![Stability_Diagram](https://upload.wikimedia.org/wikipedia/commons/3/3b/Stability_Diagram.png)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an autonomous system or autonomous differential equation is a
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s which does not explicitly depend on the
independent variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
. When the variable is time, they are also called
time-invariant system
In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...
s.
Many laws in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, where the independent variable is usually assumed to be
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
, 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.
Definition
An autonomous system is a
system of ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s of the form
where takes values in -dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
; 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 , again often interpreted as time; such systems are by definition not autonomous.
Properties
Solutions are invariant under horizontal translations:
Let
be a unique solution of the
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or oth ...
for an autonomous system
Then
solves
Denoting
gets
and
, thus
For the initial condition, the verification is trivial,
Example
The equation
is autonomous, since the independent variable (
) does not explicitly appear in the equation.
To plot the
slope field
Slope fields (also called direction fields) are a graphical representation of the solutions to a first-order differential equation of a scalar function. Solutions to a slope field are functions drawn as solid curves. A slope field shows the slope ...
and
isocline
300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''.
Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some m ...
for this equation, one can use the following code in
GNU Octave
GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a langu ...
/
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
Ffun = @(X, Y)(2 - Y) .* Y; % function f(x,y)=(2-y)y
, Y
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= 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, 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
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
, 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).
Picking up some specific values for the
initial condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
, one can add the plot of several solutions
% 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, 6; ezplot(y2, 6; ezplot(y3, 6;
ezplot(y4, 6; ezplot(y5, 6; ezplot(y6, 6;
title('Slope field, isoclines and solutions for f(x,y)=(2-y)y')
legend('Slope field', 'Isoclines', 'Solutions y_');
text( 2 3 1 1 strcat('\leftarrow', ));
text( 2 3 3 3 strcat('\leftarrow', ));
grid on;
Qualitative analysis
Autonomous systems can be analyzed qualitatively using the
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
; 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 (as described in
reduction to a first-order system), but not necessarily vice versa.
First order
The first-order autonomous equation
is
separable, so it can 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
In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
of
via the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
as
so that the original equation becomes
which is a first order equation containing no reference to the independent variable
. Solving provides
as a function of
. Then, recalling the definition of
:
which is an implicit solution.
Special case:
The special case where
is independent of
benefits from separate treatment. These types of equations are very common in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
because they are always
Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can b ...
s.
The idea is to make use of the identity
which follows from the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, barring any issues due to
division by zero
In mathematics, division by zero is division (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
.
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. 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 with respect to
has been expressed as a derivative of
. The original second order equation can now be integrated:
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:
Using the above approach, the technique can extend 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
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
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
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
behavior such as the
Lorenz attractor
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lo ...
and the
Rössler attractor
The Rössler attractor is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s... These differential equations define a continuous-time dynamical ...
.
Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum.
Multivariate case
In
, where
is an
-dimensional column vector dependent on
.
The solution is
where
is an
constant vector.
Finite durations
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,
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:
:
Admits the finite duration solution:
:
See also
*
Non-autonomous system (mathematics) In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle Q\to \mathbb R over \mathbb R. For instance, this is the case of non-autonomous mechanics.
An ' ...
References
{{DEFAULTSORT:Autonomous System (Mathematics)
Differential equations
Dynamical systems
Ordinary differential equations