Types Of Dynamical Systems Research Articles

Exploring the origins of switching dynamics in a multifunctional reservoir computer.

The concept of multifunctionality has enabled reservoir computers (RCs), a type of dynamical system that is typically realized as an artificial neural network, to reconstruct multiple attractors simultaneously using the same set of trained weights. However, there are many additional phenomena that arise when training a RC to reconstruct more than one attractor. Previous studies have found that in certain cases, if the RC fails to reconstruct a coexistence of attractors, then it exhibits a form of metastability, whereby, without any external input, the state of the RC switches between different modes of behavior that resemble the properties of the attractors it failed to reconstruct. In this paper, we explore the origins of these switching dynamics in a paradigmatic setting via the "seeing double" problem.

Real Time Prediction and Modelling of Power Grid System Stability and Load Balancing

Abstract The extensive geographical distribution of the power system determines its highly complex structure. The system includes key components such as generators, substations, transmission and distribution grids, and users. This system has characteristics such as nonlinearity, time-varying, and parameter uncertainty. Power system stability is inherently linked to its regular operational performance. This article uses Simulink models to simulate the working state of generators and circuit breakers during faults. Under the conditions of a synchronous generator with a rated power of 350MW, a voltage of 10.5KV, a three-phase parallel RLC load rated voltage of 115KV, and a power consumption of 300MW, when a short occurs on line, the speed and relative angle of the generator tend to a fixed value, and the system remains stable. When a fault occurs in the circuit within 0.1 seconds, the circuit breaker will cut off the faulty line within 0.6 seconds. Avoiding the occurrence of oscillations and unstable states in typical dynamic systems.

Pattern formation and delay-induced instability in a Leslie–Gower type prey–predator system with Smith growth function

In this article, we have investigated the spatiotemporal dynamics and delay-induced instability of a Leslie–Gower type prey–predator model under the influence of environmental toxicants with Smith growth function. This growth function is more realistic than logistic growth as it better describes the growth of the biological population. It has been used where the growth limitations are based on the proportion of available resources not utilized. A few works of Smith’s growth models are reported in the literature. Therefore, spatiotemporal dynamics and pattern formation with delay effect remain an exciting area of research, which motivates the present work. This work has studied two types of dynamical systems: (i) an ordinary differential temporal system with time delay and (ii) a reaction–diffusion system with time delay. The existence of equilibrium points and their stability conditions are discussed. Hopf bifurcation emerges in both proposed systems with respect to delay parameter. The stability and direction of Hopf bifurcation and delay–diffusion-driven instability have been investigated for the reaction–diffusion system. Numerical simulation is performed to support the analytical results and theorems. Moreover, the existence of Hopf and delay-induced instability are proved numerically. Interesting one-dimensional regular and irregular stripe patterns are obtained for increased values of the time delay parameter. Also, the presence of natural toxicants has a negative impact on the growth of prey–predator species.

The Dynamicist Landscape.

The dynamical hypothesis states that cognitive systems are dynamical systems. While dynamical systems play an important role in many cognitive phenomena, the dynamical hypothesis as stated applies to every system and so fails both to specify what makes cognitive systems distinct and to distinguish between proposals regarding the nature of cognitive systems. To avoid this problem, I distinguish several different types of dynamical systems, outlining four dimensions along which dynamical systems can vary: total-state versus partial-state, internal versus external, macroscopic versus microscopic, and systemic versus componential, and illustrate these with examples. I conclude with two illustrations of partial-state, internal, microscopic, componential dynamicism.

Forward invariant set preservation in discrete dynamical systems and numerical schemes for ODEs: application in biosciences

We present two results on the analysis of discrete dynamical systems and finite difference discretizations of continuous dynamical systems, which preserve their dynamics and essential properties. The first result provides a sufficient condition for forward invariance of a set under discrete dynamical systems of specific type, namely time-reversible ones. The condition involves only the boundary of the set. It is a discrete analog of the widely used tangent condition for continuous systems (viz. the vector field points either inwards or is tangent to the boundary of the set). The second result is nonstandard finite difference (NSFD) scheme for dynamical systems defined by systems of ordinary differential equations. The NSFD scheme preserves the hyperbolic equilibria of the continuous system as well as their stability. Further, the scheme is time reversible and, through the first result, inherits from the continuous model the forward invariance of the domain. We show that the scheme is of second order, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes. The application of these results, including some numerical simulations for invariant sets, is exemplified on a general Susceptible-Infective-Recovered/Removed (SIR)-type epidemiological model, which may have arbitrary large number of infective or recovered/removed compartments.

