Well-known Dynamical Systems Research Articles

Modeling Ordinal First Return Dynamics from Time Series

We examine the family of ordinal First Return Maps (FRMs) of a dynamical system time series to model the system’s dynamics. We consider two distinct modeling approaches using the same modeling method and compare their outcomes to ascertain the prerequisites for each to function and which approach is more effective in different circumstances. The first approach involves short-term modeling of the entire flow, followed by identifying the ordinal sections on the predicted flow. The second approach relies on locating the ordinal sections within the reference time series and using the data from each section independently to build a model that can produce a long-term prediction for each section’s FRM. We demonstrate the effectiveness of these approaches on two well-known dynamical systems, the Lorenz and Rössler equations, emulating data deficiency by selecting a low sampling rate. We consider the effect of downsampling on the shape of the FRMs to choose suitable ordinal sequences to use in creating Poincaré sections from the time series. Following that, we propose a dissimilarity measure between the reference and predicted FRMs to assess the model quality. We then compare the reference and predicted FRMs using the introduced dissimilarity criterion to determine which modeling approach performs better in each situation. Ultimately, we classify the FRMs situation into continuous and piecewise continuous scenarios. Based on our observations from the Lorenz and Rössler dynamical systems, we found that when the system’s FRMs are piecewise continuous, it is more effective to model the entire flow. However, when the FRMs are continuous, our choice of modeling approach depends on the criterion that the predicted FRM is more similar to the reference.

Characterizing the spike timing of a chaotic laser by using ordinal analysis and machine learning.

Semiconductor lasers with optical feedback are well-known nonlinear dynamical systems. Under appropriate feedback conditions, these lasers emit optical pulses that resemble neural spikes. Influenced by feedback delay and various noise sources, including quantum spontaneous emission noise, the dynamics are highly stochastic. A good understanding of the spike timing statistics is needed to develop photonic systems capable of using the fast-spiking laser output for novel applications, such as information processing or random number generation. Here we analyze experimental sequences of inter-spike intervals (ISIs) recorded when a sinusoidal signal was applied to the laser current. Different combinations of the DC value and frequency of the signal applied to the laser lead to ISI sequences with distinct statistical properties. This variability prompts an investigation into the relationship between experimental parameters and ISI sequence statistics, aiming to uncover potential encoding methods for optical spikes, since this can open a new way of encoding and decoding information in sequences of optical spikes. By using ordinal analysis and machine learning, we show that the ISI sequences have statistical ordinal properties that are similar to Flicker noise signals, characterized by a parameter α that varies with the signal that was applied to the laser current when the ISIs were recorded. We also show that for this dataset, the (α, permutation entropy) plane is more informative than the (complexity, permutation entropy) plane because it allows better differentiation of ISI sequences recorded under different experimental conditions, as well as better differentiation of original and surrogate ISI sequences.

A Note on a Fractional Extension of the Lotka–Volterra Model Using the Rabotnov Exponential Kernel

In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka–Volterra model. This mathematical model describes the dynamics of a predator and prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) kernel. In this work, we derive an approximate formula of the fractional derivative of a power function ζp in terms of the RFE kernel. Next, by using the spectral collocation method (SCM) based on the shifted Vieta–Lucas polynomials (VLPs), the fractional differential system is reduced to a set of algebraic equations. We provide a theoretical convergence analysis for the numerical approach, and the accuracy is verified by evaluating the residual error function through some concrete examples. The results are then contrasted with those derived using the fourth-order Runge-Kutta (RK4) method.

