Bendixson-Dulac theorem
Encyclopedia
In mathematics
, the Bendixson–Dulac theorem on dynamical system
s states that if there exists a function
such that
has the same sign (
) almost everywhere
in a simply connected region, then the plane autonomous system
has no periodic solutions. "Almost everywhere" means everywhere except possibly in a set of area 0, such as a point or line.
The theorem was first established by Swedish mathematician Ivar Bendixson in 1901 and further refined by French mathematician Henri Dulac
in 1933 using Green's theorem
.
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...
, the Bendixson–Dulac theorem on 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 states that if there exists a function
such thathas the same sign (
) almost everywhereAlmost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
in a simply connected region, then the plane autonomous system
has no periodic solutions. "Almost everywhere" means everywhere except possibly in a set of area 0, such as a point or line.
The theorem was first established by Swedish mathematician Ivar Bendixson in 1901 and further refined by French mathematician Henri Dulac
Henri Dulac
- Life :Born in Fayence , Dulac graduated from Ecole Polytechnique and obtained a Doctorate in Mathematics. He started to teach a class of mathematic analysis at University, in Grenoble , Algiers and Poitiers...
in 1933 using Green's theorem
Green's theorem
In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
.