Autonomous system (mathematics)

Encyclopedia

In mathematics

, an

of ordinary differential equation

s which does not explicitly depend on the independent variable

. When the variable is the time, they are also named Time-invariant system

.

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.

Autonomous systems are closely related to dynamical system

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.

where

and

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.

initial value problem

for an autonomous system.

Then solves.

Indeed, denoting we have

and , thus.

For the initial condition, the verification is trivial,.

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

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 systemSimultaneous 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

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

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.

## Definition

An**autonomous system**is a system of ordinary differential equations of the formwhere

*x*takes values in*n*-dimensional Euclidean spaceEuclidean 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.

## Properties

Let be a unique solution of theinitial 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,.

## 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

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

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

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,...