Dynamical system

Linear dynamical system Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems in general do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point. Introduction[edit] In a linear dynamical system, the variation of a state vector (an -dimensional vector denoted ) equals a constant matrix (denoted ) multiplied by varies continuously with time or as a mapping, in which varies in discrete steps These equations are linear in the following sense: if and are two valid solutions, then so is any linear combination of the two solutions, e.g where need not be symmetric. Solution of linear dynamical systems[edit] If the initial vector is aligned with a right eigenvector If ) of the matrix .

Lagrangian The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. The Lagrangian is named after Italian-French mathematician and astronomer Joseph Louis Lagrange. The concept of a Lagrangian was introduced in a reformulation of classical mechanics introduced by Lagrange known as Lagrangian mechanics. Definition[edit] In classical mechanics, the natural form of the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V.[1] In symbols, If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation. , but solving any equivalent Lagrangians will give the same equations of motion.[2][3] The Lagrangian formulation[edit] Simple example[edit] The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a (local) minimum. Importance[edit] does not depend on . . .

Dynamical systems theory 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. When difference equations are employed, the theory is called discrete dynamical systems. This theory deals with the long-term qualitative behavior of dynamical systems, and studies the solutions of the equations of motion of systems that are primarily mechanical in nature; although this includes both planetary orbits as well as the behaviour of electronic circuits and the solutions to partial differential equations that arise in biology. This field of study is also called just Dynamical systems, Mathematical Dynamical Systems Theory and Mathematical theory of dynamical systems. Overview[edit] Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. History[edit]

Limit-cycle Stable limit cycle (shown in bold) and two other trajectories spiraling into it In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854-1912). Definition[edit] We consider a two-dimensional dynamical system of the form where is a smooth function. with values in which satisfies this differential equation. such that for all . Properties[edit] By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve. Stable, unstable and semi-stable limit cycles[edit] Finding limit cycles[edit] .

Orbit (dynamics) 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. Diagram showing the periodic orbit of a mass-spring system in simple harmonic motion. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams) Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function where we define then the set is called orbit through x. for every point x on the orbit. Given a real dynamical system (R, M, Φ), I(x)) is an open interval in the real numbers, that is . 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 : Usually different notation is used : is written as where is in the above notation. acting on a probability space is a lattice inside

List of chaotic maps List of chaotic maps[edit] List of fractals[edit] Limit set 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. Types[edit] In general limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all possible limit sets as a union of fixed points and periodic orbits. Definition for iterated functions[edit] Let be a metric space, and let be a continuous function. -limit set of , denoted by , is the set of cluster points of the forward orbit of the iterated function . if and only if there is a strictly increasing sequence of natural numbers such that as . where denotes the closure of set . If is a homeomorphism (that is, a bicontinuous bijection), then the Both sets are in R so that and

Poincaré–Bendixson theorem Theorem[edit] Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countably many homoclinic orbits connecting one fixed point. A weaker version of the theorem was originally conceived by Henri Poincaré, although he lacked a complete proof which was later given by Ivar Bendixson (1901). Discussion[edit] The condition that the dynamical system be on the plane is necessary to the theorem. Applications[edit] One important implication is that a two-dimensional continuous dynamical system cannot give rise to a strange attractor. References[edit] Jump up ^ Teschl, Gerald (2012).