Lyapunov Time Research Articles

Observer Based Leader–Follower Bipartite Consensus With Intermittent Failures Using Lyapunov Functions and Time Scale Theory

This letter introduces a distributed observer-based protocol to achieve bipartite leader-follower consensus for linear multi-agent systems under intermittent failures. Under possible failures in the measurement and control input channels, the consensus protocol is based on intermittent interaction rules. Using the concept of time scales theory, the closed-loop system dynamics are converted to a switched system between continuous-time and discrete-time dynamics. Closed-loop stability is proved using Lyapunov functions and time scales theory. Numerical simulations illustrate the effectiveness of the designed distributed observer-based protocol proposed in this letter.

Orbital evolution of Mars-crossing asteroids

ABSTRACT This study is an orbital analysis of the interesting Mars-crossing asteroids (MCAs), also known as Mars-crosser (MC) asteroids or Mars-crossers (MCs). We computed that after 100 million years (Myr), approximately 66 ${{\ \rm per\ cent}}$ of all known MCs are ejected out of the Solar System by collision with the Sun, the planets, Ceres, Pallas, Vesta, or Hygiea. The rate of MC migration is high. Thus, this population of MCs would be supplied by just as many asteroids from outside the Solar System. We estimated the rate at which near-Earth objects were created from MCs throughout a 100 Myr period, with Atiras accounting for nearly 3 ${{\ \rm per\ cent}}$ of these objects, over 2 ${{\ \rm per\ cent}}$ were Atens, nearly 7.5 ${{\ \rm per\ cent}}$ were Apollos, approximately 9${{\ \rm per\ cent}}$ were Amors, and nearly 0.4 ${{\ \rm per\ cent}}$ became Centaurs. These results were calculated with 10 000 yr output intervals. Furthermore, 0.028${{\ \rm per\ cent}}$ of all the starting MCs were in retrograde orbits for at least 10 000 yr. We found that majority of the remaining MCs have migrated into the region of three asteroid families: Phocaea, Hungaria, and Flora. We calculated a small but significant influence of Ceres, Pallas, Vesta, and Hygiea on the orbital evolution of the MCs. From the AstDys catalogue, we found that the largest number of studied numbered MCs have their Lyapunov time (LT) in the range 2–4 kyr. Using the orbfit software, we computed the LT of selected MCs in retrograde orbits, and obtained an LT of between 540 yr (asteroid 2016 DR1) and 71 000 yr (asteroid 42887 1999 RV155).

Dynamic Analysis and Circuit Realization of a Novel No-Equilibrium 5D Memristive Hyperchaotic System with Hidden Extreme Multistability

In this paper, a novel no-equilibrium 5D memristive hyperchaotic system is proposed, which is achieved by introducing an ideal flux-controlled memristor model and two constant terms into an improved 4D self-excited hyperchaotic system. The system parameters-dependent and memristor initial conditions-dependent dynamical characteristics of the proposed memristive hyperchaotic system are investigated in terms of phase portrait, Lyapunov exponent spectrum, bifurcation diagram, Poincaré map, and time series. Then, the hidden dynamic attractors such as periodic, quasiperiodic, chaotic, and hyperchaotic attractors are found under the variation of its system parameters. Meanwhile, the most striking phenomena of hidden extreme multistability, transient hyperchaotic behavior, and offset boosting control are revealed for appropriate sets of the memristor and other initial conditions. Finally, a hardware electronic circuit is designed, and the experimental results are well consistent with the numerical simulations, which demonstrate the feasibility of this novel 5D memristive hyperchaotic system.

Bifurcation and chaos of the traveling membrane on oblique supports subjected to external excitation

This paper analyzes bifurcation and chaos of the traveling membrane on oblique supports subjected to external excitation. Through coordinate transformation, the Von Karman nonlinear plate theory was employed to derive the non-linear governing equations of membrane on oblique supports in terms of axial movement in the oblique coordinate system. The boundary conditions were also given in the oblique coordinate system. The approximate solution for the governing equations was found via the Galerkin method as well as the fourth-order Runge-Kutta numerical computing method. The nonlinear dynamic techniques including Lyapunov exponents diagrams, bifurcation plots, Poincare maps, phase trajectories, and time histories were introduced to analyze the impacts of the angles of oblique supports, aspect ratios and traveling velocities on dynamics responses and the various forms of vibration regarding the membrane system.

