In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As a phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems.
For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces.
Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function
where with
we define
then the set
is called orbit through x. An orbit which consists of a single point is called constant orbit. A non-constant orbit is called closed or periodic if there exists a in such that
Given a real dynamical system (R, M, Φ), I(x) is an open interval in the real numbers, that is . For any x in M
is called positive semi-orbit through x and
is called negative semi-orbit through x.
For discrete time dynamical system :
forward orbit of x is a set :
backward orbit of x is a set :
and orbit of x is a set :
where :
is an evolution function which is here an iterated function,
set is dynamical space,
is number of iteration, which is natural number and
is initial state of system and
Usually different notation is used :
is written as
where is in the above notation.
For a general dynamical system, especially in homogeneous dynamics, when one has a "nice" group acting on a probability space in a measure-preserving way, an orbit will be called periodic (or equivalently, closed) if the stabilizer is a lattice inside .
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Le cours étudie les concepts fondamentaux de l'analyse vectorielle et l'analyse de Fourier en vue de leur utilisation pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.
The course provides students with the tools to approach the study of nonlinear systems and chaotic dynamics. Emphasis is given to concrete examples and numerical applications are carried out during th
Introduction to local and global behavior of
nonlinear dynamical systems arising from maps and ordinary differential equations. Theoretical and computational aspects studied.
In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right: with the circle‐shaped symbol of function composition.
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Given a mapping f from a set X into itself, a point x in X is called periodic point if there exists an n so that where f_n is the nth iterate of f. The smallest positive integer n satisfying the above is called the prime period or least period of the point x.
In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A system that has reached its limiting set is said to be at equilibrium.
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