Sparse identification of dynamical systems by reweighted [formula omitted]-regularized least absolute deviation regression

This work proposes a generalized methodology for sparse identification of dynamical systems (SID), utilizing reweighted l1-regularized least absolute deviation (LAD) regression to recover the governing equation of dynamical systems. To programmatically implement the proposed method, a stabilized and efficient resolution scheme is developed by combining the solution method of the classical LAD-based least absolute shrinkage and selection operator (LAD-lasso) and the threshold iterative idea of the sequentially thresholded least-squares (STLS) method. The new method relaxes the assumption of normality for the error terms in the linear regression model employed by traditional SID techniques. Additionally, it overcomes the low identification accuracy deficiency of the recently proposed regularized least absolute deviation-based sparse identification of dynamics method (RLAD-SID) under some poor data conditions. The performance of the proposed approach is investigated by simulations of two well-known dynamical systems, that is, the Lorenz system and the Duffing system. The reliability of this approach in recovering the governing equation of dynamical systems from noise-contaminated datasets is firstly demonstrated. Furthermore, the advantage of the proposed method is illustrated by comparing it with four traditional techniques in relatively poor data environments. Finally, we display the robustness of the new method when the time derivative dataset is affected by different types of noise, and reveal its superiority compared to the traditional RLAD-SID.

Ordinal Poincaré sections: Reconstructing the first return map from an ordinal segmentation of time series.

We propose a robust algorithm for constructing first return maps of dynamical systems from time series without the need for embedding. A first return map is typically constructed using a convenient heuristic (maxima or zero-crossings of the time series, for example) or a computationally nuanced geometric approach (explicitly constructing a Poincaré section from a hyper-surface normal to the flow and then interpolating to determine intersections with trajectories). Our method is based on ordinal partitions of the time series, and the first return map is constructed from successive intersections with specific ordinal sequences. We can obtain distinct first return maps for each ordinal sequence in general. We define entropy-based measures to guide our selection of the ordinal sequence for a "good" first return map and show that this method can robustly be applied to time series from classical chaotic systems to extract the underlying first return map dynamics. The results are shown for several well-known dynamical systems (Lorenz, Rössler, and Mackey-Glass in chaotic regimes).

Nonequilibrium dynamics in a three-state opinion-formation model with stochastic extreme switches.

We investigate the nonequilibrium dynamics of a three-state kinetic exchange model of opinion formation, where switches between extreme states are possible, depending on the value of a parameter q. The mean field dynamical equationsare derived and analyzed for any q. The fate of the system under the evolutionary rules used in S. Biswas etal. [Physica A 391, 3257 (2012)0378-437110.1016/j.physa.2012.01.046] shows that it is dependent on the value of q and the initial state in general. For q=1, which allows the extreme switches maximally, a quasiconservation in the dynamics is obtained which renders it equivalent to the voter model. For general q values, a "frozen" disordered fixed point is obtained which acts as an attractor for all initially disordered states. For other initial states, the order parameter grows with time t as exp[α(q)t] where α=1-q/3-q for q≠1 and follows a power law behavior for q=1. Numerical simulations using a fully connected agent-based model provide additional results like the system size dependence of the exit probability and consensus times that further accentuate the different behavior of the model for q=1 and q≠1. The results are compared with the nonequilibrium phenomena in other well-known dynamical systems.

Data-Driven Discovery of Stochastic Differential Equations

Stochastic differential equations (SDEs) are mathematical models that are widely used to describe complex processes or phenomena perturbed by random noise from different sources. The identification of SDEs governing a system is often a challenge because of the inherent strong stochasticity of data and the complexity of the system’s dynamics. The practical utility of existing parametric approaches for identifying SDEs is usually limited by insufficient data resources. This study presents a novel framework for identifying SDEs by leveraging the sparse Bayesian learning (SBL) technique to search for a parsimonious, yet physically necessary representation from the space of candidate basis functions. More importantly, we use the analytical tractability of SBL to develop an efficient way to formulate the linear regression problem for the discovery of SDEs that requires considerably less time-series data. The effectiveness of the proposed framework is demonstrated using real data on stock and oil prices, bearing variation, and wind speed, as well as simulated data on well-known stochastic dynamical systems, including the generalized Wiener process and Langevin equation. This framework aims to assist specialists in extracting stochastic mathematical models from random phenomena in the natural sciences, economics, and engineering fields for analysis, prediction, and decision making.

Direct statistical simulation of the Lorenz63 system.

