Mem. Differential Equations Math. Phys. 11(1997), 170–173

N. Izobov

ESTIMATES OF THE LOWER EXPONENT OF THE TWO-DIMENSIONAL LINEAR SYSTEM UNDER PERRON PERTURBATIONS

(Reported on May 12, 1997)

Consider a linear system

ẋ = A(t)x, x ∈ R^{n}, t ≥ 0,  (1_{A})

with a piecewise continuous bounded matrix of coefficients A(·), the characteristic exponents λ_{1}(A) ≤ ⋯ ≤ λ_{n}(A), the incorrectness coefficients of Perron [1] σ_{P} (A) and Grobman [1] σ_{G}(A), and with a normal ordered system X(t) = [X_{1}(t), …, X_{n}(t)] of its solutions X_{i}(t). Along with the system (1_{A}) consider perturbed systems (1_{A+Q}) with piecewise continuous Perron perturbations Q(·) determined by the condition λ[Q] ≡

\overline{lim}_{t→+∞} \frac{1}{t} ln ∥Q(t)∥ < −σ_{P} (A).

For Perron perturbations only three following results are known: 1) the upper exponents λ_{2}(A) and λ_{2}(A+Q) of the two-dimensional systems (1_{A}) and (1_{A+Q}),respectively, coincide [1]; 2) generally speaking, the lower exponents λ_{1}(A) and λ_{1}(A+Q) of these systems have not this property [2]; 3) all characteristic exponents of three and higher-order systems (1_{A}) are, generally speaking, unstable (N. A. Izobov, S. N. Batan). Therefore, two-dimensional systems play a special role in the study of the behaviour of their characteristic exponents (lower and upper) under Perron perturbations.

For the two-dimensional system (1_{A}), introduce the angle γ(t) ≡ ∡{X_{1}(t), X_{2}(t)} between the solutions X_{1}(t) and X_{2}(t) forming its normal system of solutions.

Theorem 1. For the lower exponent λ_{1}(A+Q) of the two-dimensional system under any Perron perturbation Q(·) the following is true: 1) λ_{1}(A + Q) = λ_{1}(A) if σ_{P} (A) = σ_{G}(A); 2) λ_{2}(A) > λ_{1}(A + Q) > 2λ_{1}(A) − λ_{2}(A) and nonstrict λ_{1}(A + Q) ≥ λ[X_{1} sin γ] otherwise.

Scheme of proof. 1. The equality λ_{1}(A + Q) = λ_{1}(A) if σ_{P} (A) = σ_{G}(A) is a consequence of the Grobman theorem.

2. Supposing without loss of generality (1_{A}) to be a lower-triangular system, we transform (1_{A+Q}) by y = X(t)z to

ż = Q̃(t)z, z ∈ R^{2}, t ≥ 0,  (2)

a system of linear asymptotic balance (λ[Q̃] < 0). This allows us to prove the inequalities

λ[X_{1} sin γ] ≤ λ_{1}(A + Q) < λ_{2}.  (3)

3. In the case due to (3) λ[X_{1} sin γ] ≤ 2λ_{1}(A) − λ_{2}(A) we establish that the Perron lower-triangular perturbation Q_{T} (·) preserves the characteristic exponents of the initial system (1_{A}), as well as of the conjugate one (1_{−A^{T}}), and their Perron incorrectness coefficient is invariant. This allows to include the lower-triangular part Q_{T} (·) of the

1991 Mathematics Subject Classification. 34D10.

Key words and phrases. Lower exponent, two-dimensional linear system, Person pertarpation.

Perron perturbation Q(·) into the very matrix A(t) and to suppose that only the element q_{12}(t) with the exponent λ[q_{12}] < −σ_{P} (A) of the matrix Q(t) is nonzero.

4. Assume that the inequality λ(A + Q) ≤ 2λ_{1}(A) − λ_{2}(A), the opposite to one under investigation, is satisfied. Then the second component y_{2}(t) = x_{12}(t)z_{1}(t)+x_{22}(t)z_{2}(t) of the solution y(t) = X(t)z(t) of (1_{A+Q}), realizing the lower exponent λ_{1}(A + Q) with the appropriate solution z(t) = (z_{1}(t), z_{2}(t)) of (2) (or (1_{Q})) has the exponents of its terms λ[x_{21}z_{1}] = λ_{1}(A), λ[x_{22}z_{2}] < λ_{1}(A), and, as a result, it has the exponent λ[y_{2}] = λ_{1}(A+ Q) = λ_{1}(A), which contradicts to the assumption and to the inequality λ_{1}(A) < λ_{2}(A) following from the condition σ_{P} (A) < σ_{G}(A). ■

The exactness of the bounds of the lower exponent λ(A + Q) of (1_{A+Q}) is established by Theorems 2 and 3 below.

