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.
An autonomous system is a system of ordinary differential equations of the formwhere takes values in -dimensional Euclidean space; is often interpreted as time.
It is distinguished from systems of differential equations of the formin 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.
Solutions are invariant under horizontal translations:
Let
x1(t)
x2(t)=x1(t-t0)
s=t-t0
x1(s)=x2(t)
ds=dt
The equation
y'=\left(2-y\right)y
x
\left(2-y\right)y
x
y(x)=y(x-x0)
x0
Solving the equation symbolically in MATLAB, by running
y=0
y=2
C3
Picking up some specific values for the initial condition, one can add the plot of several solutions
1);y3 = dsolve(equation, y(3)
3);y5 = dsolve(equation, y(2)
3);% plot the solutionsezplot(y1, [0 6]); ezplot(y2, [0 6]); ezplot(y3, [0 6]);ezplot(y4, [0 6]); ezplot(y5, [0 6]); ezplot(y6, [0 6]);title('Slope field, isoclines and solutions for f(x,y)=(2-y)y')legend('Slope field', 'Isoclines', 'Solutions y_');text([1 2 3], [1 1 1], strcat('\leftarrow',));text([1 2 3], [3 3 3], strcat('\leftarrow',));grid on;
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
n
n
The first-order autonomous equationis separable, so it can be solved by rearranging it into the integral form
The second-order autonomous equationis more difficult, but it can be solved[1] by introducing the new variableand expressing the second derivative of
x
t
v
x
v
which is an implicit solution.
The special case where
f
x'
benefits from separate treatment.[2] 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.
By inverting both sides of a first order autonomous system, one can immediately integrate with respect to
x
which is another way to view the separation of variables technique. The second derivative must be expressed as a derivative with respect to
x
t
To reemphasize: what's been accomplished is that the second derivative with respect to
t
x
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.
Using the above approach, the technique can extend to the more general equation
where
n
x'
The right will carry +/− if
n
n=2
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[3] [4] (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.[5]
See main article: Matrix differential equation. In
x'(t)=Ax(t)
x(t)
n
t
The solution is
x(t)=eAc
c
n x 1
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,[7] meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. This finite-duration solutions cannot be analytical functions on the whole real line, and because they will being non-Lipschitz function at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations.
As example, the equation:
y'=-sgn(y)\sqrt{|y|},y(0)=1
y(x)= | 1 | \left(1- |
4 |
x | +\left|1- | |
2 |
x | |
2 |
\right|\right)2