An autonomous system is a system of ordinary differential equations of the form 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 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
Let be a unique solution of the initial value problem for an autonomous system Then solves Indeed, denoting we have and, thus For the initial condition, the verification is trivial,
Example
The equation is autonomous, since the independent variable, let us call it, 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 Ffun = @.* Y; % function f=y = meshgrid; % choose the plot sizes DY = Ffun; DX = ones; % generate the plot values quiver; % plot the direction field in black hold on; contour; % add the isoclines in green title=
One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e. for any shift. Solving the equation symbolically in MATLAB, by running y = dsolve; % solve the equation symbolically
we obtain two equilibrium solutions, and, and a third solution involving an unknown constant, y = - 2 /
Picking up some specific values for the initial condition, we can add the plot of several solutions y1 = dsolve*y', 'y; % solve the initial value problem symbolically y2 = dsolve*y', 'y; % for different initial conditions y3 = dsolve*y', 'y; y4 = dsolve*y', 'y; y5 = dsolve*y', 'y; y6 = dsolve*y', 'y; ezplot; ezplot; % plot the solutions ezplot; ezplot; ezplot; ezplot; title= legend; text; text; grid on;
Qualitative analysis
Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line.
Solution techniques
The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system, but not necessarily vice versa.
First order
The first-order autonomous equation is separable, so it can easily be solved by rearranging it into the integral form
Second order
The second-order autonomous equation is more difficult, but it can be solved by introducing the new variable and expressing the second derivative of as so that the original equation becomes which is a first order equation containing no reference to the independent variable and if solved provides as a function of. Then, recalling the definition of : which is an implicit solution.
Special case: ''x'''' = ''f''(''x'')
The special case where is independent of 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. Note aside then that by inverting both sides of a first order autonomous system, one can immediately integrate with respect to : which is another way to view the separation of variables technique. A natural question is then: can we do something like this with higher order equations? The answer is yes for second order equations, but there's more work to do. The second derivative must be expressed as a derivative with respect to instead of : To reemphasize: what's been accomplished is that the second derivative in has been expressed as a derivative in. The original second order equation may then finally be integrated: This is an implicit solution, and beyond that 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 mentality, we can extend the technique to the more general equation where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative: The right will carry +/- if is even. The treatment must be different if :
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. 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. With this mentality, it also isn't too surprising that general non-autonomous equations of second order can't be solved explicitly, since these can also be chaotic.