A composite decoupling method of the 2D WPT transmitter array based on the compensation topology and orthogonal coupling

Typical dynamic wireless power transfer systems use a long-track transmitter (Tx) coil or multiple segmented Tx coils, producing a limited efficient charging area. To improve charging flexibility, the two-dimensional (2D) transmitter array is adopted. However, the cross coupling between Tx coils will exist in multiple directions, resulting in the degradation of system performance. Therefore, this paper presents a composite decoupling method for the 2D wireless power supply array based on the compensation topology and orthogonal coupling. The compensation topology eliminates cross coupling between electrically connected Tx coils, while orthogonal coils decouple Tx coils without electrical connection. Compared with traditional methods, the proposed decoupling method can eliminate cross coupling between Tx coils in multiple directions. Additionally, it reduces the cost of inverters and decoupling inductors significantly. The transmitter array is configured with three operation modes to stabilize system power transmission. The effectiveness of the proposed method is verified through the experiment.

Physics informed neural network for charged particles surrounded by conductive boundaries

Molecular dynamics of charged particles in porous conductive media have received considerable attention in recent years due to their application in cutting-edge technologies such as batteries and supercapacitors. Due to the presence of long-range electrical interactions, induced charges present at the boundary, and the influence of boundary conditions, the simulation of these systems is more challenging than the simulation of typical molecular dynamic systems. Simulating these kinds of systems typically involves using a numerical solver to solve the Poisson equation, which is a very time-consuming procedure. Recently, Physics-Informed Neural Networks (PINNs) have been introduced as an alternative to numerical solutions of PDEs. In this paper, we present a new PINN-based model for predicting the potential of point-charged particles surrounded by conductive walls. As a result of the proposed PINN model, the mean square error is less than 7% and R^{2} score is more than 90% for the corresponding example simulation. Results have been compared with typical neural networks and random forest as standard machine learning algorithms. The R^{2} score of the random forest model was 70%, and a standard neural network could not be trained well.

Invertible Koopman Network and its application in data-driven modeling for dynamic systems

Koopman operator, acting on an infinite-dimensional Hilbert space of the observables, provides a global systematic linear representation of nonlinear systems, which is a leading candidate for data-driven modeling. In practical applications, the observable function is required to have an invertible property to predict or control the original system. However, two approaches for constructing the invertible observable functions in current research, introducing the full-state observables and adopting the auto-encoder, have drawbacks such as overly strong assumptions and lossy reconstruction. To address these issues, we propose the Invertible Koopman Network (IKN) to approximate the Koopman operator. With special invertible blocks, IKN achieves structure-level reversibility while retaining sufficient nonlinear mapping capability. This network results in a transformation from the complex dynamic system’s states to the observables with simpler dynamics and affords lossless reconstruction via its invertible process. Then, we use the IKN as a dynamical embedding model and adopt the Transformer based on an autoregressive task to learn the evolution pattern of the observables. Through the greedy decoding strategy, the trained IKN and Transformer model enable data-driven prediction where only initial value is provided. The method is tested on three typical chaotic dynamic systems (Lorenz system, Chen system, and Kuramoto-Sivashinsky equation) and is shown to capture the systems’ intrinsic evolution patterns, such as the structure of the strange attractor. Comparative experiments with alternative data-driven modeling approaches illustrate that our proposed method can model a longer evolutionary process with less error, further demonstrating its effectiveness and superiority.

Evolving Neuro-Fuzzy Systems-Based Design of Experiments in Process Identification

This paper presents a new design of experiment approach based on an evolving neuro-fuzzy model. The input of the process is proposed by a space-filling method that uses a sequence of step functions to maximise the coverage of the antecedent clusters in the input space of the evolving model, which is called the heuristic step sequence (HSS). When the system reaches a steady state, a new rule is added, existing rules are merged based on the similarity of the consequent transfer functions, and a new optimal excitation is calculated. The output error model structure was chosen because it assumes a noise distribution common to real processes and is suitable for identification, while the step function input signal is one of the most commonly used signals in identification and control. A filtered recursive least squares method is used to identify the consequent parameters of the output error models and the optimisation filter is adapted based on the the confidence interval of the local model. Evaluations were performed on a Hammerstein type model comparing the HSS method with a staircase excitation and on a real plate heat-exchanger pilot plant. The experiments show that the proposed HSS method outperforms the staircase excitation and can be used to identify nonlinear dynamical systems of Hammerstein-Wiener type.

