Autonomous system (mathematics)
In mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an autonomous system or autonomous differential equation is a system
Simultaneous equations
In mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations...

 of ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

s which does not explicitly depend on the independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...

. When the variable is the time, they are also named Time-invariant system
Time-invariant system
A time-invariant system is one whose output does not depend explicitly on time.This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output....


Many laws in physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, where the independent variable is usually assumed to be time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

, 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.

Autonomous systems are closely related to dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s. Any autonomous system can be transformed into a dynamical system and, using very weak assumptions, a dynamical system can be transformed into an autonomous system.


An autonomous system is a system of ordinary differential equations of the form
where x takes values in n-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 and t is usually time.

It is distinguished from systems of differential equations of the form
in which the law governing the rate of motion of a particle depends not only on the particle's location, but also on time; such systems are not autonomous.


Let be a unique solution of the
initial value problem
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...

 for an autonomous system.
Then solves.
Indeed, denoting we have
and , thus.
For the initial condition, the verification is trivial,.


The equation is autonomous, since the independent variable,
let us call it , does not explicitly appear in the equation.
To plot the slope field
Slope field
In mathematics, a slope field is a graphical representation of the solutions of a first-order differential equation. It is achieved without solving the differential equation analytically, and thus it is useful...

 and isocline
thumb|right|300px|Fig. 1: Isoclines , slope field , and some solution curves of y'=xyAn Isocline is a curve through points at which the parent function's slope will always be the same, regardless of initial conditions...

 for this equation, one can use the following
code in GNU Octave
GNU Octave
GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...

MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

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); % plot the direction field
hold on;
contour(X,Y,DY,[0 2]); %add the isoclines
title('Slope field and isoclines for f(x,y)=(2-y)y')

One can observe from the plot that the function is, of course, -invariant, and so it the shape of the solution, i.e. for any shift .

Solving the equation symbolically in MATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

, by running

y=dsolve('Dy=(2-y)*y','x'); % solve the equation symbolically

we obtain two equilibrium solutions, and ,
and a third solution involving an unknown constant ,

y(3)=-2/(exp(C3 - 2*x) - 1)

Piking up some specific values for the initial condition, we can add the plot of several solutions

y1=dsolve('Dy=(2-y)*y','y(1)=1','x'); % solve the initial value problem symbolically
y2=dsolve('Dy=(2-y)*y','y(2)=1','x'); % for different initial conditions
y3=dsolve('Dy=(2-y)*y','y(3)=1','x'); y4=dsolve('Dy=(2-y)*y','y(1)=3','x');
y5=dsolve('Dy=(2-y)*y','y(2)=3','x'); y6=dsolve('Dy=(2-y)*y','y(3)=3','x');
ezplot(y1, [0 6]); ezplot(y2, [0 6]); % plot the solutions
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')

Qualitative analysis

Autonomous systems can be analyzed qualitatively using the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

; in the one-variable case, this is the phase line
Phase line (mathematics)
In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, dy/dt = ƒ...


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 Ordinary differential equation#Reduction to a 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
Second derivative
In calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...

 of (via the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

) 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
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

 because they are always Hamiltonian system
Hamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....


The idea is to make use of the identity (barring division by zero
Division by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...


which follows from the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

. 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
Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...

 or dependence of the right side of the equation on the dependent variable only (ie, not its derivatives). This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

 behavior such as the Lorenz attractor
Lorenz attractor
The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...

 and the Rössler attractor
Rössler attractor
The Rössler attractor is the attractor for the Rössler system, a system of three non-linear ordinary differential equations. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the 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 (an example of this is a periodically forced pendulum).
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