Spiral Attractor Research Articles

Mobile Robot Motion Behavior Based on Hopfield Neural Network Dynamics Under Electromagnetic Radiation

The nonlinear characteristic of activation function is a critical aspect that affects the performance of artificial neural networks (ANNs). Despite its importance, the impact of neuronal nonlinearity variations on the dynamics of ANNs has not been thoroughly studied in previous research. This paper aims to fill this gap by exploring the influence of nonlinearity changes in activation function due to memristive electromagnetic induction on the dynamical characteristics of ANNs. We first propose a memristor model with sinusoidal conductance function. Then, the effects on the biological neuron under the electromagnetic induction are simulated by replacing the gradient of hyperbolic tangent activation function with the conductance of the proposed memristor. Furthermore, a Hopfield neural network (HNN) is constructed by taking the nonlinearity changes of activation function into consideration and its dynamical behaviors are analyzed through Lyapunov exponent spectra, bifurcation diagram, biparameter-based dynamic maps and phase portraits. The numerical simulation results show complex dynamical behaviors such as single-scroll chaotic spiral attractor, double-scroll chaotic attractors and initial-value-dependent offset boosted attractors, demonstrating that nonlinearity changes of activation function induced by the electromagnetic induction have a significant impact on the dynamics of HNN. To further verify our findings, we implement the proposed HNN in hardware and find that the results are in line with our numerical simulations. Finally, we apply the chaotic HNN in the mobile robot path planning task, which achieves higher coverage rate than the random number-based random walk method.

Complex dynamic behaviors in a small network of three ring coupled Rayleigh-Duffing oscillators: Theoretical study and circuit simulation

This work focuses on the dynamics of a small network of three ring-coupled unidirectional Rayleigh-Duffing oscillators. The equations governing the Rayleigh-Duffing oscillator, containing a cubic term, make this study a more interesting and complex case to analyze. Coupling is achieved by perturbing the amplitude of each oscillator with a signal proportional to the amplitude of the other. The sixth-order self-driven nonlinear system obtained after coupling is analyzed, and presents up to twenty seven equilibrium points. Amongst these equilibrium points, we determined which can present the Hopf bifurcation. Also, the effects of the coupling coefficients and damping coefficients are analyzed. It is shown that varying these different coefficients leads to the appearance of extremely complex dynamic phenomena such as: instability and bifurcations (i.e coexistence of bifurcation branches), coexistence of up to fifteen attractors (heterogeneous multistability) and eight spiral chaotic attractor. The investigation of the coupled system is carried out by using to both analytical and numerical tools such as Hopf bifurcation theorem, the phase portraits, bifurcation diagrams, Lyapunov exponent diagram, frequency spectrum, to name but a few. The Routh-Hurwitz criterion is also used to analyze the stability of equilibrium points. We compute basins of attraction to highlight different zones corresponding to coexisting attractors. The implementation of an analog circuit of coupled Rayleigh-Duffing oscillators has enabled us to confirm the analytical and numerical results.

Spiral attractors in a reduced mean-field model of neuron-glial interaction.

This paper investigates various bifurcation scenarios of the appearance of bursting activity in the phenomenological mean-field model of neuron-glial interactions. In particular, we show that the homoclinic spiral attractors in this system can be the source of several types of bursting activity with different properties.

Arnold Tongue-Like Structures and Coexisting Attractors in the Memristive Muthuswamy–Chua–Ginoux Circuit Model

