Stability theory

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

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[edit] Definition

Let (R, X, Φ) be a real dynamical system whith 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 \ne \emptyset \, \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 \ne \emptyset \, \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 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 so 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 so 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.

[edit] 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 \to \infty} \to M\}

and

A_{\alpha}(M) := \{x \in X : \phi(t, x)_{t \to -\infty} \to M\}

with

\phi(t, x)_{t \to \infty} \to M

meaning for every neighbourhood U of M there exists a tU so that

\phi(t,x) \in U \quad t \ge t_U.

[edit] Stability of fixed points

[edit] 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}^' = \mathbf{A} \mathbf{x} \quad \mathbf{A} \in \mathbb{R}(n,n)

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.

[edit] 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 C1-vector field

\mathbf{x}^' = \mathbf{F} (\mathbf{x})

in Rn with fixed point p and let J(F) denote the Jacobi 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

[edit] Lyapunov function

Main article: 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 proof the stability of a fixed point.

[edit] See also

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