Theorem 2. For any numbers λ_{2} > λ_{1}, σ_{0} ≥ 2(λ_{2} − λ_{1}), α ∈ (2λ_{1} − λ_{2}, λ_{1}] and β ∈ [α, α+λ_{2}−λ_{1}) there exists a two-dimensional system (1_{A}) with an infinitely differentiable bounded matrix of coefficients A(·), the characteristic exponents λ_{i}(A) = λ_{i}, i = 1, 2, and the Perron incorrectness coefficient σ_{P} (A) = σ_{0}. Moreover, for any λ ∈ [α, β] there exists an analytical Perron perturbation Q_{λ}(·) such that the system (1_{A+Q_{λ}}) has a lower exponent λ_{1}(A + Q_{λ}) = λ

Scheme of a proof. 1. Fix numbers θ > 1 and θ_{0} ∈ (1, θ^{1/4}). Using the points τ_{k} ≡ (θθ_{0}^{−1})^{k} and t_{k} ≡ (θθ_{0})^{k}, k ≥ 0, define the functions f_{1}(t) and f_{2}(t) on the halfaxis t ≥ 0 as follows. On the segments [t_{k}, τ_{k+1}] with zero and even k define them by f_{i}(t) = (i − 1)λ′_{i} + (i − 2)δ_{i}, i = 1, 2, while with odd k by f_{i}(t) = (2 − i)λ′_{i} + (1 − i)δ_{i}, i = 1, 2. Here λ′_{1} = α, λ′_{2} = λ_{2}, and the numbers δ_{1} and δ_{2} satisfy λ_{1} +δ_{2} = λ_{2} +δ_{1} = σ_{0}. On the intervals (τ_{k}, t_{k}) put

f_{i}(t) = f[t; f_{i}(τ_{k}), f_{i}(t_{k})] ≡
≡ f_{i}(τ_{k}) + [f_{i}(t_{k}) − f_{i}(τ_{k})] exp{− ln^{2}(t/τ_{k}) exp[− ln^{−2}(t/τ_{k})]},  (4)

using for this purpose an analogue of the well-known infinitely differentiable function (see B. Gelbaum and J. Olmsted “Counterexamples in Analysis”); on the interval [0, 1) these functions are continued as the constants f_{i}(1). It is easy to check that a_{ii}(t) ≡ d[tf_{i}(t)]/dt are bounded and infinitely differentiable.

2. We will build the matrix A(·) of (1_{A}) as lower-triangular with already defined diagonal coefficients a_{ii}(t) and the off-diagonal coefficient a_{21}(t, σ) = −e^{−σt}, t ≥ 0, with such σ > σ_{0} that the second component x_{21}(t, σ) of its solution x(t) = (exp tf_{1}(t), x_{21}(t, σ)) has the exponent λ[x_{21}] = λ[x] = λ_{1}. For the proof of the existence of such σ > 0 we establish the Lipschitz condition |λ[x_{21}(·, σ_{2})] − λ[x_{21}(·, σ_{1})]| ≤ η|σ_{2} − σ_{1}| with a constant η = η(ε_{0}) > 1, whose exponent λ[x_{21}(·, σ)] satisfies σ_{1}, σ_{2} ≥ σ_{0} + ε_{0}, as well as the estimates

λ_{1} + δ − ε ≥ λ[x_{21}(·, σ_{0} + ε)] ≥
≥ λ_{1} + (δ − ε)θ_{0}^{2} − (λ_{2} − λ_{1})(θ_{0}^{2} − 1)  (5)

for all ε > 0 satisfying λ_{2} − λ_{1} > δ − ε, where δ = α − (2λ_{1} − λ_{2}). Due to the proved continuity on σ > 0 of the exponent λ[x_{21}(·, σ)] and inequalities (5), we obtain now, taking θ_{0} − 1 > 0 small enough, the existence of the required σ_{1} > σ_{0} : λ[x_{21}(·, σ_{1})] = λ_{1}. The required equality σ_{P} (A) = σ_{0} is established via the proof of the inequality λ[x_{21}/(x_{1}x_{2})] ≤ δ_{2}.

3. Let B_{σ}(t) be an analytical Perron perturbation with the unique nonzero element b_{21}(t, σ) = −a_{21}(t, σ) − exp(−σt), σ > σ_{0}, t ≥ 0. Arguing as in the part 2, we establish the existence of such number σ_{2} = σ(λ) > σ_{0} that the second component of the solution y(t) = (exp tf_{1}(t), y_{21}(t, σ)) of (1_{A+B_{σ2}} ) realizing its lower exponent λ_{1}(A+B_{σ_{2}}) has the exponent λ[y_{21}(·, σ_{2})] = λ. To complete the proof, it is sufficient to put B_{σ(λ)} ≡ Q_{λ}(t), t ≥ 0.

