Asymmetric Attractors Research Articles

The effects of symmetry breaking on the dynamics of a simple autonomous jerk circuit

We investigate the dynamics of a simple jerk circuit where the symmetry is broken by forcing a dc voltage. The analysis shows that with a zero forcing dc voltage, the system displays a perfect symmetry and develops rich dynamics including period doubling, merging crisis, hysteresis, and coexisting multiple (up to six) symmetric attractors. In the presence of a non-zero forcing dc voltage, several unusual and striking nonlinear phenomena occur such as coexisting bifurcation branches, hysteresis, asymmetric double scroll strange attractors, and multiple coexisting asymmetric attractors for some appropriate sets of system parameters. In the latter case, different combinations of attractors are depicted consisting for instance of two, three, four, or five disconnected periodic and chaotic attractors depending solely on the choice of initial conditions. The investigations are carried out by using standard nonlinear analysis tools such as Lyapunov exponent plots, bifurcation diagrams, basins of attraction, and phase space trajectory plots. The theoretical results are checked experimentally and a very good agreement is found between theory and experiment.

Relaxation Oscillations in a Nonsmooth Oscillator with Slow-Varying External Excitation

The main purpose of the paper is to explore the influence of the coupling of two scales on the dynamics of a nonsmooth dynamical system. Based on a typical Chua’s circuit, by introducing a nonlinear resistor with piecewise characteristics as well as a harmonically changed electric source, a modified nonsmooth model is established, in which the coupling of two scales in frequency domain exists. Different types of bursting oscillations, appearing in the combination of large-amplitude oscillations, called spiking oscillations ([Formula: see text]), and small-amplitude oscillations or at rest, denoted by quiescent states ([Formula: see text]), can be observed with the variation of the exciting amplitude. When the exciting frequency is relatively small, by regarding the whole exciting term as a slow-varying parameter, the original system can be transformed into a generalized autonomous system. The phase space can be divided into three regions by the nonsmooth boundaries, in which the trajectory is governed by three different subsystems, respectively. Based on the analysis of the three subsystems as well as the behaviors on the nonsmooth boundaries, all the equilibrium branches and their bifurcations can be obtained, which can be employed to investigate the mechanism of the bursting oscillations. It is found that, for relatively small exciting amplitude, since no bifurcation on the equilibrium branches can be realized with the variation of the slow-varying parameter, the system behaves in periodic movement, which may evolve to bursting oscillations when a pair of fold bifurcations occurs with the increase of the exciting amplitude. Further increase of the exciting amplitude may lead to more complicated bursting oscillations, which may bifurcate into two coexisted asymmetric bursting attractors via symmetric breaking. Interaction between the two attractors may result in an enlarged symmetric bursting attractor, in which more forms of bifurcations at the transitions between the quiescent states and repetitive spiking states can be observed.

Coexistence of multiple attractors in the tree dynamics

This paper considers a specific plant (pinus family) and examines its complex behavior under the air flow while checking the real-time dynamics of the proposed analog electronic simulator with an active RC realization. It appears that the system can be chaotic and its dynamics depend on the chosen initial conditions. We show the coexistence of multiple attractors in the system and observe that their occurrence makes its chaotic character less robust. We also establish through the basin of attraction that the region of mixed-mode oscillations can be extended by increasing values of the wind amplitude ratio. Furthermore, the isospike diagram is introduced to instantly inform how the dynamics of the plant moves from periodic to chaotic motion as the main parameters of the wind augment all together. Our experimental searches yield results that are in perfect agreement with the numerical outcomes established via Matlab and Pspice environments. Those experimental surveys also display the coexistence of asymmetric and symmetric attractors that confirms the complex behavior of the plants subjected to the wind loads.

