Autonomous system (mathematics)
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.
DefinitionEdit
An autonomous system is a system of ordinary differential equations of the form <math display="block">\frac{d}{dt}x(t)=f(x(t))</math> where Template:Mvar takes values in Template:Mvar-dimensional Euclidean space; Template:Mvar is often interpreted as time.
It is distinguished from systems of differential equations of the form <math display="block">\frac{d}{dt}x(t)=g(x(t),t)</math> in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter Template:Mvar, again often interpreted as time; such systems are by definition not autonomous.
PropertiesEdit
Solutions are invariant under horizontal translations:
Let <math>x_1(t)</math> be a unique solution of the initial value problem for an autonomous system <math display="block">\frac{d}{dt}x(t)=f(x(t)) \, , \quad x(0)=x_0.</math> Then <math>x_2(t)=x_1(t-t_0)</math> solves <math display="block">\frac{d}{dt}x(t)=f(x(t)) \, , \quad x(t_0)=x_0.</math> Denoting <math>s=t-t_0</math> gets <math>x_1(s)=x_2(t)</math> and <math>ds=dt</math>, thus <math display="block">\frac{d}{dt}x_2(t) = \frac{d}{dt}x_1(t-t_0)=\frac{d}{ds}x_1(s) = f(x_1(s)) = f(x_2(t)) .</math> For the initial condition, the verification is trivial, <math display="block">x_2(t_0)=x_1(t_0-t_0)=x_1(0)=x_0 .</math>
ExampleEdit
The equation <math>y'= \left(2-y\right)y</math> is autonomous, since the independent variable (<math>x</math>) 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 <syntaxhighlight lang="matlab"> Ffun = @(X, Y)(2 - Y) .* Y; % function f(x,y)=(2-y)y [X, Y] = meshgrid(0:.2:6, -1:.2:3); % choose the plot sizes DY = Ffun(X, Y); DX = ones(size(DY)); % generate the plot values quiver(X, Y, DX, DY, 'k'); % plot the direction field in black hold on; contour(X, Y, DY, [0 1 2], 'g'); % add the isoclines(0 1 2) in green title('Slope field and isoclines for f(x,y)=(2-y)y') </syntaxhighlight> One can observe from the plot that the function <math>\left(2-y\right)y</math> is <math>x</math>-invariant, and so is the shape of the solution, i.e. <math>y(x)=y(x-x_0)</math> for any shift <math>x_0</math>.
Solving the equation symbolically in MATLAB, by running <syntaxhighlight lang="matlab"> syms y(x); equation = (diff(y) == (2 - y) * y); % solve the equation for a general solution symbolically y_general = dsolve(equation); </syntaxhighlight> obtains two equilibrium solutions, <math>y=0</math> and <math>y=2</math>, and a third solution involving an unknown constant <math>C_3</math>, <syntaxhighlight lang="matlab" inline>-2 / (exp(C3 - 2 * x) - 1)</syntaxhighlight>.
Picking up some specific values for the initial condition, one can add the plot of several solutions
<syntaxhighlight lang="matlab"> % solve the initial value problem symbolically % for different initial conditions y1 = dsolve(equation, y(1) == 1); y2 = dsolve(equation, y(2) == 1); y3 = dsolve(equation, y(3) == 1); y4 = dsolve(equation, y(1) == 3); y5 = dsolve(equation, y(2) == 3); y6 = dsolve(equation, y(3) == 3); % plot the solutions ezplot(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_{1..6}'); text([1 2 3], [1 1 1], strcat('\leftarrow', {'y_1', 'y_2', 'y_3'})); text([1 2 3], [3 3 3], strcat('\leftarrow', {'y_4', 'y_5', 'y_6'})); grid on; </syntaxhighlight>
Qualitative analysisEdit
Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line.
Solution techniquesEdit
The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order <math>n</math> is equivalent to an <math>n</math>-dimensional first-order system (as described in reduction to a first-order system), but not necessarily vice versa.
First orderEdit
The first-order autonomous equation <math display="block">\frac{dx}{dt} = f(x)</math> is separable, so it can be solved by rearranging it into the integral form <math display="block">t + C = \int \frac{dx}{f(x)}</math>
Second orderEdit
The second-order autonomous equation <math display="block">\frac{d^2x}{dt^2} = f(x, x')</math> is more difficult, but it can be solved<ref>Template:Cite book</ref> by introducing the new variable <math display="block">v = \frac{dx}{dt}</math> and expressing the second derivative of <math>x</math> via the chain rule as <math display="block">\frac{d^2x}{dt^2} = \frac{dv}{dt} = \frac{dx}{dt}\frac{dv}{dx} = v\frac{dv}{dx}</math> so that the original equation becomes <math display="block">v\frac{dv}{dx} = f(x, v)</math> which is a first order equation containing no reference to the independent variable <math>t</math>. Solving provides <math>v</math> as a function of <math>x</math>. Then, recalling the definition of <math>v</math>:
<math display="block">\frac{dx}{dt} = v(x) \quad \Rightarrow \quad t + C = \int \frac{d x}{v(x)} </math>
which is an implicit solution.
