## Dynamical system |

In **dynamical system** is a system in which a

At any given time, a dynamical system has a *evolution rule* of the dynamical system is a function that describes what future states follow from the current state. Often the function is ^{[1]}^{[2]} However, some systems are

In **dynamical system** is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives."^{[3]} In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized.

The study of dynamical systems is the focus of ^{[4]}^{[5]} ^{[6]} ^{[7]} ^{[8]}

- overview
- history
- basic definitions
- linear dynamical systems
- local dynamics
- bifurcation theory
- ergodic systems
- multidimensional generalization
- see also
- references
- further reading
- external links

The concept of a dynamical system has its origins in *solving the system* or *integrating the system*. If the system can be solved, given an initial point it is possible to determine all its future positions, a collection of points known as a * trajectory* or

Before the advent of

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 orstructural 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 theirequivalence 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 systems andsystems 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 thetransition 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 forhyperbolic systems . Understanding the probabilistic aspects of dynamical systems has helped establish the foundations ofstatistical mechanics and ofchaos .