Editing Dynamical system (section)

==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 an implicit relation that gives the state of the system for only a short time into the future.  (The relation is either a [[differential equation]], [[Recurrence relation|difference equation]] or other [[Time scale calculus|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''. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a ''[[trajectory]]'' or ''[[orbit (dynamics)|orbit]]''.

Before the advent of [[computers]], finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems.  Numerical methods implemented on electronic computing machines 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 is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their [[Equivalence relation|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 do not change under coordinate changes. [[Linear dynamical system]]s and [[Poincaré–Bendixson theorem|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 theory|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 [[Turbulence|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 theory|ergodic systems]] and a more detailed understanding has been worked out for [[Anosov diffeomorphism|hyperbolic systems]]. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of [[statistical mechanics]] and of [[chaos theory|chaos]].