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Concept
Autonomous system (mathematics)
Summary
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 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.
Solutions are invariant under horizontal translations:
Let be a unique solution of the initial value problem for an autonomous system
Then solves
Denoting gets and , thus
For the initial condition, the verification is trivial,
The equation is autonomous, since the independent variable () 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 = @(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')
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
syms y(x);
equation = (diff(y) == (2 - y) * y);
% solve the equation for a general solution symbolically
y_general = dsolve(equation);
obtains two equilibrium solutions, and , and a third solution involving an unknown constant ,
2 / (exp(C3 - 2 * x) - 1).
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