Continuous Dynamical Systems Research Articles

Reservoir computing with logistic map.

Recent studies on reservoir computing essentially involve a high-dimensional dynamical system as the reservoir, which transforms and stores the input as a higher-dimensional state for temporal and nontemporal data processing. We demonstrate here a method to predict temporal and nontemporal tasks by constructing virtual nodes as constituting a reservoir in reservoir computing using a nonlinear map, namely, the logistic map, and a simple finite trigonometric series. We predict three nonlinear systems, namely, Lorenz, Rössler, and Hindmarsh-Rose, for temporal tasks and a seventh-order polynomial for nontemporal tasks with great accuracy. Also, the prediction is made in the presence of noise and found to closely agree with the target. Remarkably, the logistic map performs well and predicts close to the actual or target values. The low values of the root mean square error confirm the accuracy of this method in terms of efficiency. Our approach removes the necessity of continuous dynamical systems for constructing the reservoir in reservoir computing. Moreover, the accurate prediction for the three different nonlinear systems suggests that this method can be considered a general one and can be applied to predict many systems. Finally, we show that the method also accurately anticipates the time series of the all the three variable of Rössler system for the future (self-prediction).

Interpersonal strategy for controlling unpredictable opponents in soft tennis

Competition in sports, unlike cooperation in everyday life, does not involve a single solution because individuals aim to behave unpredictably, thereby preventing others from predicting their actions. This study determined how individuals in court-based sports attempted to control others’ unpredictable behaviors, addressing the gap in previous studies regarding the lack of clarity around strategies employed by individuals in competitive situations. We achieved this by applying a switching hybrid dynamics model, considering external inputs to analyze individual behaviors. Consequently, the study indicates that skilled individuals, in contrast to intermediate players, exhibit greater consistency in their behaviors. These skilled individuals lead others to anticipate their consistency and subsequently employ strategies to disrupt these expectations. This strategy exploits the principles of active human inference, implying that competition involves cooperation. This was revealed by an analysis of both human decision-making and behavior in actual matches as discrete and continuous dynamical systems. This interpersonal strategy could assist policymakers in the field of everyday life to enhance competitiveness. This strategy enables policymakers to adopt new policies that promote cooperation with competitors, ultimately increasing competitiveness in various aspects of our daily lives.

Stable attractors for neural networks classification via ordinary differential equations (SA-nODE)

Abstract A novel approach for supervised classification is presented which sits at the intersection of machine learning and dynamical systems theory. At variance with other methodologies that employ ordinary differential equations for classification purposes, the untrained model is a priori constructed to accommodate for a set of pre-assigned stationary stable attractors. Classifying amounts to steer the dynamics towards one of the planted attractors, depending on the specificity of the processed item supplied as an input. Asymptotically the system will hence converge on a specific point of the explored multi-dimensional space, flagging the category of the object to be eventually classified. Working in this context, the inherent ability to perform classification, as acquired ex post by the trained model, is ultimately reflected in the shaped basin of attractions associated to each of the target stable attractors. The performance of the proposed method is here challenged against simple toy models crafted for the purpose, as well as by resorting to well established reference standards. More precisely, we achieved an accuracy of 98.06 % on the MNIST test set and 88.21% on the Fashion MNIST test set. Although this method does not reach the performance of state-of-the-art deep learning algorithms, it illustrates that continuous dynamical systems with closed analytical interaction terms can serve as high-performance classifiers.

Bridging pharmacology and neural networks: A deep dive into neural ordinary differential equations.

The advent of machine learning has led to innovative approaches in dealing with clinical data. Among these, Neural Ordinary Differential Equations (Neural ODEs), hybrid models merging mechanistic with deep learning models have shown promise in accurately modeling continuous dynamical systems. Although initial applications of Neural ODEs in the field of model-informed drug development and clinical pharmacology are becoming evident, applying these models to actual clinical trial datasets-characterized by sparse and irregularly timed measurements-poses several challenges. Traditional models often have limitations with sparse data, highlighting the urgent need to address this issue, potentially through the use of assumptions. This review examines the fundamentals of Neural ODEs, their ability to handle sparse and irregular data, and their applications in model-informed drug development.

An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems.

