Autonomous system (mathematics)
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems.
Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future.
Autonomous systems are closely related to dynamical systems. Any autonomous system can be transformed into a dynamical system and, using very weak assumptions, a dynamical system can be transformed into an autonomous system.
An autonomous system is a system of ordinary differential equations of the form
where x takes values in n-dimensional Euclidean space and t is usually time.
It is distinguished from systems of differential equations of the form
in which the law governing the rate of motion of a particle depends not only on the particle's location, but also on time; such systems are not autonomous.
Let be a unique solution of the initial value problem for an autonomous system
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