The three-dimensional Muthuswamy–Chua–Ginoux (MCG) circuit model is a generalization of the paradigmatic canonical Muthuswamy–Chua circuit, where a physical memristor assumes the role of a thermistor, and it is connected in series with a linear passive capacitor, a linear passive inductor, and a nonlinear resistor. The physical memristor presents an electrical resistance which is a function of temperature. Nowadays, the MCG circuit model has gained considerable attention due to the lack of extensive numerical explorations and their distinct dynamical properties, exemplified by phenomena such as the transition from torus breakdown to chaos, giving rise to a double spiral attractor associated to independent period-doubling cascades. In this contribution, the complex dynamics of the MCG circuit model is studied in terms of the Lyapunov exponents spectra, Kaplan–Yorke (KY) dimension, and the number of local maxima (LM) computed in one period of oscillation, as two parameters are simultaneously varied. Using the Lyapunov spectra to distinguish different dynamical regimes, KY dimension to estimate the attractors’ dimension, and the number of LM to characterize different periodic attractors, we construct high-resolution two-dimensional stability diagrams considering specific ranges of the parameter pairs [Formula: see text]. These parameters are associated with the inverse of the capacitance in the passive capacitor, and the heat capacitance and dissipation constant of the thermistor, respectively. Unexpectedly, we identify sequences of infinite self-organized generic stable periodic structures (SPSs) and Arnold tongues-like structures (ATSs) merged into chaotic dynamics domains, and the coexistence of different attracting sets (attractors) for the same parameter combinations and different initial conditions (multistability). We explore the multistable dynamics using the stability analysis and computation of Lyapunov coefficients, the inspection of the coexisting attractors, bifurcations diagrams, and basins of attraction. The periods of the ATSs and a particular sequence of shrimp-shaped SPSs obey specific generating and recurrence rules responsible for the bifurcation cascades. As the MCG circuit model has the crucial properties presented by the usual Muthuswamy–Chua circuit model, specific properties explored in our study should be helpful in real problems involving circuits with the presence of physical memristor playing the role of thermistors.

From coexisting attractors to multi-spiral chaos in a ring of three coupled excitation-free Duffing oscillators

The dynamics of a ring of three unidirectional coupled excitation-free double well Duffing oscillators is considered. The coupling technique followed in this work consists in the amplitude modulation of an oscillator with a proportion of the amplitude of its neighboring counterpart. The rate equation of this model is a sixth-order autonomous nonlinear odd symmetric system endowed with twenty seven equilibrium points eight of which undergo Hopf bifurcation. A detailed analysis of the model based on classic nonlinear analysis tools (e.g. bifurcation diagrams, phase portraits, basins of attraction) discloses attractive and interesting dynamical features such as eight parallel bifurcation threes, Hopf bifurcations, and various types of coexisting modes of oscillations (e.g. eight coexisting limit cycles, four competing double spiral chaotic attractors, two coexisting four-spiral chaotic attractors), and eight-spiral chaotic attractor, just to cite a few. The electronic circuit implementation of the three coupled Duffing oscillators system was executed based on simple available electronic components. The simulation study of the proposed analog circuit in PSpice matches well with the findings the theoretical study. Our results yield the conclusion that coupling excitation-free oscillators of Duffing type in a ring topology can be seen as a useful approach to obtain multi-spiral chaotic attractors.

Data-driven discovery of canonical large-scale brain dynamics.

Human behavior and cognitive function correlate with complex patterns of spatio-temporal brain dynamics, which can be simulated using computational models with different degrees of biophysical realism. We used a data-driven optimization algorithm to determine and classify the types of local dynamics that enable the reproduction of different observables derived from functional magnetic resonance recordings. The phase space analysis of the resulting equations revealed a predominance of stable spiral attractors, which optimized the similarity to the empirical data in terms of the synchronization, metastability, and functional connectivity dynamics. For stable limit cycles, departures from harmonic oscillations improved the fit in terms of functional connectivity dynamics. Eigenvalue analyses showed that proximity to a bifurcation improved the accuracy of the simulation for wakefulness, whereas deep sleep was associated with increased stability. Our results provide testable predictions that constrain the landscape of suitable biophysical models, while supporting noise-driven dynamics close to a bifurcation as a canonical mechanism underlying the complex fluctuations that characterize endogenous brain activity.

On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators.

We study chaotic dynamics in a system of four differential equations describing the interaction of five identical phase oscillators coupled via biharmonic function. We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of a three-dimensional Poincaré map for the system under consideration. We show that chaotic dynamics develop here near a codimension three bifurcation, when a periodic orbit (fixed point of the Poincaré map) has the triplet of multipliers ( 1 , 1 , 1 ). As it is known, the flow normal form for such bifurcation is the well-known three-dimensional Arneodó-Coullet-Spiegel-Tresser (ACST) system, which exhibits spiral attractors. According to this, we conclude that the additional zero Lyapunov exponent for orbits in the observed attractors appears due to the fact that the corresponding three-dimensional Poincaré map is very close to the time-shift map of the ACST-system.