After coarse-graining a complex system, the dynamics of its macro-state may exhibit more pronounced causal effects than those of its micro-state. This phenomenon, known as causal emergence, is quantified by the indicator of effective information. However, two challenges confront this theory: the absence of well-developed frameworks in continuous stochastic dynamical systems and the reliance on coarse-graining methodologies. In this study, we introduce an exact theoretic framework for causal emergence within linear stochastic iteration systems featuring continuous state spaces and Gaussian noise. Building upon this foundation, we derive an analytical expression for effective information across general dynamics and identify optimal linear coarse-graining strategies that maximize the degree of causal emergence when the dimension averaged uncertainty eliminated by coarse-graining has an upper bound. Our investigation reveals that the maximal causal emergence and the optimal coarse-graining methods are primarily determined by the principal eigenvalues and eigenvectors of the dynamic system's parameter matrix, with the latter not being unique. To validate our propositions, we apply our analytical models to three simplified physical systems, comparing the outcomes with numerical simulations, and consistently achieve congruent results.

Yoneyama's 3/2 test for the asymptotic stability of delay dynamic equations

We consider the linear delay dynamic equation (*) x Δ ( t ) + p ( t ) x ( τ ( t ) ) = 0 for t ∈ [ t 0 , ∞ ) T . We enhanced the results in Wu and Zhou [2007. Stability for first order delay dynamic equations on time scales. Computers and Mathematics with Applications, 53(12), 1820–1831] by removing the boundedness criterion from the delay and graininess functions to cover the result in Zhang et al. [1999. Global attractivity of difference equations with variable delay. Dynamics of Continuous, Discrete and Impulsive Systems, 6(3), 307–317]. Furthermore, we have defined new functions that will assist us in proving our main results. We have presented some examples that illustrate the essence of our results in this work.

Energetic spectral-element time marching methods for phase-field nonlinear gradient systems

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Representation of solutions to fuzzy linear fractional differential equations with a piecewise constant argument

This study marks the first exploration of fuzzy linear fractional differential equations with a piecewise constant argument (FLFDEs-PCA), incorporating the concept of Caputo’s type gH-differentiability with the order α ∈ (0, 1]. Such problems are noteworthy as they represent hybrid systems, blending the characteristics of continuous and discrete dynamical systems and integrating aspects from both differential and difference equations. The primary objective of this research is to establish a standardized framework for deriving explicit solution formulas for FLFDEs-PCA under various scenarios. Additionally, illustrative examples are provided to demonstrate the practical implications of our theoretical findings.

Periodic motions with impact chatters in an impact Duffing oscillator.

The periodic motions of discontinuous nonlinear dynamical systems are very difficult problems to solve in engineering and physics. Until now, except for numerical studies, one cannot find a better way to solve such problems. In fact, one still has difficulty obtaining periodic motions in continuous nonlinear dynamical systems. In this paper, a method is presented systematically for periodic motions in discontinuous nonlinear dynamical systems. The stability and grazing bifurcations of such periodic motions are studied. Such a method is presented through discussion on a periodically forced, impact Duffing oscillator. Thus, periodic motions with impact chatters in a periodically forced Duffing oscillator with one-sidewall constraint are studied. The analytical conditions for motion grazing at the boundary are developed from discontinuous dynamical systems. The impact Duffing oscillator is discretized to generate subimplicit mappings. With impact, the mapping structures are employed to construct specific impact periodic motions for an impact Duffing oscillator. The bifurcation trees of impact chatter periodic motions are achieved semi-analytically. The grazing and period-doubling bifurcations are obtained, and the grazing bifurcations are for the appearing and disappearance for an impact chatter periodic motion. The impact chatter periodic motions with and without grazing are presented for illustration of impact periodic motion complexity in the impact Duffing oscillator.

Analysis of Prey-Predator Scheme in Conjunction with Help and Gestation Delay

This paper presents a three-dimensional continuous time dynamical system of three species, two of which are competing preys and one is a predator. We also assume that during predation, the members of both teams of preys help each other and the rate of predation of both teams is different. The interaction between prey and predator is assumed to be governed by a Holling type II functional response and discrete type gestation delay of the predator for consumption of the prey. In this work, we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system, and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functions when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. As we introduced time delay due to the gestation of the predator, we also discuss the stability of the delayed model. It is observed that the existence of stability switching occurs around the interior equilibrium point as the gestation delay increases through a certain critical threshold. Here, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of the system is also observed. This study reveals that the delay is taken as a bifurcation parameter and also plays a significant role for the stability of the proposed model. Computer simulations of numerical examples are given to explain our proposed model. We have also addressed critically the biological implications of our analytical findings with proper numerical examples.