Physics-informed echo state networks

We propose a physics-informed echo state network (ESN) to predict the evolution of chaotic systems. Compared to conventional ESNs, the physics-informed ESNs are trained to solve supervised learning tasks while ensuring that their predictions do not violate physical laws. This is achieved by introducing an additional loss function during the training, which is based on the system's governing equations. The additional loss function penalizes non-physical predictions without the need of any additional training data. This approach is demonstrated on a chaotic Lorenz system and a truncation of the Charney–DeVore system. Compared to the conventional ESNs, the physics-informed ESNs improve the predictability horizon by about two Lyapunov times. This approach is also shown to be robust with regard to noise. The proposed framework shows the potential of using machine learning combined with prior physical knowledge to improve the time-accurate prediction of chaotic dynamical systems.

Deep learning of chaos classification

We train an artificial neural network which distinguishes chaotic and regular dynamics of the two-dimensional Chirikov standard map. We use finite length trajectories and compare the performance with traditional numerical methods which need to evaluate the Lyapunov exponent (LE). The neural network has superior performance for short periods with length down to 10 Lyapunov times on which the traditional LE computation is far from converging. We show the robustness of the neural network to varying control parameters, in particular we train with one set of control parameters, and successfully test in a complementary set. Furthermore, we use the neural network to successfully test the dynamics of discrete maps in different dimensions, e.g. the one-dimensional logistic map and a three-dimensional discrete version of the Lorenz system. Our results demonstrate that a convolutional neural network can be used as an excellent chaos indicator.

Quasi-integrable systems are slow to thermalize but may be good scramblers.

Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time-the clearest example being the Solar System-but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent, defined by the evolution of the four-point out-of-time-order correlator (OTOC), of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. (1) We find that in the semiclassical limit the quantum Lyapunov exponent is given by the classical one: it scales as ε^{1/3}, with ε being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is ∼ε^{-1}). (2) We also find that in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and (3) for sufficiently small perturbations the ε^{1/3} dependence is also suppressed-another purely quantum effect which we explain. These essential features of the problem are already present in a rotor that is kicked weakly but randomly. Concerning quantum limits on chaos, we find that quasi-integrable systems are relatively good scramblers in the sense that the ratio between the Lyapunov exponent and kT/ℏ may stay finite at a low temperature T.

Machine learning, alignment of covariant Lyapunov vectors, and predictability in Rikitake's geomagnetic dynamo model.

In this paper, the alignment of covariant Lyapunov vectors is used to train multi-layer perceptron ensembles in order to predict the duration of regimes in chaotic time series of Rikitake's geomagnetic dynamo model. The machine learning procedure reveals the relevance of the alignment of distinct covariant Lyapunov vectors for the predictions. To train multi-layer perceptron, we use a classification procedure that associates the number of maxima (or minima) inside regimes of motion with the duration of the corresponding regime. Remarkably accurate predictions are obtained, even for the longest regimes whose duration times are around 17.5 Lyapunov times. We also found long duration regimes with a distinctive statistical behavior, namely, the longest regimes are more likely to occur, a quite unusual behavior. In fact, we observed a largest regime above which no regimes were observed.

Controlling Rayleigh–Bénard convection via reinforcement learning

Thermal convection is ubiquitous in nature as well as in many industrial applications. The identification of effective control strategies to, e.g. suppress or enhance the convective heat exchange under fixed external thermal gradients is an outstanding fundamental and technological issue. In this work, we explore a novel approach, based on a state-of-the-art Reinforcement Learning (RL) algorithm, which is capable of significantly reducing the heat transport in a two-dimensional Rayleigh–Bénard system by applying small temperature fluctuations to the lower boundary of the system. By using numerical simulations, we show that our RL-based control is able to stabilise the conductive regime and bring the onset of convection up to a Rayleigh number , whereas state-of-the-art linear controllers have . Additionally, for , our approach outperforms other state-of-the-art control algorithms reducing the heat flux by a factor of about 2.5. In the last part of the manuscript, we address theoretical limits connected to controlling an unstable and chaotic dynamics as the one considered here. We show that controllability is hindered by observability and/or capabilities of actuating actions, which can be quantified in terms of characteristic time delays. When these delays become comparable with the Lyapunov time of the system, control becomes impossible.

Predicting phase and sensing phase coherence in chaotic systems with machine learning.

Recent interest in exploiting machine learning for model-free prediction of chaotic systems focused on the time evolution of the dynamical variables of the system as a whole, which include both amplitude and phase. In particular, in the framework based on reservoir computing, the prediction horizon as determined by the largest Lyapunov exponent is often short, typically about five or six Lyapunov times that contain approximately equal number of oscillation cycles of the system. There are situations in the real world where the phase information is important, such as the ups and downs of species populations in ecology, the polarity of a voltage variable in an electronic circuit, and the concentration of certain chemical above or below the average. Using classic chaotic oscillators and a chaotic food-web system from ecology as examples, we demonstrate that reservoir computing can be exploited for long-term prediction of the phase of chaotic oscillators. The typical prediction horizon can be orders of magnitude longer than that with predicting the entire variable, for which we provide a physical understanding. We also demonstrate that a properly designed reservoir computing machine can reliably sense phase synchronization between a pair of coupled chaotic oscillators with implications to the design of the parallel reservoir scheme for predicting large chaotic systems.

