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A discrete dynamical system, discrete-time dynamical system is a tuple (T, M, Φ), where M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers we call the system a semi-cascade. [14]
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems .
Supersymmetric theory of stochastic dynamics (STS) is a multidisciplinary approach to stochastic (partial) differential equations (SDEs) at the intersection of dynamical systems theory, the traditional theory of stochastic differential equations, topological field theories, and the theory of pseudo-Hermitian operators. The theory can be viewed ...
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 ...
Structural stability of the system provides a justification for applying the qualitative theory of dynamical systems to analysis of concrete physical systems. The idea of such qualitative analysis goes back to the work of Henri Poincaré on the three-body problem in celestial mechanics .
Ergodic theory is often concerned with ergodic transformations.The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set. E.g. if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not ...
Katok was also known for formulating conjectures and problems (for some of which he even offered prizes) that influenced bodies of work in dynamical systems. The best-known of these is the Katok Entropy Conjecture, which connects geometric and dynamical properties of geodesic flows. It is one of the first rigidity statements in dynamical systems.
The journal was established in 2007 with Anatole Katok as the founding editor-in-chief. [2] It covers the theory of dynamical systems with particular emphasis on the mutual interaction between dynamics and other major areas of mathematical research: number theory, symplectic geometry, differential geometry, rigidity, quantum chaos, Teichmüller theory, geometric group theory, and harmonic ...