Perturbation of a linear system

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)