- Stability theory
mathematics, stability theory deals with the stability of solutions (or sets of solutions) for differential equations and dynamical systems.
Let (R, X, Φ) be a
real dynamical systemwith R the real numbers, "X" a locally compact Hausdorff spaceand Φ the evolution function. For a Φ-invariant, non-emptyand 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 "A"ω("M") ("A"α("M")) is a neighborhood of "M" and attractive or
attractorif "A"("M") is a neighborhood of "M".
If additionally "M" is
compactwe 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 "t""U" such that:
Stability of fixed points
Linear autonomous systems
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 R"n" 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 polynomialof the corresponding matrix. A polynomial over "'R" in one variable is called a Hurwitz polynomialif the real part of all roots are negative. The Routh-Hurwitz stability criterionis 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.
Non-linear autonomous systems
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 "C"1-vector field:in R"n" 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
physical systems it is often possible to use energy conservation laws to analyze the stability of fixed points. A Lyapunov functionis a generalization of this concept and the existence of such a function can be used to prove the stability of a fixed point.
von Neumann stability analysis
* [http://demonstrations.wolfram.com/StableEquilibria/ Stable Equilibria] by Michael Schreiber,
The Wolfram Demonstrations Project.
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