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Stability theory
From Wikipedia, the free encyclopedia
In mathematics, stability theory deals with the stability of solutions (or sets of solutions) of differential equations and dynamical systems.
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Let (R, X, Φ) be a real dynamical system with R the real numbers, X a locally compact Hausdorff space and Φ the evolution function. For a Φ-invariant, non-empty and closed subset M of X we call
the ω-basin of attraction and
the α-basin of attraction and
the basin of attraction.
We call M ω-(α-)attractive or ω-(α-)attractor if Aω(M) (Aα(M)) is a neighborhood of M and attractive or attractor if A(M) is a neighborhood of M.
If additionally M is compact we call M ω-stable if for any neighborhood U of M there exists a neighbourhood V ⊂ U such that
and we call M α-stable if for any neighborhood U of M there exists a neighbourhood V ⊂ U such that
M is called asymptotically ω-stable if M is ω-stable and ω-attractive and asymptotically α-stable if M is α-stable and α-attractive.
Alternatively ω-stable is called stable, not ω-stable is called unstable, ω-attractive is called attractive and α-attractive is called repellent.
If the set M is compact, as for example in the case of fixed points or periodic orbits, the definition of the basin of attraction simplifies to
and
with
meaning for every neighbourhood U of M there exists a tU such that
The stability of fixed points of linear autonomous differential equations can be analyzed using the eigenvalues of the corresponding linear transformation.
Given a linear vector field
in Rn then the null vector is
- asymptotically ω-stable if and only if for all eigenvalues λ of A: Re( λ) < 0
- asymptotically α-stable if and only if for all eigenvalues λ of A: Re( λ) > 0
- unstable if there exists one eigenvalue λ of A with Re( λ) > 0
The eigenvalues of a linear transformation are the roots of the characteristic polynomial of the corresponding matrix. A polynomial over 'R in one variable is called a Hurwitz polynomial if the real part of all roots are negative. The Routh-Hurwitz stability criterion is a necessary and sufficient condition for a polynomial to be a Hurwitz polynomial and thus can be used to decide if the null vector for a given linear autonomous differential equation is asymptotically ω-stable.
The stability of fixed points of non-linear autonomous differential equations can be analyzed by linearisation of the system if the associated vector field is sufficiently smooth.
Given a C1-vector field
in Rn with fixed point p and let J(F) denote the Jacobian matrix of F at point p, then p is
- asymptotically ω-stable if and only if for all eigenvalues λ of J(F) : Re( λ) < 0
- asymptotically α-stable if and only if for all eigenvalues λ of J(F) : Re( λ) > 0
In physical systems it is often possible to use energy conservation laws to analyze the stability of fixed points. A Lyapunov function is a generalization of this concept and the existence of such a function can be used to proof the stability of a fixed point.