# 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)