Orbit (dynamics)

From Academic Kids

In the study of dynamical systems, an orbit is a collection of points related by time evolution. If the dynamical system is a map, the orbit is a sequence generated by iterating the map. This is a discrete-time dynamical system.

If the dynamical system is a flow, the orbit is a curve generated by a function F(t,x0). With an initial value x0, this function returns the value for all times t. This is called a continuous-time dynamical system.

An orbit is called closed if a point of the orbit evolves to itself. This means that the orbit will repeat itself. Such orbits are also called periodic. The simplest closed orbit is a fixed point, where the orbit is a single point.

An orbit is asymptotically periodic if the orbit converges to a periodic orbit. Such orbits are not closed because they never truly repeat, but they become arbitrarily close to a repeating orbit.

The most interesting orbits are those that are chaotic. These orbits are not closed or asymptotically periodic. They exhibit sensitive dependence on initial conditions, meaning that small differences in the initial value will cause large differences in future points of the orbit.

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