In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables. In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized. The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept. (Wikipedia).
Discrete-Time Dynamical Systems
This video shows how discrete-time dynamical systems may be induced from continuous-time systems. https://www.eigensteve.com/
From playlist Data-Driven Dynamical Systems
The Anatomy of a Dynamical System
Dynamical systems are how we model the changing world around us. This video explores the components that make up a dynamical system. Follow updates on Twitter @eigensteve website: eigensteve.com
From playlist Research Abstracts from Brunton Lab
Topics in Dynamical Systems: Fixed Points, Linearization, Invariant Manifolds, Bifurcations & Chaos
This video provides a high-level overview of dynamical systems, which describe the changing world around us. Topics include nonlinear dynamics, linearization at fixed points, eigenvalues and eigenvectors, bifurcations, invariant manifolds, and chaos!! @eigensteve on Twitter eigensteve.co
From playlist Dynamical Systems (with Machine Learning)
Data-Driven Dynamical Systems Overview
This video provides a high-level overview of this new series on data-driven dynamical systems. In particular, we explore the various challenges in modern dynamical systems, along with emerging techniques in data science and machine learning to tackle them. The two chief challenges are 1)
From playlist Data-Driven Dynamical Systems with Machine Learning
Mathematical modeling of evolving systems
Discover the multidisciplinary nature of the dynamical principles at the core of complexity science. COURSE NUMBER: CAS 522 COURSE TITLE: Dynamical Systems LEVEL: Graduate SCHOOL: School of Complex Adaptive Systems INSTRUCTOR: Enrico Borriello MODE: Online SEMESTER: Fall 2021 SESSION:
From playlist What is complex systems science?
Reactive Systems use a high-performance software architecture. They are resilient under stress, and their reactive design allows them to scale elastically to meet demand. The reactive design approach allows the creation of more complex, more flexible systems and forms the basis for some of
From playlist Software Engineering
Differential Equations and Dynamical Systems: Overview
This video presents an overview lecture for a new series on Differential Equations & Dynamical Systems. Dynamical systems are differential equations that describe any system that changes in time. Applications include fluid dynamics, elasticity and vibrations, weather and climate systems,
From playlist Engineering Math: Differential Equations and Dynamical Systems
This video introduces chaotic dynamical systems, which exhibit sensitive dependence on initial conditions. These systems are ubiquitous in natural and engineering systems, from turbulent fluids to the motion of objects in the solar system. Here, we discuss how to recognize chaos and how
From playlist Engineering Math: Differential Equations and Dynamical Systems
Musimathics: Dynamical Systems (Part 9)
Welcome to the Musimathics series! Musimathics gives an overview of some of the most interesting topics in the field of mathematical music theory! You are watching the ninth video in the series. In this video, Chloe goes over the basics of dynamics, as well as talking about some interesti
From playlist Musimathics: Music & Math
Deep Learning to Discover Coordinates for Dynamics: Autoencoders & Physics Informed Machine Learning
Joint work with Nathan Kutz: https://www.youtube.com/channel/UCoUOaSVYkTV6W4uLvxvgiFA Discovering physical laws and governing dynamical systems is often enabled by first learning a new coordinate system where the dynamics become simple. This is true for the heliocentric Copernican syste
From playlist Data-Driven Dynamical Systems with Machine Learning
Steve Brunton: "Dynamical Systems (Part 1/2)"
Watch part 2/2 here: https://youtu.be/HgeC0-VIUtc Machine Learning for Physics and the Physics of Learning Tutorials 2019 "Dynamical Systems (Part 1/2)" Steve Brunton, University of Washington Institute for Pure and Applied Mathematics, UCLA September 5, 2019 For more information: http
From playlist Machine Learning for Physics and the Physics of Learning 2019
Serhiy Yanchuk - Adaptive dynamical networks: from multiclusters to recurrent synchronization
Recorded 02 September 2022. Serhiy Yanchuk of Humboldt-Universität presents "Adaptive dynamical networks: from multiclusters to recurrent synchronization" at IPAM's Reconstructing Network Dynamics from Data: Applications to Neuroscience and Beyond. Abstract: Adaptive dynamical networks is
From playlist 2022 Reconstructing Network Dynamics from Data: Applications to Neuroscience and Beyond
Machine learning analysis of chaos and vice versa - Edward Ott, University of Maryland
About the talk In this talk we first consider the situation where one is interested in gaining understanding of general dynamical properties of a chaotically time evolving system solely through access to time series measurements that depend on the evolving state of an, otherwise unknown,
From playlist Turing Seminars
Koopman Spectral Analysis (Overview)
In this video, we introduce Koopman operator theory for dynamical systems. The Koopman operator was introduced in 1931, but has experienced renewed interest recently because of the increasing availability of measurement data and advanced regression algorithm. https://www.eigensteve.com
From playlist Koopman Analysis
Sparse Identification of Nonlinear Dynamics (SINDy)
This video illustrates a new algorithm for the sparse identification of nonlinear dynamics (SINDy). In this work, we combine machine learning, sparse regression, and dynamical systems to identify nonlinear differential equations purely from measurement data. From the Paper: Discovering
From playlist Research Abstracts from Brunton Lab
Dynamical systems evolving – Lai-Sang Young – ICM2018
Plenary Lecture 8 Dynamical systems evolving Lai-Sang Young Abstract: I will discuss a number of results taken from a cross-section of my work in Dynamical Systems theory and applications. The first topics are from the ergodic theory of chaotic dynamical systems. They include relation be
From playlist Plenary Lectures
Ludovic Rifford : Geometric control and dynamics
Abstract: The geometric control theory is concerned with the study of control systems in finite dimension, that is dynamical systems on which one can act by a control. After a brief introduction to controllability properties of control systems, we will see how basic techniques from control
From playlist Dynamical Systems and Ordinary Differential Equations