In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Grigori Yakovlevich Perelman is a Russian mathematician. He has made contributions to Riemannian geometry and geometric topology. In 1994, Perelman proved the soul conjecture. In 2003, he proved Thurston's geometrization conjecture. The proof was confirmed in 2006. This consequently solved in the affirmative the Poincaré conjecture.
In control theory, the discrete Lyapunov equation is of the form
Pierre-Louis Lions is a French mathematician. He is the recipient of the 1994 Fields Medal.
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.
Aleksandr Aleksandrovich Andronov was a Soviet physicist and member of the Soviet Academy of Sciences (1946). He worked extensively on the theory of stability of dynamical systems, introducing the notion of structural stability. In that context, he also contributed to the mathematical theory of self-oscillation by establishing a link between the generation of oscillations and the theory of Lyapunov stability. He developed the comprehensive theory of self-oscillations by linking it with the qualitative theory of differential equations, topology, and with the general theory of stability of motion. The crater Andronov on the Moon is named after him.
In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.
John Willard Morgan is an American mathematician, with contributions to topology and geometry.
Bernard Malgrange is a French mathematician who works on differential equations and singularity theory. He proved the Ehrenpreis–Malgrange theorem and the Malgrange preparation theorem, essential for the classification theorem of the elementary catastrophes of René Thom. He received his Ph.D. from Université Henri Poincaré in 1955. His advisor was Laurent Schwartz. He was elected to the Académie des sciences in 1988. In 2012 he gave the Łojasiewicz Lecture at the Jagiellonian University in Kraków.
Martin Hairer is an Austrian mathematician working in the field of stochastic analysis, in particular stochastic partial differential equations. He is Professor of Mathematics at Imperial College London, having previously held appointments at the University of Warwick and the Courant Institute of New York University. In 2014 he was awarded the Fields Medal, one of the highest honours a mathematician can achieve.
Cédric Patrice Thierry Villani is a French mathematician and politician working primarily on partial differential equations, Riemannian geometry and mathematical physics. He was awarded the Fields Medal in 2010 and he was the director of Sorbonne University's Institut Henri Poincaré from 2009 to 2017.
In mathematics, the Lyapunov–Schmidt reduction or Lyapunov–Schmidt construction is used to study solutions to nonlinear equations in the case when the implicit function theorem does not work. It permits the reduction of infinite-dimensional equations in Banach spaces to finite-dimensional equations. It is named after Aleksandr Lyapunov and Erhard Schmidt.
Sylvie Benzoni-Gavage is a French mathematician known for her research in partial differential equations, fluid dynamics, traffic flow, shock waves, and phase transitions. In 2017 she was named as the director of the Institut Henri Poincaré.
Nikolay Gur'yevich Chetaev is a Russian Soviet mechanician and mathematician. He was born in Karaduli, Laishevskiy uyezd, Kazan province, Russian Empire and died in Moscow, USSR. He belongs to the Kazan school of mathematics.
