Autonomous system (mathematics)

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 system.

Definition

An autonomous system is a system of ordinary differential equations of the form:$frac\{d\}\{dt\}x(t)=f(x(t))$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:$frac\{d\}\{dt\}x(t)=g(x(t),t)$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 :$frac\{d\}\{dt\}x(t)=f(x(t))\; ,\; mathrm\{,\}\; quad\; x(t\_0)=y\_0$is equivalent to:$frac\{d\}\{dt\}x(t)=f(x(t))\; ,\; mathrm\{,\}\; quad\; x(0)=y\_0^\{\text{'}\}$for some "y"₀′ provided the translation is well defined.

Counterexample

Consider the following problem::$frac\{d\}\{dt\}x(t)=-x(t)^2\; ,\; mathrm\{,\}\; quad\; x(1)=1$whose solution is given by::$x(t)=frac\{1\}\{t\}$Obviously, this function is not well defined at $t=0,$

Solution techniques

The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order $n$ is equivalent to an $n$-dimensional first-order system (as described in Ordinary differential equation#Reduction to a first order system), but not necessarily vice versa.

First order

The first-order autonomous equation:$frac\{dx\}\{dt\}\; =\; f(x)$is separable, so it can easily be solved by rearranging it into the integral form:$int\; frac\{dx\}\{f(x)\}\; =\; int\; dt$

Second order

The second-order autonomous equation:$frac\{d^2x\}\{dt^2\}\; =\; f(x,\; x\text{'})$is more difficult, but it can be solved by introducing the new variable:$v\; =\; frac\{dx\}\{dt\}$and expressing the second derivative of $x$ (via the chain rule) as:$frac\{d^2x\}\{dt^2\}\; =\; frac\{dv\}\{dt\}\; =\; frac\{dx\}\{dt\}frac\{dv\}\{dx\}\; =\; vfrac\{dv\}\{dx\}$This eliminates all reference to the independent variable $t$ and gives a first-order equation that, if solved, provides $v$ as a function of $x$. Then the separable equation:$frac\{dx\}\{dt\}\; =\; v(x)$can easily be solved to give $x$ as a function of $t$. [cite book |last=Boyce |first=William E. |coauthors=Richard C. DiPrima |title=Elementary Differential Equations and Boundary Volume Problems |edition=8th ed. |year=2005 |publisher=John Wiley & Sons |isbn=0-471-43338-1 |pages=page 133]

Higher orders

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

See also

* Time-invariant system

References