Difference between revisions of "Perturbation of a linear system"
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| + | The mapping $ f $ | ||
| + | in the system of ordinary differential equations | ||
| + | |||
| + | $$ \tag{1 } | ||
| + | \dot{x} = A ( t) x + f ( x, t). | ||
| + | $$ | ||
A perturbation is usually assumed to be small in some sense, for example | A perturbation is usually assumed to be small in some sense, for example | ||
| − | + | $$ \tag{2 } | |
| − | + | \frac{| f ( x, t) | }{| x | } | |
| + | \rightarrow 0 \ \ | ||
| + | \textrm{ if } | x | \rightarrow 0. | ||
| + | $$ | ||
| − | + | The solution $ \phi ( t) $ | |
| + | of the perturbed system (1) and the solution $ \Psi ( t) $ | ||
| + | of the linear system | ||
| − | + | $$ \tag{3 } | |
| + | \dot{y} = A ( t) y | ||
| + | $$ | ||
| − | + | with the same initial value $ y _ {0} $ | |
| + | at $ t = t _ {0} $, | ||
| + | are connected by the relation | ||
| − | + | $$ | |
| + | \phi ( t) = \Psi ( t) \left ( y _ {0} + | ||
| + | \int\limits _ {t _ {0} } ^ { t } | ||
| + | \Psi ^ {-} 1 ( \tau ) f ( \phi ( \tau ), \tau ) d \tau \right ) , | ||
| + | $$ | ||
| − | + | known as the formula of variation of constants, where $ \Psi ( t) $ | |
| + | is the [[Fundamental matrix|fundamental matrix]] of the linear system (3). | ||
| − | The study of the periodic solution | + | It was shown by A.M. Lyapunov [[#References|[1]]] that the trivial solution of the system (1) is asymptotically stable (cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]) if relation (2) is valid uniformly in $ t $, |
| + | the matrix $ A( t) $ | ||
| + | is constant and all real parts of the eigen values of $ A( t) $ | ||
| + | are negative; if only one such real part is positive, the trivial solution is not stable. | ||
| + | |||
| + | The study of the periodic solution $ \phi $ | ||
| + | of the system $ \dot{x} = P( x, t) $, | ||
| + | describing an oscillating process, reduces in the general case by the transformation $ x = \phi ( t) + y $ | ||
| + | to the study of a perturbed linear system, the right-hand side of which is periodic in $ t $[[#References|[3]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)</TD></TR></table> | ||
| − | |||
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====Comments==== | ====Comments==== | ||
Revision as of 08:05, 6 June 2020
The mapping $ f $
in the system of ordinary differential equations
$$ \tag{1 } \dot{x} = A ( t) x + f ( x, t). $$
A perturbation is usually assumed to be small in some sense, for example
$$ \tag{2 } \frac{| f ( x, t) | }{| x | } \rightarrow 0 \ \ \textrm{ if } | x | \rightarrow 0. $$
The solution $ \phi ( t) $ of the perturbed system (1) and the solution $ \Psi ( t) $ of the linear system
$$ \tag{3 } \dot{y} = A ( t) y $$
with the same initial value $ y _ {0} $ at $ t = t _ {0} $, are connected by the relation
$$ \phi ( t) = \Psi ( t) \left ( y _ {0} + \int\limits _ {t _ {0} } ^ { t } \Psi ^ {-} 1 ( \tau ) f ( \phi ( \tau ), \tau ) d \tau \right ) , $$
known as the formula of variation of constants, where $ \Psi ( t) $ is the fundamental matrix of the linear system (3).
It was shown by A.M. Lyapunov [1] that the trivial solution of the system (1) is asymptotically stable (cf. Asymptotically-stable solution) if relation (2) is valid uniformly in $ t $, the matrix $ A( t) $ is constant and all real parts of the eigen values of $ A( t) $ are negative; if only one such real part is positive, the trivial solution is not stable.
The study of the periodic solution $ \phi $ of the system $ \dot{x} = P( x, t) $, describing an oscillating process, reduces in the general case by the transformation $ x = \phi ( t) + y $ to the study of a perturbed linear system, the right-hand side of which is periodic in $ t $[3].
References
| [1] | A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian) |
| [2] | B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian) |
| [3] | L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian) |
Comments
Results of this type are usually called Poincaré–Lyapunov theorems. There are several extensions, described, e.g., in [a1]. A recent tutorial text containing these matters is [a2].
References
| [a1] | M. Roseau, "Vibrations non linéaires et théorie de la stabilité" , Springer (1966) |
| [a2] | F. Verhulst, "Nonlinear differential equations and dynamical systems" , Springer (1989) |
How to Cite This Entry:
Perturbation of a linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perturbation_of_a_linear_system&oldid=48168
Perturbation of a linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perturbation_of_a_linear_system&oldid=48168
This article was adapted from an original article by L.E. Reizin' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article