Complex bifurcation analysis and synchronization optimal control for Hindmarsh-Rose neuron model under magnetic flow effect.

In this contribution, the complex behaviour of the Hindmarsh-Rose neuron model under magnetic flow effect (mHR) is investigated in terms of bifurcation diagrams, Lyapunov exponent plots and time series when varying only the electromagnetic induction strength. Some exciting phenomena are found including, for instance, various firings patterns by applying appropriate magnetic strength and Hopf-fold bursting through fast-slow bifurcation. In addition to this, the interesting phenomenon of Hopf bifurcation is examined in the model. Thus, we prove that Hopf bifurcation occurs in this memristor-based HR neuron model when an appropriately chosen magnetic flux varies and reaches its critical value. Furthermore, one of the main results of this work was the optimal control approach to realize the synchronization of two mHR. The main advantage of the proposed optimal master-slave synchronization from a control point of view is that, in the practical application, the electrical activities (quiescent, bursting, spiking, period and chaos states) of a neuron can be regulated by a pacemaker (master) associated with biological neuron (slave) to treat some diseases such as epilepsy. A suitable electronic circuit is designed and used for the investigations. PSpice based simulation results confirm that the electrical activities and synchronization between coupled neurons can be modulated by electromagnetic flux.

Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: A case study with the Lorenz 96 model

A novel method, based on the combination of data assimilation and machine learning is introduced. The new hybrid approach is designed for a two-fold scope: (i) emulating hidden, possibly chaotic, dynamics and (ii) predicting their future states. The method consists in applying iteratively a data assimilation step, here an ensemble Kalman filter, and a neural network. Data assimilation is used to optimally combine a surrogate model with sparse noisy data. The output analysis is spatially complete and is used as a training set by the neural network to update the surrogate model. The two steps are then repeated iteratively. Numerical experiments have been carried out using the chaotic 40-variables Lorenz 96 model, proving both convergence and statistical skill of the proposed hybrid approach. The surrogate model shows short-term forecast skill up to two Lyapunov times, the retrieval of positive Lyapunov exponents as well as the more energetic frequencies of the power density spectrum. The sensitivity of the method to critical setup parameters is also presented: the forecast skill decreases smoothly with increased observational noise but drops abruptly if less than half of the model domain is observed. The successful synergy between data assimilation and machine learning, proven here with a low-dimensional system, encourages further investigation of such hybrids with more sophisticated dynamics.

Classification strategies in machine learning techniques predicting regime changes and durations in the Lorenz system

In this paper, we use machine learning strategies aiming to predict chaotic time series obtained from the Lorenz system. Such strategies prove to be successful in predicting the evolution of dynamical variables over a short period of time. Transitions between the regimes and their duration can be predicted with great accuracy by means of counting and classification strategies, for which we train multi-layer perceptron ensembles. Even for the longest regimes the occurrences and duration can be predicted. We also show the use of an echo state network to generate data of the time series with an accuracy of up to a few hundreds time steps. The ability of the classification technique to predict the regime duration of more than 11 oscillations corresponds to around 10 Lyapunov times.

An empirical assessment of the universality of ANNs to predict oscillatory time series

Artificial neural networks (ANNs) are universal function approximators, therefore suitable to be trained as predictors of oscillatory time series. Though several ANN architectures have been tested to predict both synthetic and real-world time series, the universality of their predictive power remained unexplored. Here we empirically test this universality across five well-known chaotic oscillators, limiting the analysis to the simplest architecture, namely multi-layer feed-forward ANN trained to predict one sampling step ahead. To compare different predictors, data are sampled according to their frequency content and the ANN structure scales with the characteristic dimensions of the oscillator. Moreover, the quality of recursive multi-step-ahead predictions are compared in terms of the system’s (largest) Lyapunov exponent (LLE), i.e., the predictive power is measured in terms of the number of Lyapunov times (LT, the LLE inverse) predicted within a prescribed (relative) error. The results confirm the rather uniform predictive power of the proposed ANN architecture.