A minimal three-term chaotic flow with coexisting routes to chaos, multiple solutions, and its analog circuit realization

We introduce a unique chaotic flow with three terms expressed by $$ \dddot x + abx = d \sinh \left( {\dot{x} + c\ddot{x}} \right) $$. This minimal Jerk model is presented as a novel 3-D chaotic system consisting uniquely of two linear terms and single hyperbolic sine nonlinearity. The presence of coexisting routes to chaos in the numerical studies justifies the appearance of multiple solutions in some ranges of parameters. For instance, up to six different coexisting solutions (a pair of period-5 limit cycles and two pairs of chaotic attractors) have been tracked. Furthermore, the generalized form of the introduced Jerk system is synthesized with a parametric nonlinearity of the form $$ \phi_{k} \left( X \right) = 0.5\left( {exp\left( {k\left( {\dot{x} + c\ddot{x}} \right)} \right) - exp\left( { - \left( {\dot{x} + c\ddot{x}} \right)} \right)} \right) $$. Based on this generalized form, and judiciously adjusting the parameter $$ k $$, an in-depth description of the dynamics of the system at the symmetry boundaries is carried out. Hysteresis and parallel bifurcation behaviors reflect the presence of multiple asymmetric solutions (e.g. the coexistence of four asymmetric attractors) in the new model. It should be mention that the presence of six coexisting attractors reported in this work is rarely shown in third-order systems and therefore represents an enriching contribution to the study of dynamic systems in general. Finally, some PSpice simulations and laboratory tests of the proposed circuit are included.

Effects of symmetric and asymmetric nonlinearity on the dynamics of a novel chaotic jerk circuit: Coexisting multiple attractors, period doubling reversals, crisis, and offset boosting

Abstract A novel self-driven RC chaotic jerk circuit with the singular feature of having a smoothly adjustable nonlinearity and symmetry is proposed and investigated. The novel chaotic circuit is mathematically modeled by a third order system with a single nonlinear term in the form ϕ k ( x ) = 0.5 ( exp ( k x ) − exp ( − x ) ) where parameter k maps a smoothly adjustable control resistor. Obviously for k = 1 , the system is point symmetric with respect to the origin of the system coordinates since the nonlinear term reduces to the hyperbolic sine function. The case k ≠ 1 corresponds to a non-symmetric system. The numerical experiment reveals a plethora of events including period doubling route to chaos, hysteresis, periodic windows, asymmetric double scroll chaos, symmetric double scroll chaos, and coexisting bifurcations branches as well. This latter phenomenon induces multiple coexisting attractors consisting of two, three, four, five, or six disconnected symmetric or asymmetric attractors for the same set of parameter values when monitoring solely the initial conditions. Laboratory experimental measurements are carried out to confirm the theoretical predictions.

Bipolar Pulse-Induced Coexisting Firing Patterns in Two-Dimensional Hindmarsh–Rose Neuron Model

In this paper, a bipolar pulse (BP) current is taken to mimic a periodic stimulus effect on the membrane potential in the axon of a neuron. By introducing the BP current to substitute the externally applied constant current, a BP-forced two-dimensional Hindmarsh–Rose (HR) neuron model is proposed. Based on the proposed neuron model, the BP-switched equilibrium point and its stability evolution with the periodic variation in time are explored. Furthermore, coexisting asymmetric attractors (or coexisting firing patterns) with bistability are revealed by phase plane orbits, time sequences, and attraction basins, as well as the BP-induced coexisting asymmetric attractors’ behaviors are then elaborated through bifurcation analysis. The research results exhibit that, with the increase of the time, the stabilities of the neuron model are continually switched between an unstable node-focus and a stable point, resulting in the coexisting behaviors of numerous asymmetric attractors under the specified initials. Consequently, the newly introduced BP current stimulus, instead of the original constant current stimulus, allows the two-dimensional HR neuron model to possess complex dynamical behaviors for the membrane potential. Additionally, a hardware breadboard is fabricated and circuit experiments are carried out to validate the numerical simulations.

