Predicting nonsmooth chaotic dynamics by reservoir computing.

Reservoir computing (RC) has been widely applied to predict the chaotic dynamics in many systems. Yet much broader areas related to nonsmooth dynamics have seldom been touched by the RC community which have great theoretical and practical importance. The generalization of RC to this kind of system is reported in this paper. The numerical work shows that the conventional RC with a hyperbolic tangent activation function is not able to predict the dynamics of nonsmooth systems very well, especially when reconstructing attractors (long-term prediction). A nonsmooth activation function with a piecewise nature is proposed. A kind of physics-informed RC scheme is established based on this activation function. The feasibility of this scheme has been proven by its successful application to the predictions of the short- and long-term (reconstructing chaotic attractor) dynamics of four nonsmooth systems with different complexity, including the tent map, piecewise linear map with a gap, both noninvertible and discontinuous compound circle maps, and Lozi map. The results show that RC with the new activation function is efficient and easy to run. It can make perfectly both short- and long-term predictions. The precision of reconstructing attractors depends on their complexity. This work reveals that, to make efficient predictions, the activation function of an RC approach should match the smooth or nonsmooth nature of the dynamical systems.

Predicting chaotic dynamics from incomplete input via reservoir computing with (D+1)-dimension input and output.

Predicting future evolution based on incomplete information of the past is still a challenge even though data-driven machine learning approaches have been successfully applied to forecast complex nonlinear dynamics. The widely adopted reservoir computing (RC) can hardly deal with this since it usually requires complete observations of the past. In this paper, a scheme of RC with (D+1)-dimension input and output (I/O) vectors is proposed to solve this problem, i.e., the incomplete input time series or dynamical trajectories of a system, in which certain portion of states are randomly removed. In this scheme, the I/O vectors coupled to the reservoir are changed to (D+1)-dimension, where the first D dimensions store the state vector as in the conventional RC, and the additional dimension is the corresponding time interval. We have successfully applied this approach to predict the future evolution of the logistic map and Lorenz, Rössler, and Kuramoto-Sivashinsky systems, where the inputs are the dynamical trajectories with missing data. The dropoff rate dependence of the valid prediction time (VPT) is analyzed. The results show that it can make forecasting with much longer VPT when the dropoff rate θ is lower. The reason for the failure at high θ is analyzed. The predictability of our RC is determined by the complexity of the dynamical systems involved. The more complex they are, the more difficult they are to predict. Perfect reconstructions of chaotic attractors are observed. This scheme is a pretty good generalization to RC and can treat input time series with regular and irregular time intervals. It is easy to use since it does not change the basic architecture of conventional RC. Furthermore, it can make multistep-ahead prediction just by changing the time interval in the output vector into a desired value, which is superior to conventional RC that can only do one-step-ahead forecasting based on complete regular input data.

Impact of time delays and environmental noise on the extinction of a population dynamics model.

ABSTRACT: In this paper, we examine a population model with carrying capacity, time delay, and sources of additive and multiplicative environmental noise. We find that time delay, noise sources and their correlation induce regime shifts and transitions between the population survival state and the extinction state. To explore the transition mechanism between these two states, we analyzed the shift time to extinction, or the delayed extinction time, of populations. The main finding is that the extinction transition time as a function of the noise intensity shows a maximum, indicating the existence of an appropriate noise intensity leading to a maximal delayed extinction. This nonmonotonic behavior, with a maximum, is a signature of the noise-enhanced stability phenomenon, observed in many physical and complex metastable systems. In particular, this maximum increases (or decreases) as the cross-correlation intensity or the delay time in the death process increases. Furthermore, the signal-to-noise ratio as a function of noise intensity shows a maximum, which is a signature of the stochastic resonance phenomenon in the population dynamics model investigated in the presence of time delay and environmental noise.

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.

Synchronous slowing down in coupled logistic maps via random network topology.

The speed and paths of synchronization play a key role in the function of a system, which has not received enough attention up to now. In this work, we study the synchronization process of coupled logistic maps that reveals the common features of low-dimensional dissipative systems. A slowing down of synchronization process is observed, which is a novel phenomenon. The result shows that there are two typical kinds of transient process before the system reaches complete synchronization, which is demonstrated by both the coupled multiple-period maps and the coupled multiple-band chaotic maps. When the coupling is weak, the evolution of the system is governed mainly by the local dynamic, i.e., the node states are attracted by the stable orbits or chaotic attractors of the single map and evolve toward the synchronized orbit in a less coherent way. When the coupling is strong, the node states evolve in a high coherent way toward the stable orbit on the synchronized manifold, where the collective dynamics dominates the evolution. In a mediate coupling strength, the interplay between the two paths is responsible for the slowing down. The existence of different synchronization paths is also proven by the finite-time Lyapunov exponent and its distribution.

Cyclic synchronous patterns in coupled discontinuous maps.

Cyclic collective behaviors are commonly observed in biological and neuronal systems, yet the dynamical origins remain unclear. Here, by models of coupled discontinuous map lattices, we investigate the cyclic collective behaviors by means of cluster synchronization. Specifically, we study the synchronization behaviors in lattices of coupled periodic piecewise-linear maps and find that in the nonsynchronous regime the maps can be synchronized into different clusters and, as the system evolves, the synchronous clusters compete with each other and present the recurring process of cluster expanding, shrinking, and switching, i.e., showing the cyclic synchronous patterns. The dynamical mechanisms of cyclic synchronous patterns are explored, and the crucial roles of basin distribution are revealed. Moreover, due to the discontinuity feature of the map, the cyclic patterns are found to be very sensitive to the system initial conditions and parameters, based on which we further propose an efficient method for controlling the cyclic synchronous patterns.

Surface enhanced fluorescence of physically polished nanostructured metal surface: the effect of the native oxide layer.

The fluorescence emission of Rhodamine 6G molecules at the physically polished nanostructured copper surface with varying thickness of the native oxide layer was investigated. The quenching effect was observed when the dye molecule was directly adsorbed onto the substrate surface without the formative oxide layer. However, the fluorescence was enhanced obviously when the native oxide layer was formed at the substrate surface. The experimental observations were discussed by taking into account the formation of the native oxide layer, the non-radiative energy transfer process and the local surface plasmon resonance at rough copper surface. The study highlights the importance of the native oxide layer formed on the metal substrate in surface enhanced fluorescence.

Hot electrons and electron-phonon coupling in a cylindrical nanoshell.

We use a standard model for the low-temperature electron-phonon interaction in metals to calculate the rate of thermal energy transfer between electrons and acoustic phonons in suspended metallic nanoshells. The electrons are treated as three-dimensional and noninteracting, whereas the vibrational modes are that of an thin cylindrical elastic shell of radius R with a free surface and thickness h. Disorder is neglected. The temperature dependence of the thermal power is obtained analytically for this model, and a crossover from the T3 dependence expected for one-dimensional phonons to a T3/(1 - v2) + 9gammaT4/[T*(1 - v2)(3/2)] dependence is obtained.