Dynamical system Totally Explained

Everything about Dynamic System totally explained

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient 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 spring in a lake.
A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state.

Overview

The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. Once the system can be solved, given an initial point it's possible to determine all its future points, a collection known as a trajectory or orbit.
Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems. Numerical methods executed on computers have simplified the task of determining the orbits of a dynamical system.
For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:

The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there's a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that don't change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.
The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.
The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos. It was in the work of Poincaré that these dynamical systems themes developed.

Basic definitions

A dynamical system is a manifold M called the phase (or state) space and a smooth evolution function Φ^t that for any element of t ∈ T, the time, maps a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it's a cascade or a map; and the restriction to the non-negative integers is a semi-cascade.

Examples

The evolution function Φ^t is often the solution of a differential equation of motion ť

dotsigma ) = mu(sigma)

. Combining the above, a map τ is said to be a measure-preserving transformation of X , if it's a map from X to itself, it's Σ-measurable, and is measure-preserving. The quadruple

(X,Sigma,mu,	au)

, for such a τ, is then defined to be a dynamical system.
The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates

au^n=	au circ 	au circ ldotscirc	au

for integer n are studied. For continuous dynamical systems, the map τ is understood to be finite time evolution map and the construction is more complicated.

Examples of dynamical systems

Wikipedia links

Arnold's cat map

Baker's map is an example of a chaotic piecewise linear map

Circle map

Double pendulum

Billiards and Outer Billiards

Henon map

Rossler map ==

Further Information

Get more info on 'Dynamic System'.

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