Chaotic behaviors, exotic solitons and exact solutions of a nonlinear Schrödinger-type equation

In this paper, qualitative analysis is used to study a nonlinear Schrödinger-type equation. By applying traveling wave transform and choosing different parameter values, we obtain two types of dynamic systems. We apply the qualitative method to show that there are periodic solutions, optical solitary waves and exotic solitons for the equation. Then we perform the quantitative analysis to these different dynamic systems and obtain the corresponding exact solutions, which verifies the correctness of our previous conclusions. In addition, some new solutions are also constructed, such as, rational solutions and double period elliptic function solution. Finally, we add different perturbed forms to the dynamic system and obtain the largest Lyapunov exponents and corresponding phase diagrams, which demonstrates the existence of chaotic behaviors of the equation. To the best of our knowledge, the chaotic behaviors of the equation we obtained are firstly proposed.

Decision-making on the solution of non-linear dynamical systems of Kannan non-expansive type in Nakano sequence space of fuzzy numbers

Decision-making on the solution of non-linear dynamical systems of Kannan non-expansive type in Nakano sequence space of fuzzy numbers

Persistent Subspaces of Reaction-Based Dynamical Systems

Various types of dynamical systems, such as ordinary differential equations (ODEs) or partial differential equations (PDEs), are widely applied not only in chemistry but also in many scientific disciplines to model the dynamics arising from interactions described by reactions between molecules, individuals, or species. This study provides an overview of how Chemical Organization Theory (COT) can be used to analyze such systems by identifying all potentially persistent species solely from the underlying reaction network, without the need for simulations or even knowledge of reaction constants or kinetic laws. Two minimalist examples with only three resp. four species are used to introduce all fundamental definitions including a new, naturally arising concept of persistence, and to illustrate the fore-mentioned technique without mathematical details such as proofs. Thereby, COT is shown to provide measures to analyze, compare, and construct very complex systems on an abstract level and thus to complement other powerful techniques for the analysis of complex systems such as deficiency, RAF theory, elementary modes, graph theory, Lyapunov functions, and bifurcation theory.

New finite-time stability for fractional-order time-varying time-delay linear systems: A Lyapunov approach

The primary goal of this paper is to examine the finite-time stability and finite-time contractive stability of the linear systems in fractional domain with time-varying delays. We develop some sufficient criteria for finite-time contractive stability and finite-time stability utilizing fractional-order Lyapunov-Razumikhin technique. To validate the proposed conditions, two different types of dynamical systems are taken into account, one is general time-delay fractional-order system and another one is fractional-order linear time-varying time-delay system, furthermore the efficacy of the stability conditions is demonstrated numerically.

Revealing spatiotemporal interaction patterns behind complex cities.

Cities are typical dynamic complex systems that connect people and facilitate interactions. Revealing general collective patterns behind spatiotemporal interactions between residents is crucial for various urban studies, of which we are still lacking a comprehensive understanding. Massive cellphone data enable us to construct interaction networks based on spatiotemporal co-occurrence of individuals. The rank-size distributions of dynamic population of locations in all unit time windows are stable, although people are almost constantly moving in cities and hot-spots that attract people are changing over time in a day. A larger city is of a stronger heterogeneity as indicated by a larger scaling exponent. After aggregating spatiotemporal interaction networks over consecutive time windows, we reveal a switching behavior of cities between two states. During the "active" state, the whole city is concentrated in fewer larger communities, while in the "inactive" state, people are scattered in smaller communities. Above discoveries are universal over three cities across continents. In addition, a city stays in an active state for a longer time when its population grows larger. Spatiotemporal interaction segregation can be well approximated by residential patterns only in smaller cities. In addition, we propose a temporal-population-weighted-opportunity model by integrating a time-dependent departure probability to make dynamic predictions on human mobility, which can reasonably well explain the observed patterns of spatiotemporal interactions in cities.

Analyzing Pattern Formation in the Gray--Scott Model: An XPPAUT Tutorial

The Gray--Scott model is a widely studied autocatalytic model that exhibits a range of interesting pattern formation behavior, as well as a rich structure of dynamics that includes many ideas from a typical undergraduate dynamical systems course, and some from beyond. Gaining an understanding of the solutions to this model is arguably most easily conducted via a bifurcation analysis of corresponding ODE problems within the software XPPAUT. In this paper, we provide an introductory XPPAUT tutorial, through which we begin to expose the range of intricate patterns that the Gray--Scott model emits.

