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We scrutinize the use of machine learning, based on reservoir computing, to build data-driven effective models of multiscale chaotic systems. We show that, for a wide scale separation, machine learning generates effective models akin to…
Chaotic systems which are due to nonlinearity have attracted a great concern in the current world and chaotic models. Systems for a wide range of operation conditions have their application in almost all branches of engineering and science.…
A model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the physical processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have…
In the era of big data, one of the key challenges is the development of novel optimization algorithms that can accommodate vast amounts of data while at the same time satisfying constraints and limitations of the problem under study. The…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
Time delays are a common perturbation in systems with many states, such as networked, distributed, or decentralized systems. Current methods analyzing the stability of large systems with time delay typically produce very conservative…
The Lyapunov exponent is well-known in deterministic dynamical systems as a measure for quantifying chaos and detecting coherent regions in physically evolving systems. In this Letter, we show how the Lyapunov exponent can be unified with…
The Lyapunov exponent is used to characterize the stability of the dynamic response of the system, and it is often employed to verify if a system is chaotic. Since its discovery in the nineteenth century, various methods have been proposed…
If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is…
Numerical solutions to Newton's equations of motion for chaotic self gravitating systems of more than 2 bodies are often regarded to be irreversible. This is due to the exponential growth of errors introduced by the integration scheme and…
In chaotic dynamical systems, an infinitesimal perturbation is exponentially amplified at a time-rate given by the inverse of the maximum Lyapunov exponent $\lambda$. In fully developed turbulence, $\lambda$ grows as a power of the Reynolds…
We study the chaoticity and the predictability of a turbulent flow on the basis of high-resolution direct numerical simulations at different Reynolds numbers. We find that the Lyapunov exponent of turbulence, which measures the exponential…
Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their…
Cancer is a disease that takes millions of lives every year. Then, to propose treatments, avoid recurrence, and improve the patient's life quality, we need to analyze this disease from a biophysical perspective with a solid mathematical…
In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on…
Using 1200 CPUs of the National Supercomputer TH-A1 and a parallel integral algorithm based on the 3500th-order Taylor expansion and the 4180-digit multiple precision data, we have done a reliable simulation of chaotic solution of Lorenz…
A time-stepping scheme with adaptivity in both the step size and the integration order is presented in the context of non-equilibrium dynamics described via Kadanoff-Baym equations. The accuracy and effectiveness of the algorithm are…
Proper states' representations are the key to the successful dynamics modeling of chaotic systems. Inspired by recent advances of deep representations in various areas such as natural language processing and computer vision, we propose the…
The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete…
Linear dynamical systems are canonical models for learning-based control of plants with uncertain dynamics. The setting consists of a stochastic differential equation that captures the state evolution of the plant understudy, while the true…