Scenario to chaos and multistability in a modified Coullet system: effects of broken symmetry

Recently, the study of nonlinear systems with multiple attractors has drained the attention of researchers worldwide owing to its fundamental and technological relevance. In this work, we consider a modified Coullet system with a cubic polynomial nonlinearity in the form $$ \varphi_{m} \left( x \right) = x - mx^{2} - x^{3} $$ where $$ m $$ is a control parameter. The new system bridges the gap between chaotic systems with symmetry and those without symmetry. In fact for $$ m = 0 $$ , the system displays a perfect symmetry which is reflected on the location of equilibria, the type of attractors, and the shape of basins of attraction as well. In this special case, multistability involves a pair of mutually symmetric periodic or chaotic attractors, a pair of chaotic attractors with a pair of limit cycles, two pairs of limit cycles with a pair of chaotic attractors, and so on. For $$ m \ne 0 $$ , the system is non symmetric and more complex nonlinear phenomena are observed such as parallel bifurcation branches, coexisting multiple asymmetric attractors, and hysteresis. For instance, the coexistence of a stable fixed point with a limit cycle, two different non symmetric limit cycles or chaotic attractors, three or four different non symmetric periodic and chaotic attractors are reported when monitoring the system parameters and initial conditions as well. Basins of attraction with non-trivial basin boundaries structures are computed to visualize the magnetization of the state space in the presence of multiple attractors. Experimental results based on a suitable electronic implementation of the proposed jerk system are included.

The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems

The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems

Complex dynamics of a 4D Hopfield neural networks (HNNs) with a nonlinear synaptic weight: Coexistence of multiple attractors and remerging Feigenbaum trees

This contribution investigates the nonlinear dynamics of a model of a 4D Hopfield neural networks (HNNs) with a nonlinear synaptic weight. The investigations show that the proposed HNNs model possesses three equilibrium points (the origin and two nonzero equilibrium points) which are always unstable for the set of synaptic weights matrix used to analyze the equilibria stability. Numerical simulations, carried out in terms of bifurcation diagrams, Lyapunov exponents graph, phase portraits and frequency spectra, are used to highlight the rich and complex phenomena exhibited by the model. These rich nonlinear dynamic behaviors include period doubling bifurcation, chaos, periodic window, antimonotonicity (i.e. concurrent creation and annihilation of periodic orbits) and coexistence of asymmetric self-excited attractors (e.g. coexistence of two and three disconnected periodic and chaotic attractors). Finally, PSpice simulations are used to confirm the results of the theoretical analysis.

A new oscillator with infinite coexisting asymmetric attractors

In this paper, we introduce a new two dimensional nonlinear oscillator with infinite number of coexisting asymmetric limit cycles and stable equilibria. Some of these attractors are self-excited, while some others are hidden attractors. Adding a forcing term to this new system, we introduce a new chaotic system with infinite number of coexisting asymmetric strange attractor, torus and limit cycles, depending on the parameters.

Dynamic analysis of a unique jerk system with a smoothly adjustable symmetry and nonlinearity: Reversals of period doubling, offset boosting and coexisting bifurcations

We investigated the dynamics of a novel jerk system generalized with a single parametric nonlinearity in the form φk(x)=0.5(exp(kx)-exp(-x)). The form of nonlinearity is 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 shape of 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 symmetric about the origin. For k≠1, the system is non-symmetric. To the best of the authors’ knowledge, this interesting feature is unique and has not yet been discussed in chaotic systems. The dynamic analysis of the model involves a preliminary study of basic properties such as the nature of the fixed point, the transitions to chaos (bifurcation diagrams), the phase portraits as well as the Lyapunov exponent diagrams. When monitoring the system parameters, some striking phenomena are found including period doubling cascade bifurcation, reverse bifurcations, merging crises, offset boosting, coexisting bifurcations and hysteresis. Several windows in the parameters’ space are depicted in which the novel jerk system displays a plethora of coexisting asymmetric and symmetric attractors (i.e. coexistence of two, three or four different attractors) depending only on the selection of initial states. The magnetization of state space due to the presence of multiple competing solutions is illustrated by means of basins of attraction. More importantly, multistability in the symmetry boundary is discussed by monitoring the bifurcation parameter k. Laboratory experimental results based on the suitably designed electronic analogue of the model confirm the numerical findings.

