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.
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge at a rate given by
James A. Yorke is a Distinguished University Research Professor of Mathematics and Physics and former chair of the Mathematics Department at the University of Maryland, College Park.
Grigory Aleksandrovich Margulis is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978, a Wolf Prize in Mathematics in 2005, and an Abel Prize in 2020, becoming the fifth mathematician to receive the three prizes. In 1991, he joined the faculty of Yale University, where he is currently the Erastus L. De Forest Professor of Mathematics.
Yakov Grigorevich Sinai is a Russian-American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dynamical systems and connected the world of deterministic (dynamical) systems with the world of probabilistic (stochastic) systems. He has also worked on mathematical physics and probability theory. His efforts have provided the groundwork for advances in the physical sciences.
David Pierre Ruelle is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term strange attractor, and developed a new theory of turbulence.
In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1.
Oscar Eramus Lanford III was an American mathematician working on mathematical physics and dynamical systems theory.
Elon Lindenstrauss is an Israeli mathematician, and a winner of the 2010 Fields Medal.
Amie Wilkinson is an American mathematician and Professor of Mathematics at the University of Chicago. Her research topics include smooth dynamical systems, ergodic theory, chaos theory and semisimple Lie groups. Wilkinson, in collaboration with Christian Bonatti and Sylvain Crovisier, partially resolved the twelfth problem on Stephen Smale's list of mathematical problems for the 21st Century.
Anatoly Borisovich Katok was an American mathematician with Russian-Jewish origins. Katok was the director of the Center for Dynamics and Geometry at the Pennsylvania State University. His field of research was the theory of dynamical systems.
Caroline Mary Series is an English mathematician known for her work in hyperbolic geometry, Kleinian groups and dynamical systems.
Lai-Sang Lily Young is a Hong Kong-born American mathematician who holds the Henry & Lucy Moses Professorship of Science and is a professor of mathematics and neural science at the Courant Institute of Mathematical Sciences of New York University. Her research interests include dynamical systems, ergodic theory, chaos theory, probability theory, statistical mechanics, and neuroscience. She is particularly known for introducing the method of Markov returns in 1998, which she used to prove exponential correlation delay in Sinai billiards and other hyperbolic dynamical systems.
The Michael Brin Prize in Dynamical Systems, abbreviated as the Brin Prize, is awarded to mathematicians who have made outstanding advances in the field of dynamical systems and are within 14 years of their PhD. The prize is endowed by and named after Michael Brin, whose son Sergey Brin, is a co-founder of Google. Michael Brin is a retired mathematician at the University of Maryland and a specialist in dynamical systems.
Viviane Baladi is a mathematician who works as a director of research at the Centre national de la recherche scientifique (CNRS) in France. Originally Swiss, she has become a naturalized citizen of France. Her research concerns dynamical systems.
Yakov Borisovich Pesin was born in Moscow, Russia on December 12, 1946. Pesin is currently a Distinguished Professor in the Department of Mathematics and the Director of the Anatole Katok Center for Dynamical Systems and Geometry at the Pennsylvania State University (PSU). His primary areas of research are the theory of dynamical systems with an emphasis on smooth ergodic theory, dimension theory in dynamical systems, and Riemannian geometry, as well as mathematical and statistical physics.
Luís Barreira is a Portuguese mathematician and a professor in the department of mathematics of Instituto Superior Técnico.
Pierre Collet is a French mathematical physicist, specializing in statistical mechanics, stochastic processes, and chaos theory.
François Ledrappier is a French mathematician.
In the mathematical discipline of ergodic theory, a Sinai–Ruelle–Bowen (SRB) measure is an invariant measure that behaves similarly to, but is not an ergodic measure. In order to be ergodic, the time average would need to be equal the space average for almost all initial states , with being the phase space. For an SRB measure , it suffices that the ergodicity condition be valid for initial states in a set of positive Lebesgue measure.