Special case: Template:MathEdit
The special case where <math>f</math> is independent of <math>x'</math>
<math display="block">\frac{d^2 x}{d t^2} = f(x)</math>
benefits from separate treatment.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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
<math display="block">\frac{d x}{d t} = \left(\frac{d t}{d x}\right)^{-1}</math>
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 <math>x</math>:
<math display="block">\frac{d x}{d t} = f(x) \quad \Rightarrow \quad \frac{d t}{d x} = \frac{1}{f(x)} \quad \Rightarrow \quad t + C = \int \frac{dx}{f(x)}</math>
which is another way to view the separation of variables technique. The second derivative must be expressed as a derivative with respect to <math>x</math> instead of <math>t</math>:
<math display="block">\begin{align} \frac{d^2 x}{d t^2} &= \frac{d}{d t}\left(\frac{d x}{d t}\right) = \frac{d}{d x}\left(\frac{d x}{d t}\right) \frac{d x}{d t} \\[4pt] &= \frac{d}{d x}\left(\left(\frac{d t}{d x}\right)^{-1}\right) \left(\frac{d t}{d x}\right)^{-1} \\[4pt] &= - \left(\frac{d t}{d x}\right)^{-2} \frac{d^2 t}{d x^2} \left(\frac{d t}{d x}\right)^{-1} = - \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} \\[4pt] &= \frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right) \end{align}</math>
To reemphasize: what's been accomplished is that the second derivative with respect to <math>t</math> has been expressed as a derivative of <math>x</math>. The original second order equation can now be integrated:
<math display="block">\begin{align} \frac{d^2 x}{d t^2} &= f(x) \\ \frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right) &= f(x) \\ \left(\frac{d t}{d x}\right)^{-2} &= 2 \int f(x) dx + C_1 \\ \frac{d t}{d x} &= \pm \frac{1}{\sqrt{2 \int f(x) dx + C_1}} \\ t + C_2 &= \pm \int \frac{dx}{\sqrt{2 \int f(x) dx + C_1}} \end{align}</math>
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: Template:MathEdit
Using the above approach, the technique can extend to the more general equation
<math display="block">\frac{d^2 x}{d t^2} = \left(\frac{d x}{d t}\right)^n f(x)</math>
where <math>n</math> is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of <math>x'</math>. Rewriting the second derivative, rearranging, and expressing the left side as a derivative:
<math display="block">\begin{align} &- \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} = \left(\frac{d t}{d x}\right)^{-n} f(x) \\[4pt] &- \left(\frac{d t}{d x}\right)^{n - 3} \frac{d^2 t}{d x^2} = f(x) \\[4pt] &\frac{d}{d x}\left(\frac{1}{2 - n}\left(\frac{d t}{d x}\right)^{n - 2}\right) = f(x) \\[4pt] &\left(\frac{d t}{d x}\right)^{n - 2} = (2 - n) \int f(x) dx + C_1 \\[2pt] &t + C_2 = \int \left((2 - n) \int f(x) dx + C_1\right)^{\frac{1}{n - 2}} dx \end{align}</math>
The right will carry +/− if <math>n</math> is even. The treatment must be different if <math>n = 2</math>:
<math display="block">\begin{align} - \left(\frac{d t}{d x}\right)^{-1} \frac{d^2 t}{d x^2} &= f(x) \\ -\frac{d}{d x}\left(\ln\left(\frac{d t}{d x}\right)\right) &= f(x) \\ \frac{d t}{d x} &= C_1 e^{-\int f(x) dx} \\ t + C_2 &= C_1 \int e^{-\int f(x) dx} dx \end{align}</math>
Higher ordersEdit
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<ref>Third order autonomous equation at eqworld.</ref><ref>Fourth order autonomous equation at eqworld.</ref> (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.<ref>Template:Cite book</ref>
Multivariate caseEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In <math>\mathbf x'(t) = A \mathbf x(t)</math>, where <math>\mathbf x(t)</math> is an <math>n</math>-dimensional column vector dependent on <math>t</math>.
The solution is <math>\mathbf x(t) = e^{A t} \mathbf c</math> where <math>\mathbf c</math> is an <math>n \times 1</math> constant vector.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Finite durationsEdit
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,<ref>Template:Cite book</ref> 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 standTemplate:Clarification needed uniqueness of solutions of Lipschitz differential equations.
As example, the equation:
- <math>y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1</math>
Admits the finite duration solution:
- <math>y(x)=\frac{1}{4}\left(1-\frac{x}{2}+\left|1-\frac{x}{2}\right|\right)^2</math>
See alsoEdit
ReferencesEdit
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