Andronov–Pontryagin criterion
Encyclopedia
The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical system
s in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.
where v is a C1-vector field
on the plane, x∈R2 ,
is orbitally topologically stable if and only if the following two conditions hold:
The same statement holds if the vector field v is defined on the unit disk and is transversal to the boundary.
close to the identity map which transforms the orbits of the original dynamical system to the orbits of the perturbed system (cf structural stability
).
The first condition of the theorem is known as global hyperbolicity. A zero of a vector field v, i.e. a point x0 where v(x0)=0, is said to be hyperbolic if none of the eigenvalues of the linearization of v at x0 is purely imaginary. A periodic orbit of a flow is said to be hyperbolic if none of the eigenvalues of the Poincaré return map
at a point on the orbit has absolute value one.
Finally, saddle connection refers to a situation where an orbit from one saddle point enters the same or another saddle point, i.e. the unstable and stable separatrices
are connected (cf homoclinic orbit
and heteroclinic orbit
).
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 in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.
Statement
A dynamical system
where v is a C1-vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
on the plane, x∈R2 ,
is orbitally topologically stable if and only if the following two conditions hold:
- All equilibrium points and periodic orbits are hyperbolic.
- There are no saddle connections.
The same statement holds if the vector field v is defined on the unit disk and is transversal to the boundary.
Clarifications
Orbital topological stability of a dynamical system means that for any sufficiently small perturbation (in the C1-metric), there exists a homeomorphismHomeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
close to the identity map which transforms the orbits of the original dynamical system to the orbits of the perturbed system (cf structural stability
Structural stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by C1-small perturbations....
).
The first condition of the theorem is known as global hyperbolicity. A zero of a vector field v, i.e. a point x0 where v(x0)=0, is said to be hyperbolic if none of the eigenvalues of the linearization of v at x0 is purely imaginary. A periodic orbit of a flow is said to be hyperbolic if none of the eigenvalues of the Poincaré return map
Poincaré map
In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower dimensional subspace, called the Poincaré section, transversal to...
at a point on the orbit has absolute value one.
Finally, saddle connection refers to a situation where an orbit from one saddle point enters the same or another saddle point, i.e. the unstable and stable separatrices
Separatrix
Separatrix may refer to:*The slash , also called a "separatrix"*separatrix , an equation to determine the borders of a system*A mechanism for magnetically limiting a plasma, and hence for controlling the nuclear fusion in a tokamak...
are connected (cf homoclinic orbit
Homoclinic orbit
In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium.Homoclinic orbits and homoclinic points...
and heteroclinic orbit
Heteroclinic orbit
In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit is a path in phase space which joins two different equilibrium points...
).