On the convergence of the trajectories of the dynamical Moudafi’s viscosity approximation system

We study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associated with the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi (J Math Anal Appl 241:46–55, 2000). We establish that the trajectories x(t) of the continuous dynamical system (CDS) has an asymptotic behavior similar to the behavior of the sequences (xn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x_{n})$$\\end{document} generated by the discrete viscosity approximation (DDS)

Nonlinear optical dynamics in Heisenberg space: Directional curves and recursive inquiry

This paper delves into the exploration of directional recursion operators within the realm of regular space curves modeled by Heisenberg systems. The central objective is to introduce a myriad of recursive flows, encompassing ferromagnetic and antiferromagnetic solutions, alongside a family of general normalization operators in the normal and binormal directions. The study employs the extended compatible and inextensible flow model of curves to examine the evolution models, providing a comprehensive understanding of their dynamics. A significant aspect of the investigation involves elucidating the evolution model in terms of anholonomy shapes and their density. The directional recursive operator, a focus of this study, demonstrates distinct results compared to traditional approaches. The reliability and applicability of the obtained results extend to the examination of various linear and nonlinear continuous dynamical systems.

Knotted toroidal sets, attractors and incompressible surfaces

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for discrete or continuous dynamical systems globally defined in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}^3$$\\end{document} which arise naturally when considering toroidal sets.

Estimation of the Impact of Vaccination Intervention on Recovered Coronavirus Patients

This work estimated the impact of vaccination intervention on coronavirus patients who have recovered from the disease and the vulnerability index of the recovered population due to the impact of vaccine was also investigated. This work adopted a numerical solution to study the continuous dynamical system of linear first order differential equations describing a SEIR (Susceptible, Exposed, Infected, Recovered) model on the spread of Coronavirus Disease – 2019 (COVID-19). To tackle this problem, MATLAB ordinary differential equation of order 45 (ODE45) numerical method was adopted for the analysis. The vulnerability index of the recovered population was low due to the impact of vaccine meaning that the recovered population will gain immunity and they will not be re-infected. The study recommended that coronavirus patients who have recovered from the disease should ensure that they have vaccination administered to them to avoid re-occurrence of the virus attack as an intervention strategy.

Data-driven Lie point symmetry detection for continuous dynamical systems

Symmetry detection, the task of discovering the underlying symmetries of a given dataset, has been gaining popularity in the machine learning community, particularly in science and engineering applications. Most previous works focus on detecting ‘canonical’ symmetries such as translation, scaling, and rotation, and cast the task as a modeling problem involving complex inductive biases and architecture design of neural networks. We challenge these assumptions and propose that instead of constructing biases, we can learn to detect symmetries from raw data without prior knowledge. The approach presented in this paper provides a flexible way to scale up the detection procedure to non-canonical symmetries, and has the potential to detect both known and unknown symmetries alike. Concretely, we focus on predicting the generators of Lie point symmetries of partial differential equations, more specifically, evolutionary equations for ease of data generation. Our results demonstrate that well-established neural network architectures are capable of recognizing symmetry generators, even in unseen dynamical systems. These findings have the potential to make non-canonical symmetries more accessible to applications, including model selection, sparse identification, and data interpretability.

Finite Difference Models of Dynamical Systems with Quadratic Right-Hand Side

Difference schemes that approximate dynamic systems are considered discrete models of the same phenomena that are described by continuous dynamic systems. Difference schemes with t-symmetry and midpoint and trapezoid schemes are considered. It is shown that these schemes are dual to each other, and, from this fact, we derive theorems on the inheritance of quadratic integrals by these schemes (Cooper’s theorem and its dual theorem on the trapezoidal scheme). Using examples of nonlinear oscillators, it is shown that these schemes poses challenges for theoretical research and practical application due to the problem of extra roots: these schemes do not allow one to unambiguously determine the final values from the initial values and vice versa. Therefore, we consider difference schemes in which the transitions from layer to layer in time are carried out using birational transformations (Cremona transformations). Such schemes are called reversible. It is shown that reversible schemes with t-symmetry can be easily constructed for any dynamical system with a quadratic right-hand side. As an example of such a dynamic system, a top fixed at its center of gravity is considered in detail. In this case, the discrete theory repeats the continuous theory completely: (1) the points of the approximate solution lie on some elliptic curve, which at Δt→0 turns into an integral curve; (2) the difference scheme can be represented using quadrature; and (3) the approximate solution can be represented using an elliptic function of a discrete argument. The last section considers the general case. The integral curves are replaced with closures of the orbits of the corresponding Cremona transformation as sets in the projective space over R. The problem of the dimension of this set is discussed.