We use direct statistical simulation to find the low-order statistics of the well-known dynamical system, the Lorenz63 model. Instead of accumulating statistics from numerical simulation of the dynamical system or solving the Fokker-Planck equation for the full probability distribution of the dynamical system, we directly solve the equations of motion for the low-order statistics after closing them by making several different choices for the truncation. Fixed points of the statistics are obtained either by time evolving or by iterative methods. The stability and statistical realizability of the fixed points of the statistics are analyzed, and the statistics so obtained are compared to those found by the traditional approach. Low-order statistics of the chaotic Lorenz63 system can be obtained from cumulant expansions more efficiently than by accumulation via direct numerical simulation or by solution of the Fokker-Planck equation.

Analysis and pattern formation scenarios in the superdiffusive system of predation described with Caputo operator

The concept of a fractional derivative is introduced to the predator-prey system to describe the species anomalous superdiffusive process. To achieve this, a new class of predator-prey model with the Beddington-DeAngelis functional response is formulated in the sense of the Caputo fractional order operator. This work aims to give a mathematical basis for computational studies of a two-variable fractional reaction-diffusion system in one and two dimensions from biological and numerical perspectives. As a result, some details of the local and global dynamics of the reaction-diffusion system are provided by using the idea of the linear stability analysis and well-known dynamical systems theory to derive conditions on the parameters which can guarantee biologically meaningful equilibria also serve as a guide in ensuring the correct choice of parameters when numerically experimenting with the solutions of the full fractional reaction-diffusion model. The behavior of the new dynamical system is examined for both diffusive and non-diffusive systems at different instances of fractional order.

Probability Distributions for Analog-To-Target Distances

AbstractSome properties of chaotic dynamical systems can be probed through features of recurrences, also called analogs. In practice, analogs are nearest neighbors of the state of a system, taken from a large database called the catalog. Analogs have been used in many atmospheric applications including forecasts, downscaling, predictability estimation, and attribution of extreme events. The distances of the analogs to the target state usually condition the performances of analog applications. These distances can be viewed as random variables, and their probability distributions can be related to the catalog size and properties of the system at stake. A few studies have focused on the first moments of return-time statistics for the closest analog, fixing an objective of maximum distance from this analog to the target state. However, for practical use and to reduce estimation variance, applications usually require not just one but many analogs. In this paper, we evaluate from a theoretical standpoint and with numerical experiments the probability distributions of the K shortest analog-to-target distances. We show that dimensionality plays a role on the size of the catalog needed to find good analogs and also on the relative means and variances of the K closest analogs. Our results are based on recently developed tools from dynamical systems theory. These findings are illustrated with numerical simulations of well-known chaotic dynamical systems and on 10-m wind reanalysis data in northwest France. Practical applications of our derivations are shown for forecasts of an idealized chaotic dynamical system and for objective-based dimension reduction using the 10-m wind reanalysis data.

Constant bias and weak second periodic forcing : tools to mitigate extreme events

We propose two potentially viable non-feedback methods, namely (i) constant bias and (ii) weak second periodic forcing as tools to mitigate extreme events. We demonstrate the effectiveness of these two tools in suppressing extreme events in two well-known nonlinear dynamical systems, namely (i) Liénard system and (ii) a non-polynomial mechanical system with velocity-dependent potential. As far as the constant bias is concerned, in the Liénard system, the suppression occurs due to the decrease in large amplitude oscillations and in the non-polynomial mechanical system the suppression occurs due to the destruction of chaos into a periodic orbit. As far as the second periodic forcing is concerned, in both the examples, extreme events get suppressed due to the increase in large amplitude oscillations. We also demonstrate that by introducing a phase in the second periodic forcing one can decrease the probability of occurrence of extreme events even further in the non-polynomial system. To provide a support to the complete suppression of extreme events, we present the two parameter probability plot for all the cases. In addition to the above, we examine the feasibility of the aforementioned tools in a parametrically driven version of the non-polynomial mechanical system. Finally, we investigate how these two methods influence the multistability nature in the Liénard system.

