- Stability theory
In

mathematics ,**stability theory**deals with the stability of solutions (or sets of solutions) fordifferential equation s anddynamical system s.**Definition**Let (

**R**, X, Φ) be areal dynamical system with**R**thereal number s, "X" alocally compact Hausdorff space and Φ theevolution function . For aΦ-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**orif "A"("M") is a neighborhood of "M".attractor 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 equation s can be analyzed using theeigenvalue s 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 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) > 0The 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 aHurwitz polynomial if the real part of all roots are negative. TheRouth-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 equation s 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 λ of "J"("F") : Re( λ) < 0

* asymptotically α-stable if and only if for all eigenvalues λ of "J"("F") : Re( λ) > 0**Lyapunov function**In

physical system s it is often possible to useenergy conservation law s to analyze the stability of fixed points. ALyapunov 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 .

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