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Encyclopedia > Autonomous system (mathematics)

In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not depend on the independent variable. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In an experimental design, the independent variable (also known as predictor or regressor) is the variable which is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ...


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. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Two distinct views exist on the meaning of time. ... The Laws of Nature are claimed in the United States Declaration of Independence to be the work of the Creator of unalienable rights identified as Natures God. ...


Autonomous systems are closely related to dynamical systems. Any autonomous system can be transformed into a dynamical system and, using very weak assumptions, a dynamical system can be transformed into an autonomous systems. A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ...


Definition

An autonomous system is a system of differential equations of the form

frac{d}{dt}x(t)=f(x(t))

where x takes values in n-dimensional Euclidean space and t is usually time. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


It is distinguished from systems of differential equations of the form

frac{d}{dt}x(t)=g(x(t),t)

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

Every initial value problem for an autonomous system In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution. ...

frac{d}{dt}x(t)=f(x(t)) , mathrm{,} quad x(t_0)=y_0

is equivalent to

frac{d}{dt}x(t)=f(x(t)) , mathrm{,} quad x(0)=y_0^{'}

for some y0′.


See also


  Results from FactBites:
 
Springer Online Reference Works (951 words)
in the autonomous system (1) yields the system
For a Hamiltonian system, a consequence of (3) is the conservation of the phase volume by the phase flow.
For a linear autonomous system the structure of phase trajectories in a neighbourhood of an equilibrium point is known, since the autonomous system is integrable [5].
truth in mathematics (930 words)
Mathematical propositions are not true because they deal in eternal or idealized entities, but because their denial is logically impossible.
Indeed, mathematics begins with a languageless activity of the mind which moves on from one thing to another but keeps a memory of the first as the empty form of a common substratum of all such moves.
Mathematics arises out of its practice, and must ultimately be a free creation of the human mind, not an exercise in logic or a discovery of preexisting fundamentals.
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