Mathematics

R. E. Vinograd

ESTIMATE OF THE JUMP OF THE LARGEST CHARACTERISTIC EXPONENT UNDER SMALL PERTURBATIONS

(Presented by Academician I. G. Petrovskii, December 6, 1956)

Let two systems of differential equations of order (n) be given in matrix-vector form: the unperturbed one
[
\frac{dx}{dt}=A(t)x
\tag{1}
]
and the perturbed one
[
\frac{dx}{dt}=A(t)x+f(t,x).
\tag{2}
]

Here (A(t)) is a continuous or piecewise-continuous bounded matrix for (0\leq t<\infty); (|A(t)|\leq M); (x) and (f) are vectors, with (f(t,0)\equiv 0); (f) is continuous in (t,x) and satisfies, with respect to (x), a Lipschitz condition with small constant (\delta), independent of (t), or the condition
[
|f(t,x)|\leq \delta |x|.
\tag{3}
]

The class of vector-additions (f(t,x)) possessing Lipschitz constant (\leq \delta) will be denoted by (L_\delta).

Let (\lambda_0) and (\lambda_f) be the largest characteristic exponents of the solutions of systems (1) and (2), respectively* . It is known ((^{1,2})) that (\lambda_f), while being (\geq \lambda_0), does not always tend to (\lambda_0) as the additions (f) decrease, i.e., if one sets
[
\Lambda=\lim_{\delta\to 0}\sup_{f\in L_\delta}\lambda_f,
\tag{4}
]
then the case (\Lambda>\lambda_0) is possible (for exceptions see ((^{3-5}))).

Thus, in passing from (1) to (2), the largest exponent may undergo a finite jump (\Lambda-\lambda_0>0) (up to a quantity tending to zero together with (\delta)). In this paper an upper estimate for this jump—or, equivalently, for (\Lambda)—is constructed from system (1), and in a certain sense cannot be improved.

1. Let (X(t)) be the matrix of a fundamental system of solutions of (1). Consider a bounded function (F(t)) such that, for all (t) and (\tau<t), the estimate
   [
   |X(t)X^{-1}(\tau)|\leq C\exp\left[\int_{\tau}^{t}F(\xi)\,d\xi\right]
   \tag{5}
   ]
   holds, where (C) depends on the choice of (F(t)), but not on (t) and (\tau). The class of functions (F(t)) is nonempty; it includes all sufficiently large constants ((\geq M\geq |A(t)|)).

* The characteristic exponent of a vector (x(t)) is defined as
[
\lambda=\overline{\lim}_{t\to\infty}t^{-1}\ln|x(t)|;
]
(\lambda_f) is understood as the supremum of (\lambda) over all solutions (x(t)) of system (2).

Set

[
\omega_F=\overline{\lim_{t\to\infty}}\, t^{-1}\int_0^t F(\xi)\,d\xi,\qquad
\Omega=\inf \omega_F,
\tag{6}
]

where the infimum is taken over all functions of the indicated class. The number (\Omega) so defined does not depend on the choice of the fundamental matrix (X(t)) and is not changed under Lyapunov transformations ((x=S(t)y) with bounded matrices (S) and (S^{-1})) of system (1). Therefore one may transform systems (1) and (2) by Perron’s method ((^6,^7)), i.e., regard the matrix (A(t)) as triangular.

Lemma. The number (\Omega) is determined only by the elements of the main diagonal of the matrix (A(t)), if the latter is triangular.

Without dwelling on the proof, we note the following construction. Let (a_{11}(t), a_{22}(t),\ldots,a_{nn}(t)) be the elements of the main diagonal of (A(t)). Divide the semiaxis (0\le t<\infty) into equal intervals (\Delta_i) of length (T), and consider the function (\alpha_T(t)) which on each (\Delta_i) coincides with that one of the functions (a_{kk}(t)) which has the largest integral over (\Delta_i). Put

[
\omega_T^=\overline{\lim_{t\to\infty}}\, t^{-1}\int_0^t \alpha_T(\xi)\,d\xi,\qquad
\Omega^=\inf_{0<T<\infty}\omega_T^* .
]

Then one can prove that (\Omega^*) coincides with (\Omega).

Theorem 1. For any (\varepsilon>0) there exists a sufficiently small (\delta>0) such that every solution (x(t)) of system (2) with (f\in L_\delta) admits the estimate

[
|x(t)|\le |x(0)|\,B_\varepsilon e^{(\Omega+\varepsilon)t},
\tag{7}
]

where (B_\varepsilon) depends only on (\varepsilon).

Proof. Take the matrix (X(t)) of solutions of (1) with the initial condition (X(0)=E), and choose the function (F(t)=F_\varepsilon(t)) from (5) so that (\omega_F<\Omega+\sigma), where (\sigma=\varepsilon/3); denote the corresponding constant by (C=C_\varepsilon). Replacing (2) by the integral equation

[
x(t)=X(t)x_0+\int_0^t X(t)X^{-1}(\tau)f(\tau,x(\tau))\,d\tau,\qquad x_0=x(0),
]

we shall solve it by Picard’s method (whose convergence is known).

Let the first approximation be (x_1(t)=X(t)x_0). Then from (5) we obtain

[
|x_1(t)|\le |x_0|\,C\exp\left[\int_0^t F\,d\xi\right].
\tag{8}
]

Suppose that, for (r=1,2,\ldots,k), the estimate

[
|x_r(t)|\le |x_0|\,C\exp\left[\int_0^t (F+\sigma)\,d\xi\right]
\tag{9}
]

has been established.

Then for the ((k+1))-st approximation, using (8), (5), (9), and the Lipschitz condition for (f), we find

[
|x_{k+1}(t)|\le |X(t)x_0|+\int_0^t |X(t)X^{-1}(\tau)|\,\delta\,|x_k(\tau)|\,d\tau \le
]

[
\le |x_0|\,C\exp\left[\int_0^t F\,d\xi\right]
+\int_0^t C\exp\left[\int_\tau^t F\,d\xi\right]\delta\,|x_0|\,C\exp\left[\int_0^\tau (F+\sigma)\,d\xi\right]d\tau =
]

[
=|x_0|\,C\exp\left[\int_0^t F\,d\xi\right]\left[1+C\delta\int_0^t e^{\sigma\tau}\,d\tau\right]
=|x_0|\,C\exp\left[\int_0^t F\,d\xi\right]\left[1+\frac{C\delta}{\sigma}\left(e^{\sigma t}-1\right)\right].
]

We see that if (\delta) is so small that (C\delta/\sigma<1), then (x_{k+1}(t)) also satisfies inequality (9). Therefore it is also valid for the desired solution (x(t)=\lim\limits_{k\to\infty}x_k(t)):

[
|x(t)|\leq |x_0|C\exp\left[\int_0^t(F+\sigma)\,d\xi\right].
\tag{10}
]

Taking into account that (\omega_F<\Omega+\sigma), we have

[
\int_0^t F\,d\xi