Dynamics of a class of Chua’s oscillator with a smooth periodic nonlinearity: Occurrence of infinitely many attractors

In this article, a class of Chua’s system with smooth periodic nonlinear term is considered. This kind of systems exhibits strange dynamical behavior. Except for the existence of infinitely many equilibrium points, period-doubling bifurcation, and double-scroll attractors, this type of systems has strange dynamics different from the classical Chua’s system: (i) the coexistence of (infinitely) many non-chaotic strange attractors; (ii) the coexistence of (infinitely) many spiral chaotic attractors; (iii) the coexistence of multi-scroll attractors; (iv) “growing”-scroll attractors, where the number of scrolls is an increasing function with respect to the time variable.

Numerical Analysis of a Novel 3D Chaotic System with Period-Subtracting Structures

In this work, we present a novel three-dimensional chaotic system with only two cubic nonlinear terms. Dynamical behavior of the system reveals a period-subtracting bifurcation structure containing all [Formula: see text]th-order ([Formula: see text]) periods that are found in the dynamical evolution of the novel system concerning different values of parameters. The new system could be evolved into different states such as point attractor, limit cycle, strange attractor and butterfly strange attractor by changing the parameters. Also, the system is multistable, which implies another feature of a chaotic system known as the coexistence of numerous spiral attractors with one limit cycle under different initial values. Furthermore, bifurcation analysis reveals interesting phenomena such as period-doubling route to chaos, antimonotonicity, periodic solutions, and quasi-periodic motion. In the meantime, the existence of periodic solutions is confirmed via constructed Poincaré return maps. In addition, by studying the influence of system parameters on complexity, it is confirmed that the chaotic system has high spectral entropy. Numerical analysis indicates that the system has a wide variety of strong dynamics. Finally, a message coding application of the proposed system is developed based on periodic solutions, which indicates the importance of studying periodic solutions in dynamical systems.

Wild pseudohyperbolic attractor in a four-dimensional Lorenz system

We present an example of a new strange attractor which, as we show, belongs to a class of wild pseudohyperbolic spiral attractors. We find this attractor in a four-dimensional system of differential equations which can be represented as an extension of the Lorenz system.

Dynamics of Fourier Modes in Torus Generative Adversarial Networks

Generative Adversarial Networks (GANs) are powerful machine learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable, and typically, it is necessary to implement several accessory heuristics to the networks to reach acceptable convergence of the model. In this paper, we introduce a novel method to analyze the convergence and stability in the training of generative adversarial networks. For this purpose, we propose to decompose the objective function of the adversary min–max game defining a periodic GAN into its Fourier series. By studying the dynamics of the truncated Fourier series for the continuous alternating gradient descend algorithm, we are able to approximate the real flow and to identify the main features of the convergence of GAN. This approach is confirmed empirically by studying the training flow in a 2-parametric GAN, aiming to generate an unknown exponential distribution. As a by-product, we show that convergent orbits in GANs are small perturbations of periodic orbits so the Nash equillibria are spiral attractors. This theoretically justifies the slow and unstable training observed in GANs.

Cascade of torus birth bifurcations and inverse cascade of Shilnikov attractors merging at the threshold of hyperchaos.

We study the hyperchaos formation scenario in the modified Anishchenko-Astakhov generator. The scenario is connected with the existence of sequence of secondary torus bifurcations of resonant cycles preceding the hyperchaos emergence. This bifurcation cascade leads to the birth of the hierarchy of saddle-focus cycles with a two-dimensional unstable manifold as well as of saddle hyperchaotic sets resulting from the period-doubling cascades of unstable resonant cycles. Hyperchaos is born as a result of an inverse cascade of bifurcations of the emergence of discrete spiral Shilnikov attractors, accompanied by absorbing the cycles constituting this hierarchy.

On the Organization of Homoclinic Bifurcation Curves in Systems with Shilnikov Spiral Attractors

We study spiral chaos in the classical Rössler and Arneodo – Coullet – Tresser systems. Special attention is paid to the analysis of bifurcation curves that correspond to the appearance of Shilnikov homoclinic loop of saddle-focus equilibrium states and, as a result, spiral chaos. To visualize the results, we use numerical methods for constructing charts of the maximal Lyapunov exponent and bifurcation diagrams obtained using the MatCont package.

