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 where takes values in -dimensional Euclidean space; is often interpreted as time.

in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous.

Properties

Solutions are invariant under horizontal translations:

be a unique solution of the initial value problem

Example

) does not explicitly appear in the equation. To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB .

Solving the equation symbolically in MATLAB, by running obtains two equilibrium, .

Picking up some specific values for the initial condition, one can add the plot of several solutions

Qualitative analysis

Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line.

Solution techniques

-dimensional first-order system (as described in reduction to a first-order system), but not necessarily vice versa.

First order

is separable

Second order

is more difficult, but it can be solved and expressing the second derivative via the chain rule:

which is an implicit solution.

Special case: x″ = f(x)

benefits from separate treatment. These types of equations are very common in classical mechanics because they are always Hamiltonian systems.

The idea is to make use of the identity

which follows from the chain rule, barring any issues due to division by zero.

:

:

. The original second order equation can now be integrated:

This is an implicit solution. The greatest potential problem is inability to simplify the integrals, which implies difficulty or impossibility in evaluating the integration constants.

Special case: x″ = x′n f(x)

Using the above approach, the technique can extend to the more general equation

. Rewriting the second derivative, rearranging, and expressing the left side as a derivative:

:

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 or dependence of the right side of the equation on the dependent variable only (i.e., not its derivatives). 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.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum.

Multivariate case

.

constant vector.

Finite durations

For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stay there in zero forever after. These finite-duration solutions cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations.

As example, the equation:

Admits the finite duration solution:

See also

• Non-autonomous system (mathematics)

References