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## Dynamical systemAdynamical system is a deterministic process in which a variable's value changes over time according to a well-defined rule which only involves the variable's current value.
## Dynamical systems and chaos theoryThis branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"
An important goal is to describe the fixed points, or steady states of a given dynamical systems; these are values of the variable which won't change over time. Some of these fixed points are
Similarly, one is interested in
Even simple nonlinear dynamical systems often exhibit almost random, completely unpredictable behavior that has been called ## Types of dynamical systemsA dynamical system is calleddiscrete if time is measured in discrete steps; these are modeled as recursive relations as for instance in the logistic map
n denotes the discrete time steps and x is the variable changing over time. If time is measured continuously, the resulting continuous dynamical systems are expressed as ordinary differential equations, for instance
x is the variable that changes with time t.
The changing variable
We distinguish between
The two examples given earlier are nonlinear systems. These are much harder to analyze and often exhibit a phenomenon known as chaos which marks complete unpredictability; see also nonlinearity. ## Examples of dynamical systemsSee also: List of dynamical system topics | |||||||

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