Coordination and control of multi-channel supply chain driven by consumers’ channel preference and sales effort

Abstract The Stackelberg Dynamic Game Model for a multi-channel supply chain which consists of one manufacturer and two retailers has been analyzed in this research as manufacturer provides a single product to traditional retailer, online retailer and direct channel. The complexity of the multi-channel system of such is therefore analyzed under the aid of the methods including stability domain, bifurcation diagram, the largest Lyapunov exponent, attractor, and time series. The results indicate that a great adjustment of supply order quantity in direct channels, retailer's order quantity, consumers’ channel preference, and sales effort will cause the system to lose stability and trap into complexity. On condition that customer didn't prefer the channel, the speed to adjust the order quantity would influence less on the system. Otherwise, when consumers’ channel preference too strong, the players on supply chain players would not see a profit increase, no matter how hard they'd improved their sales effort. And, so will it exhibit such a meagre influence on the profits of the whole supply players chain. During the period when sales efforts being adjusted, retailers, as the rule followers, will be confronted by more limitations than the rule maker the manufacturer. Then, to further analyze, parameters adjustment control method is utilized to control the chaos of the multi-channel supply chain. Furthermore, to investigate the influence of the delay decision on the stability domain and the profits. The research findings of this study can provide a great reference for managers and decision-making process in their supply chain management.

Study of Dynamic Behaviors in a Spin Valve System Modeled by the Landau-Lifshitz-Slonczewski Equation

In this work, the dynamic behavior of a spin valve oscillator with a Nickel-free layer, modeled by the Landau-Lifshitz-Slonczewski equation is studied. It is considered a constant applied field and a spin current with two components, a constant term and a term with a time-dependent harmonic modulation. Techniques to characterize dynamic behaviors of systems, such as Lyapunov exponents, bifurcation diagram, phase portraits, time series, and Fourier spectra were used. It is demonstrated that the system presents multiple transitions between chaotic and regular states when the constant magnetic field, the magnitude, and frequency of the alternating current are varied. Furthermore, it is found that the effect of the magnetic field and the amplitude of the currents play a meaningful role in the chaotic behavior start.

Dynamical glass in weakly nonintegrable Klein-Gordon chains.

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a nonintegrable perturbation creates a coupling network in action space which can be short or long ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale T_{E} on which these distributions converge to δ distributions. We relate T_{E} to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ_{τ}^{+} dominating the means μ_{τ}^{+} and establish that T_{E}∼(σ_{τ}^{+})^{2}/μ_{τ}^{+}. The Lyapunov time T_{Λ} (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks T_{Λ}≈σ_{τ}^{+}, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a dynamical glass, where T_{E} grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies T_{Λ}≲μ_{τ}^{+}. This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of nonchaotic regions. These structures persist up to the ergodization time T_{E}.

Nonlinear Dynamics of Three-Neurons-Based Hopfield Neural Networks (HNNs): Remerging Feigenbaum Trees, Coexisting Bifurcations and Multiple Attractors

In this work, the dynamics of a simplified model of three-neurons-based Hopfield neural networks (HNNs) is investigated. The simplified model is obtained by removing the synaptic weight connection of the third and second neuron in the original Hopfield networks introduced in Ref. 11 . The investigations have shown that the simplified model possesses three equilibrium points among which origin of the systems coordinates. It is found that the origin is always unstable while the symmetric pair of fixed points with conditional stability has values depending on synaptic weight between the second and the first neuron that is used as bifurcation control parameter. Numerical simulations, carried out in terms of bifurcation diagrams, graph of Lyapunov exponents, phase portraits, Poincaré section, time series and frequency spectra are employed to highlight the complex dynamical behaviors exhibited by the model. The results indicate that the modified model of HNNs exhibits rich nonlinear dynamical behaviors including symmetry breaking, chaos, periodic window, antimonotonicity (i.e., concurrent creation and annihilation of periodic orbits) and coexisting self-excited attractors (e.g., coexistence of two, four and six disconnected periodic and chaotic attractors) which have not been reported in previous works focused on the dynamics of HNNs. Finally, PSpice simulations verify the results of theoretical analyses of the simplified model of three-neurons-based HNNs.

Time Recurrence Analysis of a Near Singular Billiard

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.

N-body chaos, phase-space transport and relaxation in numerical simulations

AbstractUsing direct N-body simulations of self-gravitating systems we study the dependence of dynamical chaos on the system size N. We find that the N-body chaos quantified in terms of the largest Lyapunov exponent Λmax decreases with N. The values of its inverse (the so-called Lyapunov time tλ) are found to be smaller than the two-body collisional relaxation time but larger than the typical violent relaxation time, thus suggesting the existence of another collective time scale connected to many-body chaos.