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EspañolAUTONOMOUS SYSTEM (MATHEMATICS)
(Redirected from Autonomous differential equation)
In mathematics, an 'autonomous system' or 'autonomous differential equation' is a system of ordinary differential equations which does not depend on the independent variable.
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 systems.
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.
Every initial value problem for an autonomous system
:
is equivalent to
:
for some ''y''0′.
★ Time-invariant system
In mathematics, an 'autonomous system' or 'autonomous differential equation' is a system of ordinary differential equations which does not depend on the independent variable.
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 systems.
| Contents |
| Definition |
| Properties |
| See also |
Definition
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.
Properties
Every initial value problem for an autonomous system
:
is equivalent to
:
for some ''y''0′.
See also
★ Time-invariant system
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