Dark Type Dynamical Systems: The Integrability Algorithm and Applications

Based on a devised gradient-holonomic integrability testing algorithm, we analyze a class of dark type nonlinear dynamical systems on spatially one-dimensional functional manifolds possessing hidden symmetry properties and allowing their linearization on the associated cotangent spaces. We described main spectral properties of nonlinear Lax type integrable dynamical systems on periodic functional manifolds particular within the classical Floquet theory, as well as we presented the determining functional relationships between the conserved quantities and related geometric Poisson and recursion structures on functional manifolds. For evolution flows on functional manifolds, parametrically depending on additional functional variables, naturally related with the classical Bellman-Pontriagin optimal control problem theory, we studied a wide class of nonlinear dynamical systems of dark type on spatially one-dimensional functional manifolds, which are both of diffusion and dispersion classes and can have interesting applications in modern physics, optics, mechanics, hydrodynamics and biology sciences. We prove that all of these dynamical systems possess rich hidden symmetry properties, are Lax type linearizable and possess finite or infinite hierarchies of suitably ordered conserved quantities.

Invariant measures of torus piecewise isometries

We study measure-theoretical aspects of torus piecewise isometries. Not much is known about this type of dynamical systems, except for the special case of one-dimensional interval exchange mappings. The last case is fundamentally different from the general situation in the presence of an invariant measure (Lebesgue measure), which helps a lot in the analysis. Due to the absence of good methods of analysis of general systems with discontinuities, even the existence of invariant measures of the torus piecewise isometries was an open question. We establish sufficient conditions for the existence/absence of invariant measures for this class of systems. Technically, our results are based on the approximation of the maps under study by weakly periodic ones.

Design and numerical analysis of fuzzy nonstandard computational methods for the solution of rumor based fuzzy epidemic model

This model extends the classical epidemic model for cyber consumerism by introducing fuzziness to the model. Fuzziness arises due to insufficient knowledge, experimental errors, operating conditions and parameters that provide inaccurate information. The concepts of confused, escapers and recovered consumers are uncertain due to the different degrees of confusion, escaping and recovery among the individuals of the cyber consumers. The differences can arise, when the cyber consumers under the consideration having distinct habits, customs and different age groups have different degrees of resistance, etc. The chance of transmission of rumors and recovery rates are considered as fuzzy numbers. A rumor-free and two rumor existing-endemic equilibrium points have been derived for the studied model. The model is then solved numerically with fuzzy forward Euler and fuzzy nonstandard finite difference (FNSFD) methods respectively. The numerical and simulation results show that the proposed FNSFD technique is an efficient and reliable tool to deal with such type of dynamical system.

On a High-Precision Method for Studying Attractors of Dynamical Systems and Systems of Explosive Type

The author of this article considers a numerical method that uses high-precision calculations to construct approximations to attractors of dynamical systems of chaotic type with a quadratic right-hand side, as well as to find the vertical asymptotes of solutions of systems of explosive type. A special case of such systems is the population explosion model. A theorem on the existence of asymptotes is proved. The extension of the numerical method for piecewise smooth systems is described using the Chua system as an example, as well as systems with hysteresis.

Junk-neuron-deletion strategy for hyperparameter optimization of neural networks

With the complexity of problems in reality increasing, the sizes of deep learning neural networks, including the number of layers, neurons, and connections, are increasing in an explosive way. Optimizing hyperparameters to improve the prediction performance of neural networks has become an important task. In literatures, the methods of finding optimal parameters, such as sensitivity pruning and grid search, are complicated and cost a large amount of computation time. In this paper, a hyperparameter optimization strategy called junk neuron deletion is proposed. A neuron with small mean weight in the weight matrix can be ignored in the prediction, and is defined subsequently as a junk neuron. This strategy is to obtain a simplified network structure by deleting the junk neurons, to effectively shorten the computation time and improve the prediction accuracy and model the generalization capability. The LSTM model is used to train the time series data generated by Logistic, Henon and Rossler dynamical systems, and the relatively optimal parameter combination is obtained by grid search with a certain step length. The partial weight matrix that can influence the model output is extracted under this parameter combination, and the neurons with smaller mean weights are eliminated with different thresholds. It is found that using the weighted mean value of 0.1 as the threshold, the identification and deletion of junk neurons can significantly improve the prediction efficiency. Increasing the threshold accuracy will gradually fall back to the initial level, but with the same prediction effect, more operating costs will be saved. Further reduction will result in prediction ability lower than the initial level due to lack of fitting. Using this strategy, the prediction performance of LSTM model for several typical chaotic dynamical systems is improved significantly.