Three‐Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Since the electrical activities of neurons are closely related to complex electrophysiological environment in neuronal system, a novel three‐dimensional memristive Hindmarsh–Rose (HR) neuron model is presented in this paper to describe complex dynamics of neuronal activities with electromagnetic induction. The proposed memristive HR neuron model has no equilibrium point but can show hidden dynamical behaviors of coexisting asymmetric attractors, which has not been reported in the previous references for the HR neuron model. Mathematical model based numerical simulations for hidden coexisting asymmetric attractors are performed by bifurcation analyses, phase portraits, attraction basins, and dynamical maps, which just demonstrate the occurrence of complex dynamical behaviors of electrical activities in neuron with electromagnetic induction. Additionally, circuit breadboard based experimental results well confirm the numerical simulations.

Coexisting Behaviors of Asymmetric Attractors in Hyperbolic-Type Memristor based Hopfield Neural Network.

A new hyperbolic-type memristor emulator is presented and its frequency-dependent pinched hysteresis loops are analyzed by numerical simulations and confirmed by hardware experiments. Based on the emulator, a novel hyperbolic-type memristor based 3-neuron Hopfield neural network (HNN) is proposed, which is achieved through substituting one coupling-connection weight with a memristive synaptic weight. It is numerically shown that the memristive HNN has a dynamical transition from chaotic, to periodic, and further to stable point behaviors with the variations of the memristor inner parameter, implying the stabilization effect of the hyperbolic-type memristor on the chaotic HNN. Of particular interest, it should be highly stressed that for different memristor inner parameters, different coexisting behaviors of asymmetric attractors are emerged under different initial conditions, leading to the existence of multistable oscillation states in the memristive HNN. Furthermore, by using commercial discrete components, a nonlinear circuit is designed and PSPICE circuit simulations and hardware experiments are performed. The results simulated and captured from the realization circuit are consistent with numerical simulations, which well verify the facticity of coexisting asymmetric attractors' behaviors.

Coexistence of asymmetric hidden chaotic attractors in a new simple 4-D chaotic system with curve of equilibria

This paper reports a new 4-D dissipative chaotic system. When compared with the available 4-D chaotic/hyperchaotic systems, the unique properties of the proposed system are (i) the system exhibits coexistence of different kind of asymmetric hidden attractors, (ii) the system has a curve of equilibria and (ii) the system is simple. This system does not satisfy Shilnikov criterion for the existence of chaos. The system has a total of eight terms including only one nonlinear term. The coexistence of rich chaotic dynamics in the system is investigated through Lyapunov spectrum, bifurcation diagram, Poincaré map, frequency spectrum and attractors plot. The coexistence of the asymmetric hidden dynamics is shown using chaotic attractors, 3- torus and 2-torus plots. The system is developed from the type-IV 3-D Rossler chaotic system.

Periodicity, chaos, and multiple attractors in a memristor-based Shinriki's circuit.

In this contribution, a novel memristor-based oscillator, obtained from Shinriki's circuit by substituting the nonlinear positive conductance with a first order memristive diode bridge, is introduced. The model is described by a continuous time four-dimensional autonomous system with smooth nonlinearities. The basic dynamical properties of the system are investigated including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponents' spectrum. It is found that in addition to the classical period-doubling and symmetry restoring crisis scenarios reported in the original circuit, the memristor-based oscillator experiences the unusual and striking feature of multiple attractors (i.e., coexistence of a pair of asymmetric periodic attractors with a pair of asymmetric chaotic ones) over a broad range of circuit parameters. Results of theoretical analyses are verified by laboratory experimental measurements.

