In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. 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.
Floris Takens was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems.
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems and discrete systems.
The Instituto Nacional de Matemática Pura e Aplicada is widely considered to be the foremost research and educational institution of Brazil in the area of mathematics. It is located in the city of Rio de Janeiro, and was formerly known simply as Instituto de Matemática Pura e Aplicada (IMPA), whose abbreviation remains in use.
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations.
In the study of dynamical systems, a homoclinic orbit is a path through phase space which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium. It is a heteroclinic orbit–a path between any two equilibrium points–in which the endpoints are one and the same.
In mathematics, Smale's axiom A defines a class of dynamical systems which have been extensively studied and whose dynamics is relatively well understood. A prominent example is the Smale horseshoe map. The term "axiom A" originates with Stephen Smale. The importance of such systems is demonstrated by the chaotic hypothesis, which states that, 'for all practical purposes', a many-body thermostatted system is approximated by an Anosov system.
Smale's problems are a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list of Hilbert's problems that had been published at the beginning of the 20th century.
Paul Glendinning is a professor of Applied Mathematics, in the School of Mathematics at the University of Manchester who is known for his work on dynamical systems, specifically models of the time-evolution of complex mathematical or physical processes. His main areas of research are bifurcation theory ; synchronization and blowout bifurcations; low-dimensional maps; and quasi-periodically forced systems.
Oscar Erasmus Lanford III was an American mathematician working on mathematical physics and dynamical systems theory.
Artur Avila Cordeiro de Melo is a Brazilian and naturalized French mathematician working primarily in the fields of dynamical systems and spectral theory. He is one of the winners of the 2014 Fields Medal, being the first Latin American and lusophone to win such an award. He has been a researcher at both the IMPA and the CNRS. He has been a professor at the University of Zurich since September 2018.
Welington Celso de Melo was a Brazilian mathematician. Known for his contributions to dynamical systems theory, he served as full professor at Instituto Nacional de Matemática Pura e Aplicada from 1980 to 2016. Melo wrote numerous papers, one being a complete description of the topological behavior of 1-dimensional real dynamical systems . He proved the global hyperbolicity of renormalization for Cr unimodal maps. He was a recipient of the 2003 TWAS Prize.
Valentin Afraimovich was a Soviet, Russian and Mexican mathematician. He made contributions to dynamical systems theory, qualitative theory of ordinary differential equations, bifurcation theory, concept of attractor, strange attractors, space-time chaos, mathematical models of non-equilibrium media and biological systems, travelling waves in lattices, complexity of orbits and dimension-like characteristics in dynamical systems.
Jean-Pierre Eckmann is a Swiss mathematical physicist in the department of theoretical physics at the University of Geneva and a pioneer of chaos theory and social network analysis.
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.
Marcelo Miranda Viana da Silva is a Brazilian mathematician working in dynamical systems theory.
Ricardo Mañé Ramirez was a Uruguayan mathematician, known for his contributions to dynamical systems and ergodic theory. He was a doctoral student of Jacob Palis at IMPA.
Enrique Ramiro Pujals is a Brazilian mathematician known for his contributions to the understanding of dynamical systems. Since fall of 2018, he has been a professor at the Graduate Center at the City University of New York.
Albert Fathi is an Egyptian-French mathematician. He specializes in dynamical systems and is currently a professor at the Georgia Institute of Technology.
Zhihong "Jeff" Xia is a Chinese-American mathematician.