Sparse identification of nonlinear dynamical systems via reweighted [formula omitted]-regularized least squares

This work proposes an iterative sparse-regularized regression method to recover governing equations of nonlinear dynamical systems from noisy state measurements. The method is inspired by the Sparse Identification of Nonlinear Dynamics (SINDy) approach of Brunton et al. (2016), which relies on two main assumptions: the state variables are known a priori and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. The aim of this work is to improve the accuracy and robustness of SINDy in the presence of state measurement noise. To this end, a reweighted ℓ1-regularized least squares solver is developed, wherein the regularization parameter is selected from the corner point of a Pareto curve. The idea behind using weighted ℓ1-norm for regularization – instead of the standard ℓ1-norm – is to better promote sparsity in the recovery of the governing equations and, in turn, mitigate the effect of noise in the state variables. We also present a method to recover single physical constraints from state measurements. Through several examples of well-known nonlinear dynamical systems, we demonstrate empirically the accuracy and robustness of the reweighted ℓ1-regularized least squares strategy with respect to state measurement noise, thus illustrating its viability for a wide range of potential applications.

The Formulations of Classical Mechanics with Foucault’s Pendulum

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical system used to exemplify and illustrate modern concepts that are crucial in order to understand more complicated dynamical systems. The Foucault’s pendulum first installed in 2005 in the collegiate church of Sainte-Waudru (Mons, Belgium) will allow us to numerically estimate the different quantities introduced.

Data-Driven Detection and Classification of Regimes in Chaotic Systems Via Hidden Markov Modeling

AbstractChaotic dynamical systems are essentially nonlinear and are highly sensitive to variations in initial conditions and process parameters. Chaos may appear both in natural (e.g., heartbeat rhythms and weather fluctuations) and human-engineered (e.g., thermo-fluid, urban traffic, and stock market) systems. For prediction and control of such systems, it is often necessary to be able to distinguish between non-chaotic and chaotic behavior; several methods exist to detect the presence (or absence) of chaos, specially in noisy signals. A dynamical system may exhibit multiple chaotic regimes, and apparently, there exist no methods, reported in open literature, to classify these regimes individually. This paper demonstrates an application of standard hidden Markov modeling (HMM), which is a commonly used supervised method, as a technique to classify multiple regimes from a time series of dynamical systems, where classified regimes could be chaotic or non-chaotic. The proposed HMM-based method of regime classification has been tested using numerical data obtained from several well-known chaotic dynamical systems (e.g., Hénon, forced Duffing, Rössler, and Lorenz attractor). It is apparently well-suited to serve as a bench mark for the development of alternative data-driven methods to enhance the performance (e.g., accuracy and computational speed) of regime classification in chaotic dynamical systems.

Using Local Dynamics to Explain Analog Forecasting of Chaotic Systems

AbstractAnalogs are nearest neighbors of the state of a system. By using analogs and their successors in time, one is able to produce empirical forecasts. Several analog forecasting methods have been used in atmospheric applications and tested on well-known dynamical systems. Such methods are often used without reference to theoretical connections with dynamical systems. Yet, analog forecasting can be related to the dynamical equations of the system of interest. This study investigates the properties of different analog forecasting strategies by taking local approximations of the system’s dynamics. We find that analog forecasting performances are highly linked to the local Jacobian matrix of the flow map, and that analog forecasting combined with linear regression allows us to capture projections of this Jacobian matrix. Additionally, the proposed methodology allows us to efficiently estimate analog forecasting errors, an important component in many applications. Carrying out this analysis also makes it possible to compare different analog forecasting operators, helping us to choose which operator is best suited depending on the situation. These results are derived analytically and tested numerically on two simple chaotic dynamical systems. The impact of observational noise and of the number of analogs is evaluated theoretically and numerically.