A physical memristor based Muthuswamy\u2013Chua\u2013Ginoux system

In 1976, Leon Chua showed that a thermistor can be modeled as a memristive device. Starting from this statement we designed a circuit that has four circuit elements: a linear passive inductor, a linear passive capacitor, a nonlinear resistor and a thermistor, that is, a nonlinear “locally active” memristor. Thus, the purpose of this work was to use a physical memristor, the thermistor, in a Muthuswamy–Chua chaotic system (circuit) instead of memristor emulators. Such circuit has been modeled by a new three-dimensional autonomous dynamical system exhibiting very particular properties such as the transition from torus breakdown to chaos. Then, mathematical analysis and detailed numerical investigations have enabled to establish that such a transition corresponds to the so-called route to Shilnikov spiral chaos but gives rise to a “double spiral attractor”.

On the classification of homoclinic attractors of three-dimensional flows

For three-dimensional dynamical systems with continuous time (flows), a classification of strange homoclinic attractors containing an unique saddle equilibrium state is constructed. The structure and properties of such attractors are determined by the triple of eigenvalues of the equilibrium state. The method of a saddle charts is used for the classification of homoclinic attractors. The essence of this method is in the construction of an extended bifurcation diagram for a wide class of three-dimensional flows (whose linearization matrix is written in the Frobenius form). Regions corresponding to different configurations of eigenvalues are marked in this extended bifurcation diagram. In the space of parameters defining the linear part of the considered class of three-dimensional flows bifurcation surfaces bounding 7 regions are constructed. One region corresponds to the stability of the equilibrium states while other 6 regions correspond to various homoclinic attractors of the following types: Shilnikov attractor, 2 types of spiral figure-eight attractors, Lorenz- like attractor, saddle Shilnikov attractor and attractor of Lyubimov-Zaks-Rovella. The paper also discusses questions related to the pseudohyperbolicity of homoclinic attractors of three-dimensional flows. It is proved that only homoclinic attractors of two types can be pseudohyperbolic: Lorenz-like attractors containing a saddle equilibrium with a two-dimensional stable manifold whose saddle value is positive and saddle Shilnikov attractors containing a saddle equilibrium state with a two-dimensional unstable manifold.

Effects of memristor-based coupling in the ensemble of FitzHugh–Nagumo elements

In this paper, we study the impact of electrical and memristor-based couplings on some neuron-like spiking regimes, previously observed in the ensemble of two identical FitzHugh-Nagumo elements with chemical excitatory coupling. We demonstrate how increasing strength of these couplings affects on such stable periodic regimes as spiking in-phase, anti-phase and sequential activity. We show that the presence of electrical and memristor-based coupling does not essentially affect regimes of in-phase activity. Such regimes do not changes remaining stable ones. However, it is not the case for regimes of anti-phase and sequential activity. All such regimes can transform into periodic or chaotic ones which are very similar to the regimes of in-phase activity. Concerning the regimes of sequential activity, this transformation depends continuously on the coupling parameters, whereas some anti-phase regimes can disappear via a saddle-node bifurcation and nearby orbits tend to regimes of in-phase activity. Also, we show that new interesting neuron-like phenomena can appear from the regimes of sequential activity when increasing the strength of electrical and/or memristor-based coupling. The corresponding regimes can be associated with the appearance of spiral attractors containing a saddle-focus equilibrium with homoclinic orbit and, thus, they correspond to chaotic motions near the invariant manifold of synchronization, which contains all in-phase limit cycles. Such new regimes can lead to the emergence of extreme events in the system of coupled neurons. In particular, the interspike intervals can become arbitrarily large when orbit pass very close to the saddle-focus. Finally, we show that the further increase in the strength of electrical coupling and/or memristor-based coupling leads to decreasing interspike intervals and, thus, it helps to avoid such extreme behavior.

Dynamical Effects of Neuron Activation Gradient on Hopfield Neural Network: Numerical Analyses and Hardware Experiments

Hyperbolic tangent function, a bounded monotone differentiable function, is usually taken as a neuron activation function, whose activation gradient, i.e. gain scaling parameter, can reflect the response speed in the neuronal electrical activities. However, the previously published literatures have not yet paid attention to the dynamical effects of the neuron activation gradient on Hopfield neural network (HNN). Taking the neuron activation gradient as an adjustable control parameter, dynamical behaviors with the variation of the control parameter are investigated through stability analyses of the equilibrium states, numerical analyses of the mathematical model, and experimental measurements on a hardware level. The results demonstrate that complex dynamical behaviors associated with the neuron activation gradient emerge in the HNN model, including coexisting limit cycle oscillations, coexisting chaotic spiral attractors, chaotic double scrolls, forward and reverse period-doubling cascades, and crisis scenarios, which are effectively confirmed by neuron activation gradient-dependent local attraction basins and parameter-space plots as well. Additionally, the experimentally measured results have nice consistency to numerical simulations.

