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(Redirected from Basin of attraction)
In mathematics, 'stability theory' deals with the stability of solutions (or sets of solutions) of differential equations and dynamical systems.
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 ''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
:
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
:
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 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 ''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
Main articles: 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.
★ von Neumann stability analysis
★ Lyapunov stability
★ structural stability
In mathematics, 'stability theory' deals with the stability of solutions (or sets of solutions) of differential equations and dynamical systems.
| Contents |
| Definition |
| Notes |
| Stability of fixed points |
| Linear autonomous systems |
| Non-linear autonomous systems |
| Lyapunov function |
| See also |
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
:
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 '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.
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
:
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 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
:
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
Main articles: 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.
See also
★ von Neumann stability analysis
★ Lyapunov stability
★ structural stability
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