Systematic template extraction from chaotic attractors: II. Genus-one attractors with multiple unimodal folding mechanisms

Asymmetric and symmetric chaotic attractors produced by the simplest jerk equivariant system are topologically characterized. In the case of this system with an inversion symmetry, it is shown that symmetric attractors bounded by genus-one tori are conveniently analyzed using a two-components Poincaré section. Resulting from a merging attractor crisis, these attractors can be easily described as being made of two folding mechanisms (here described as mixers), one for each of the two attractors co-existing before the crisis: symmetric attractors are thus described by a template made of two mixers. We thus developed a procedure for concatenating two mixers (here associated with unimodal maps) into a single one, allowing the description of a reduced template, that is, a template simplified under an isotopy. The so-obtained reduced template is associated with a description of symmetric attractors based on one-component Poincaré section as suggested by the corresponding genus-one bounding torus. It is shown that several reduced templates can be obtained depending on the choice of the retained one-component Poincaré section.

How to compress sequential memory patterns into periodic oscillations: general reduction rules.

A neural network with symmetric reciprocal connections always admits a Lyapunov function, whose minima correspond to the memory states stored in the network. Networks with suitable asymmetric connections can store and retrieve a sequence of memory patterns, but the dynamics of these networks cannot be characterized as readily as that of the symmetric networks due to the lack of established general methods. Here, a reduction method is developed for a class of asymmetric attractor networks that store sequences of activity patterns as associative memories, as in a Hopfield network. The method projects the original activity pattern of the network to a low-dimensional space such that sequential memory retrievals in the original network correspond to periodic oscillations in the reduced system. The reduced system is self-contained and provides quantitative information about the stability and speed of sequential memory retrieval in the original network. The time evolution of the overlaps between the network state and the stored memory patterns can also be determined from extended reduced systems. The reduction procedure can be summarized by a few reduction rules, which are applied to several network models, including coupled networks and networks with time-delayed connections, and the analytical solutions of the reduced systems are confirmed by numerical simulations of the original networks. Finally, a local learning rule that provides an approximation to the connection weights involving the pseudoinverse is also presented.

Compound multiattractor built around asymmetric chaotic attractors

A technique for constructing chaotic multiattractors combining several copies of an initial asymmetric chaotic attractor is presented.

Modeling the Forming of Public Opinion: An approach from Sociophysics

This paper reviews a sociophysics two-state model for opinion forming that has proven heuristic power. The dynamics are driven by repeated small-group discussions; within each group, a local majority rule is applied to update the opinions of agents. Iterating the dynamics leads towards one of two opposite attractors at which every agent shares the same opinion. The successful attractor is a function of the initial support with respect to a certain threshold, the value of which depends on the size distribution of the local update groups. While odd-sized groups yield a threshold at fifty percent, even-sized groups, which allow the inclusion of doubt in the case of an opinion tie, produce a threshold shift toward either one of the two attractors, giving rise to minority opinion spreading. In addition, agents can be heterogeneous in their cognitive nature, obeying different rules to update their opinion. While floater agents are open to changing their mind, contrarians chose to oppose whatever opinion was held by the majority of agents in their vicinity, and inflexibles never change their mind. Contrarians and inflexibles have drastic and counter-intuitive effects on the opinion dynamics. Beyond certain critical proportions, contrarians trigger an upside change of the dynamics, making it threshold-less with only one attractor at precisely 50/50 regardless of the initial conditions. Inflexibles produce the same threshold-less dynamics, except with an asymmetric single attractor that favors a specific opinion, even when they start with very low support. The results are used to shed new and unexpected light on controversial issues such as global warming.

Multi-stability and basin crisis in synchronized parametrically driven oscillators

This paper studies the synchronization dynamics of two linearly coupled parametrically excited oscillators. The Lyapunov stability theory is employed to obtain some sufficient algebraic criteria for global asymptotic stability of the synchronization of the systems, and an estimated critical coupling, k cr, for which synchronization could be observed is determined. The synchronization transition is found to be associated with the boundary crisis of the chaotic attractor. In the bistable states, where two asymmetric T-periodic attractors co-exist, we show that the coupled oscillators can attain multi-stability via a new dynamical transition—the basin crisis wherein two co-existing attractors are destroyed while new co-existing attractors are created. The stability of the steady states is examined and the possible bifurcation routes identified.