Stability theory

In mathematics, stability theory deals with the stability of solutions (or sets of solutions) for differential equations and dynamical systems.

Definition

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:$A\_\{omega\}(M)\; :=\; \{x\; in\; X\; :\; lim\_omega\; gamma\_x\; e\; varnothing\; ,\; mathrm\{\; and\; \}\; ,\; lim\_omega\; gamma\_x\; subset\; M\}\; cup\; M$the ω-basin of attraction and:$A\_\{alpha\}(M)\; :=\; \{x\; in\; X\; :\; lim\_alpha\; gamma\_x\; e\; varnothing\; ,\; mathrm\{\; and\; \}\; ,\; lim\_alpha\; gamma\_x\; subset\; M\}\; cup\; M$the α-basin of attraction and:$A(M):=\; A\_\{omega\}(M)\; cup\; A\_\{alpha\}(M)$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 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:$Phi(t,v)\; in\; V\; quad\; v\; in\; V,\; t\; ge\; 0$and we call "M" α-stable if for any neighborhood "U" of "M" there exists a neighbourhood "V" ⊂ "U" such that:$Phi(t,v)\; in\; V\; quad\; v\; in\; V,\; t\; le\; 0.$

"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.

Notes

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:$A\_\{omega\}(M)\; :=\; \{x\; in\; X\; :\; phi(t,\; x)\_\{t\; o\; infty\}\; o\; M\}$and:$A\_\{alpha\}(M)\; :=\; \{x\; in\; X\; :\; phi(t,\; x)\_\{t\; o\; -infty\}\; o\; M\}$with:$phi(t,\; x)\_\{t\; o\; infty\}\; o\; M$meaning for every neighbourhood "U" of "M" there exists a "t"_"U" such that:$phi(t,x)\; in\; U\; quad\; t\; ge\; t\_U.$

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:$mathbf\{x\}^\text{'}\; =\; mathbf\{A\}\; mathbf\{x\}\; quad\; mathbf\{A\}\; in\; mathbb\{R\}(n,n)$in R^"n" then the null vector is
* asymptotically ω-stable if and only if for all eigenvalues &lambda; of "A": Re( &lambda;) < 0
* asymptotically α-stable if and only if for all eigenvalues &lambda; of "A": Re( &lambda;) > 0
* unstable if there exists one eigenvalue &lambda; of "A" with Re( &lambda;) > 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.

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:$mathbf\{x\}^\text{'}\; =\; mathbf\{F\}\; (mathbf\{x\})$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 &lambda; of "J"("F") : Re( &lambda;) < 0
* asymptotically α-stable if and only if for all eigenvalues &lambda; of "J"("F") : Re( &lambda;) > 0

Lyapunov function

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 prove the stability of a fixed point.

See also

* von Neumann stability analysis
* Lyapunov stability
* structural stability
* Hyperstability

References

External links

* [http://demonstrations.wolfram.com/StableEquilibria/ Stable Equilibria] by Michael Schreiber, The Wolfram Demonstrations Project.