Myopic control of neural dynamics.

Manipulating the dynamics of neural systems through targeted stimulation is a frontier of research and clinical neuroscience; however, the control schemes considered for neural systems are mismatched for the unique needs of manipulating neural dynamics. An appropriate control method should respect the variability in neural systems, incorporating moment to moment “input” to the neural dynamics and behaving based on the current neural state, irrespective of the past trajectory. We propose such a controller under a nonlinear state-space feedback framework that steers one dynamical system to function as through it were another dynamical system entirely. This “myopic” controller is formulated through a novel variant of a model reference control cost that manipulates dynamics in a short-sighted manner that only sets a target trajectory of a single time step into the future (hence its myopic nature), which omits the need to pre-calculate a rigid and computationally costly neural feedback control solution. To demonstrate the breadth of this control’s utility, two examples with distinctly different applications in neuroscience are studied. First, we show the myopic control’s utility to probe the causal link between dynamics and behavior for cognitive processes by transforming a winner-take-all decision-making system to operate as a robust neural integrator of evidence. Second, an unhealthy motor-like system containing an unwanted beta-oscillation spiral attractor is controlled to function as a healthy motor system, a relevant clinical example for neurological disorders.

Chaotic Dynamics and Multistability in the Nonholonomic Model of a Celtic Stone

We study dynamic properties of a Celtic stone moving along a plane. We consider two-parameter families of the corresponding nonholonomic models in which bifurcations leading to changing the types of stable motions of the stone, as well as the chaotic-dynamics onset are analyzed. It shown that the multistability phenomena are observed in such models when stable regimes various types (regular and chaotic) can coexist in the phase space of the system. We also show that chaotic dynamics of the nonholonomic model of a Celtic stone can be rather diverse. In this model, in the corresponding parameter regions, one can observe both spiral strange attractors various types, including the so-called discrete Shilnikov attractors, and mixed dynamics, when an attractor and a repeller intersect and almost coincide. A new scenario of instantaneous transition to the mixed dynamics as a result of the reversible bifurcation of merging of the stable and unstable limit cycles is found.

Dynamic reconstruction of chaotic system based on exponential weighted online sequential extreme learning machine with kernel

For the dynamic reconstruction of the chaotic dynamical system, a method of identifying an exponential weighted online sequential extreme learning machine with kernel(EW-KOSELM) is proposed. The kernel recursive least square (KRLS) algorithm is directly extended to an online sequential ELM framework, and weakens the effect of old data by introducing a forgetting factor. Meanwhile, the proposed algorithm can deal with the ever-increasing computational difficulties inherent in online kernel learning algorithms based on the ‘fixed-budget’ memory technique. The employed EW-KOSELM identification method is firstly applied to the numerical example of Duffing-Ueda oscillator for chaotic dynamical system based on simulated data, the qualitative and quantitative analysis for various validation tests of the dynamical properties of the original system as well as the identification model are carried out. A set of qualitative validation criteria is implemented by comparing chaotic attractors i.e. embedding trajectories, computing the corresponding Poincare mapping, plotting the bifurcation diagram, and plotting the steady-state trajectory i.e. the limit cycle between the original system and the identification model. Simultaneously, the quantitative validation criterion which includes computing the largest positive Lyapunov exponent and the correlation dimension of the chaotic attractors is also calculated to measure the closeness i.e. the approximation error between the original system and the identification model. The employed method is further applied to a practical implementation example of Chua's circuit based on the experimental data which are generated by sampling and recording the measured voltage across a capacitor, the inductor current from the double-scroll attractor, the measured voltage across a capacitor from the Chua's spiral attractor and an experimental time series from a chaotic circuit. The digital filtering technique is then used as a preprocessing approach, on the basis of wavelet denoising the measured data with lower signal-noise ratio (SNR) which can produce the double-scroll attractor or the spiral attractor, the reconstruction attractor of the identification model is compared with the reconstruction attractor from the experimental data for original system. The above experimental results confirm that the EW-FB-KOSELM identification method has a better performance of dynamic reconstruction, which can produce an accurate nonlinear model of process exhibiting chaotic dynamics. The identification model is dynamically equivalent or system approximation to the original system.