Dynamics of a random Hopfield neural lattice model with adaptive synapses and delayed Hebbian learning

UDC 517.9 A Dong–Hopfield neural lattice model with random external forcing and delayed response to the evolution of interconnection weights is developed and studied. The interconnection weights evolve according to the Hebbian learning rule with a decay term and contribute to changes in the states after a short delay. The lattice system is first reformulated as a coupled functional-ordinary differential equation system on an appropriate product space. Then the solution of the system is shown to exist and be unique. Furthermore it is shown that the system of equations generates a continuous random dynamical system. Finally, the existence of random attractors for the random dynamical system generated by the Dong–Hopfield model is established.

A Continuous Time Dynamical Turing Machine.

Continuous time recurrent neural networks (CTRNNs) are systems of coupled ordinary differential equations (ODEs) inspired by the structure of neural networks in the brain. CTRNNs are known to be universal dynamical approximators: given a large enough system, the parameters of a CTRNN can be tuned to produce output that is arbitrarily close to that of any other dynamical system. However, in practice, both designing systems of CTRNN to have a certain output, and the reverse-understanding the dynamics of a given system of CTRNN-can be nontrivial. In this article, we describe a method for embedding any specified Turing machine in its entirety into a CTRNN. As such, we describe in detail a continuous time dynamical system that performs arbitrary discrete-state computations. We suggest that in acting as both a continuous time dynamical system and as a computer, the study of such systems can help refine and advance the debate concerning the Computational Hypothesis that cognition is a form of computation and the Dynamical Hypothesis that cognitive systems are dynamical systems.

Application of zeroed neural networks to stability analysis of continuous dynamic systems

Abstract Modern production processes frequently require steady-state analysis of continuous dynamic systems. Traditional numerical approaches, however, fall short in efficiency when tasked with addressing large-scale or dynamic problems. To tackle the inverse problem inherent in stability analysis, this study presents an innovative approach by integrating a combined excitation function into the foundational zeroing neural network (ZNN) model. This integration constrains the ZNN model, evolving it into an enhanced EZNN model specifically designed for solving the inverse of dynamic complex matrices. Additionally, this paper conducts a rigorous theoretical analysis of the robust performance of the EZNN model when excited by the combined function, both in the presence and absence of noise interference. The model solution process is promoted by using a class of high-dimensional continuous dynamic systems as an example, and numerical simulation experiments are used for validation. Considering the dynamic system satisfying { A ( t ) = ( 4 + sin ( 2 t ) 4 - cos ( 2 t ) 5 + sin ( 2 t ) ) C ( t ) = ( cos ( t ) sin ( t ) - cos ( t ) - sin ( t ) cos ( t ) sin ( t ) ) b ( t ) = 4 + cos ( 4 t ) , d ( t ) = ( cos ( 2 t ) , cos ( 2 t ) ) T \left\{ \matrix{ A(t) = \left( {4 + \sin (2t)\quad 4 - \cos (2t)\quad 5 + \sin (2t)} \right) \hfill \cr C(t) = \left( {\matrix{{\cos (t)} & {\sin (t)} & { - \cos (t)} \cr { - \sin (t)} & {\cos (t)} & {\sin (t)} \cr } } \right) \hfill \cr b(t) = 4 + \cos (4t),d(t) = {(\cos (2t),\cos (2t))^T} \hfill \cr} \right. , the error E 1(x(t),t) obtained by the EZNN model with combinatorial function excitation always remains negative or tends rapidly to 0. The x (t) obtained by the model converges rapidly to an exact solution of the system. Through the discussion of parametric conditions, it is also found that increasing the value of parameter γ increases the rate of convergence of the ZNN model.

Synchronization of time-delay systems with impulsive delay via an average impulsive estimation approach.

We investigated synchronization of dynamic systems with mixed delays and delayed impulses. Using impulsive control method and the average impulsive interval approach, several Lyapunov sufficient conditions were given for ensuring synchronization in terms of impulsive perturbation and impulsive control, respectively. The derived conditions indicated that delays in continuous dynamical systems were flexible under impulsive perturbation and were not strictly dependent on the size of impulsive delays, and they may have a potential impact on synchronization of the considered system. In addition, applying the proposed concepts of average positive impulsive estimation and average impulsive estimation, we integrated the information in impulsive delay into the rate coefficient to eliminate the limitation of having the same threshold at each impulse point, while the impulsive delay maintained the synchronization effect. This was an improvement on the previous results obtained. Finally, we provided two numerical examples to illustrate the validity of our results.