Autonomous system (mathematics)
mathematical equations
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.
Definition
An autonomous system is a system of ordinary differential equations of the form
d
d
t
x
(
t
)
=
f
(
x
(
t
)
)
{\displaystyle {\frac {d}{dt}}x(t)=f(x(t))}
where x takes values in n-dimensional Euclidean space; t is often interpreted as time.
It is distinguished from systems of differential equations of the form
d
d
t
x
(
t
)
=
g
(
x
(
t
)
,
t
)
{\displaystyle {\frac {d}{dt}}x(t)=g(x(t),t)}
in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter t, again often interpreted as time; such systems are by definition not autonomous.
Properties
Solutions are invariant under horizontal translations:
Let
x
1
(
t
)
{\displaystyle x_{1}(t)}
be a unique solution of the initial value problem for an autonomous system
d
d
t
x
(
t
)
=
f
(
x
(
t
)
)
,
x
(
0
)
=
x
0
.
{\displaystyle {\frac {d}{dt}}x(t)=f(x(t))\,,\quad x(0)=x_{0}.}
Then
x
2
(
t
)
=
x
1
(
t
−
t
0
)
{\displaystyle x_{2}(t)=x_{1}(t-t_{0})}
solves
d
d
t
x
(
t
)
=
f
(
x
(
t
)
)
,
x
(
t
0
)
=
x
0
.
{\displaystyle {\frac {d}{dt}}x(t)=f(x(t))\,,\quad x(t_{0})=x_{0}.}
Denoting
s
=
t
−
t
0
{\displaystyle s=t-t_{0}}
gets
x
1
(
s
)
=
x
2
(
t
)
{\displaystyle x_{1}(s)=x_{2}(t)}
and
d
s
=
d
t
{\displaystyle ds=dt}
, thus
d
d
t
x
2
(
t
)
=
d
d
t
x
1
(
t
−
t
0
)
=
d
d
s
x
1
(
s
)
=
f
(
x
1
(
s
)
)
=
f
(
x
2
(
t
)
)
.
{\displaystyle {\frac {d}{dt}}x_{2}(t)={\frac {d}{dt}}x_{1}(t-t_{0})={\frac {d}{ds}}x_{1}(s)=f(x_{1}(s))=f(x_{2}(t)).}
For the initial condition, the verification is trivial,
x
2
(
t
0
)
=
x
1
(
t
0
−
t
0
)
=
x
1
(
0
)
=
x
0
.
{\displaystyle x_{2}(t_{0})=x_{1}(t_{0}-t_{0})=x_{1}(0)=x_{0}.}
Example
The equation
y
′
=
(
2
−
y
)
y
{\displaystyle y'=\left(2-y\right)y}
is autonomous, since the independent variable (
x
{\displaystyle x}
) does not explicitly appear in the equation.
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