Neural machine-based forecasting of chaotic dynamics

Chaotic dynamics is ubiquitous in nature. Traditionally, a good model representation of a certain system can help in predicting this system’s future behavior. However, for a complex system, a physics-based model may not be easy to construct given the complexity of a system, in particular, those that exhibit chaotic behavior. Furthermore, due to the aperiodic nature of the motion and finite precision, a model-based prediction may only have relatively high accuracy over a short-time horizon, before significant growth of error occurs in the prediction. In this article, the authors explore an alternate modeling approach, which is based on data-driven modeling, to explore forecasting viability for systems that display chaotic dynamics. Specifically, a deep recurrent neural network architecture, a neural machine, is constructed for forecasting temporal evolution of different chaotic systems. Data obtained from simulations with well-known nonlinear dynamical system prototypes serve as training data for the chosen neural network. In practice, this simulation data may be replaced with field data. The trained system is studied to examine the forecasting ability. Two ordinary differential dynamical systems, namely the Lorenz’63 system and the Lorenz’96 system, and a partial differential system, the Kuramoto–Sivashinsky equation, are studied, and the numerical experiments conducted are presented here to demonstrate the forecasting viability of the constructed neural network.

Building Logistic Spiking Neuron Models Using Analytical Approach

Spiking neuron models are inspired by biological neurons. They can simulate the neuronal activities of the mammalian brains, such as spiking (integrator) and periodic oscillation (resonator). A spiking neural network consisting of a cluster of spiking neurons can be used to simulate the collective dynamic behaviors of a brain neural network. This paper presents step-by-step analyses for the non-linear dynamics of mathematical spiking neuron models and sets forth a novel spiking model based on logistic function using an analytical approach. The logistic function is a well-known one-dimensional dynamical system and can generate spiking or periodic oscillation based on the system parameter. The novel spiking neural model is a combination of the integrate-and-fire and the quadratic integrate-and-fire neuron models with an added parameter to control the neural dynamics in order to generate stable, periodic, or chaotic neural behavior with flexibility. The analytical approach presented in this paper can be applied extensively to the design and analyses of multi-dimensional neuron models. The goal of this research project is to understand the dynamical behaviors of biological neurons in order to design biologically inspired spiking neuron model for building artificial intelligence, treating cognitive disorders, and advancing the scientific frontiers of brain research.

Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

Web geometry laboratory: case studies in Portugal and Serbia

ABSTRACTThe role of information and communication technologies (ICT) in education is well recognised – learning environments where the ICT features included are being proposed for many years now. The Web Geometry Laboratory (WGL) innovates in proposing a blended learning, collaborative and adaptive learning Web-environment for geometry. It integrates a well-known dynamic geometry system providing an adaptive environment where face-to-face activities can be combined with computer-mediated activities, in a collaborative or stand-alone, synchronous or asynchronous, classroom or homework way. The WGL collaborative environment allows each student to capitalise on one another's resources and skills. Its adaptive module gives to each teacher the ability to assess the geometric level of each student, allowing the teacher to build the individual profile and/or learning paths. In this paper we begin describing what are the goals that support the development of such an environment, the goals that support our beliefs in the usefulness of such an environment for the teaching and learning of geometry, after which we describe the environment and the case studies already conducted.

Detection of early warning signals in paleoclimate data using a genetic time series segmentation algorithm

This paper proposes a time series segmentation algorithm combining a clustering technique and a genetic algorithm to automatically find segments sharing common statistical characteristics in paleoclimate time series. The segments are transformed into a six-dimensional space composed of six statistical measures, most of which have been previously considered in the detection of warning signals of critical transitions. Experimental results show that the proposed approach applied to paleoclimate data could effectively analyse Dansgaard–Oeschger (DO) events and uncover commonalities and differences in their statistical and possibly their dynamical characterisation. In particular, warning signals were robustly detected in the GISP2 and NGRIP \(\delta ^{18}\hbox {O}\) ice core data for several DO events (e.g. DO 1, 4, 8 and 12) in the form of an order of magnitude increase in variance, autocorrelation and mean square distance from a linear approximation (i.e. the mean square error). The increase in mean square error, suggesting nonlinear behaviour, has been found to correspond with an increase in variance prior to several DO events for \(\sim \)90 % of the algorithm runs for the GISP2 \(\delta ^{18}\hbox {O}\) dataset and for \(\sim \)100 % of the algorithm runs for the NGRIP \(\delta ^{18}\hbox {O}\) dataset. The proposed approach applied to well-known dynamical systems and paleoclimate datasets provides a novel visualisation tool in the field of climate time series analysis.