculating by numerical methods the necessary conditions for the existence of chaos, reconstructing an attractor in a time series, searching for symmetric attractor fragments under conditions of weak symmetry breaking, determining the form of nonlinearities, and parametric identification. The result of the algorithm is a system of finite-difference equations in the state space Criteria for assessing symmetry breaking based on estimates of the divergence of fragments of phase trajectories are introduced. The results of modeling systems with chaotic dynamics are presented.