Автономды дифференциал теңдеулер жүйесі

System of ordinary differential equations whose current state solely determines its evolution

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

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

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.

In , where is an dimensional column vector dependent on
The solution is where is an constant vector.