Related papers: An Accurate Numerical Method and Algorithm for Con…
This paper explores backward error analysis for numerical solutions of ordinary differential equations, particularly focusing on chaotic systems. Three approaches are examined: residual assessment, the method of modified equations, and…
Discrete numerical methods with finite time-steps represent a practical technique to solve initial-value problems involving nonlinear differential equations. These methods seem particularly useful to the study of chaos since no analytical…
We investigate functions that are exact solutions to chaotic dynamical systems. A generalization of these functions can produce truly random numbers. For the first time, we present solutions to random maps. This allows us to check,…
We discuss several numerical methods for calculating Lyapunov exponents (a quantitative measure of chaos) in systems of ordinary differential equations. We pay particular attention to constrained systems, and we introduce a variety of…
Stiff and chaotic differential equations are challenging for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution…
This paper is the second in a series of two, and describes the current state of the art in modelling and prediction of chaotic time series. Sampled data from deterministic non-linear systems may look stochastic when analysed with linear…
Stochastic dynamical systems are fundamental in state estimation, system identification and control. System models are often provided in continuous time, while a major part of the applied theory is developed for discrete-time systems.…
In dealing with nonlinear systems, it is common to use numerical solutions. Unlike the careful behavior towards the numerical results in chaotic regions, the validity of numerical results in regions of transient chaos might not always be…
We develop an extension of the fast method of angles for hyperbolicity verification in chaotic systems with an arbitrary number of time-delay feedback loops. The adopted method is based on the theory of covariant Lyapunov vectors and…
In chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real-time knowledge about the error growth could enable strategies to adjust the modelling and forecasting infrastructure…
Many complex phenomena, from weather systems to heartbeat rhythm patterns, are effectively modeled as low-dimensional dynamical systems. Such systems may behave chaotically under certain conditions, and so the ability to detect chaos based…
In this paper, we discuss the Lyapunov exponent definition of chaos and how it can be used to quantify the chaotic behavior of a system. We derive a way to practically calculate the Lyapunov exponent of a one-dimensional system and use it…
The effective numerical method is developed performing the test of the hyperbolicity of chaotic dynamics. The method employs ideas of algorithms for covariant Lyapunov vectors but avoids their explicit computation. The outcome is a…
Chaos and unpredictability are traditionally synonymous, yet large-scale machine learning methods recently have demonstrated a surprising ability to forecast chaotic systems well beyond typical predictability horizons. However, recent works…
We introduce new machine-learning techniques for analyzing chaotic dynamical systems. The primary objectives of the study include the development of a new and simple method for calculating the Lyapunov exponent using only two trajectory…
By the example of a mathematical model of a biochemical process, the structural instability of dynamical systems is studied by calculating the full spectrum of Lyapunov indices with the use of the generalized Benettin algorithm. For the…
This paper demonstrates the application of Bayesian Artificial Neural Networks to Ordinary Differential Equation (ODE) inverse problems. We consider the case of estimating an unknown chaotic dynamical system transition model from state…
Identification of the unknown parameters and orders of fractional chaotic systems is of vital significance in controlling and synchronization of fractional-order chaotic systems. However there exist basic hypotheses in traditional…
The applicability of machine learning for predicting chaotic dynamics relies heavily upon the data used in the training stage. Chaotic time series obtained by numerically solving ordinary differential equations embed a complicated noise of…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…