The condition α = λ[X_{1} sin γ] > 2λ_{1}(A) − λ_{2}(A) is carried out for the system (1_{A}) constructed in the proof of Theorem 2, with normal system of solutions X = [X_{1}, X_{2}]. There is the question: what will occur to a lower exponent λ_{1}(A + Q) of perturbed system in case of Perron perturbation Q(·) and of the fulfilment of an opposite condition α = λ[X_{1} sin ∡{X_{1}, X_{2}}] ≤ 2λ_{1}(A) − λ_{2}(A). The partial answer to it is given by the following

Theorem 3. For any numbers α ≤ 2λ_{1} − λ_{2} < λ_{1} < β < λ_{2} and σ_{0} > λ_{2} − α there exists a two-dimensional system (1_{A}) with a bounded infinitely differentiable matrix of coefficients A(·), a normal ordered system of solutions [X_{1}(t), X_{2}(t)] satisfying λ[X_{1} sin ∡{X_{1}, X_{2}}] = α, the characteristic exponents λ_{i}(A) = λ_{i}, i = 1, 2, the Perron incorrectness coefficient σ_{P} (A) = σ_{0} and such, that for any λ ∈ [λ_{1}, β] there exists a system (1_{A+Q}) with infinitely differentiable Perron perturbation Q(·) and lower exponent λ_{1}(A + Q) = λ.

Scheme of a proof. 1. Fix a parameter θ satisfying 1 < \sqrt{θ} < min\{[σ_{0} + (i − 1)(α − β)]/(σ_{0} + α − λ_{i})}, i = 1, 2, and introduce the points t_{k} = θ^{k}, t_{ik} = t_{k}θ^{i/4}, t_{ik}^{∓} = t_{ik}θ^{∓1/16}, i = 0, 1, 2, 3, and k ≥ 0. Define the functions f_{1}(t) = α for all t ≥ 0 and f_{2}(t) as follows: 1) f_{2}(t) = λ_{1}−σ_{0} = −δ_{2} for t ∈ [0, t_{00}^{−}]; 2) on the segments [t_{ik}^{+}, t_{i+1,k}^{−}] for i = 0, 1, 2, 3 and t_{4k}^{−} = t_{0,k+1}^{−} put f_{2}(t) = −δ_{2} for t ∈ [t_{1k}^{+}, t_{2k}^{−}] ⋃[t_{3k}^{+}, t_{0,k+1}^{−}], f_{2}(t) = λ_{2} for t ∈ [t_{0k}^{+}, t_{1k}^{−}] and f_{2}(t) = c_{k} for t ∈ [t_{2k}^{+}, t_{3k}^{−}], k ≥ 0, with a number c_{k}, defined below; 3) on the remaining intervals (t_{ik}^{−}, t_{ik}^{+}) put (see (4)) f_{2}(t) = f[t; f_{2}(t_{ik}^{−}), f_{2}(t_{ik}^{+})], i = 0, 1, 2, 3, k ≥ 0. We define the diagonal coefficients of the required lower-triangular matrix A(t) by a_{ii}(t) = [tf_{i}(t)]′, t ≥ 0.

Let the coefficient a_{21}(t) be equal to 1 on the segments [t_{1k}^{+}, t_{2k}^{−}], k ≥ 0, on the segments [t_{3k}^{+}, t_{0,k+1}^{−}], let it be equal to a constant b_{k} ∈ (−1, 0), which is equal to the ratio of the integrals of the function exp(α + δ_{2})τ over the segments [t_{1k}^{+}, t_{2k}^{−}] and [t_{3k}^{+}, t_{0,k+1}^{−}] and on all remaining intervals of the half-axis t ≥ 0 let it be equal to 0.

2. Using the form of the constructed functions a_{ii}(t) and a_{21}(t), for the second component x_{21}(t) of the solution x(t) of the solution lower-triangular system (1_{A}) with initial value x(0) = (1, 0) we establish the existence of a bounded sequence ({c_{k}} the numbers c_{k} are used in the definition of f_{2}(t) on the intervals [t_{k}, t_{k+1})), that the equal-

ities x_{21}(τ_{k}) exp(−λ_{1}τ_{k}) = max_{t_{k}≤t≤t_{k+1}}[x_{21}(t) × exp(−λ_{1}t)] = (2α + 2δ_{2})^{−1}, k ≥ k_{0}, are

fulfilled. From them we have λ[x] = λ[x_{21}] = λ_{1}. Let q_{12}(t) be equal to exp(−στ_{k}) on the segments [τ_{k}, τ_{k} + 1], ∀k ≥ k_{1}, and be equal to 0 on the complement with respect to the half-axis t ≥ 0. Put all other elements of the matrix Q(·) to be zero. By the transformation y = X(t)z we transform (1_{A+Q}) to (1Q̃) of the form (2) with λ[Q̃] < 0.

3. Let a solution y(t) = (y_{1}(t), y_{2}(t)) of (1_{A+Q}) realize the lower exponent λ_{1}(A+Q). For an appropriate solution z(t) = (z_{1}(t), z_{2}(t)) of (1Q̃) we have: z_{1}(t) → d_{1} ≠ 0 as t → +∞, λ[z_{2}] < 0. We detect the exponent λ[x_{22}z_{2}] from the parameter σ > σ_{0} and the existence of such its value σ = σ_{1} > σ_{0}, for which λ[x_{22}z_{2}] = λ ∈ (λ_{1}, β]. Then we receive the required relations λ[y_{2}] = λ[x_{22}z_{2}] = λ > α = λ[y_{1}] from the representation of y_{2}(t).

4. We transform the constructed piecewise-constant functions a_{21}(t) and q_{12}(t) into infinitely-differentiable ones a_{21}^{∗}(t) and q_{12}^{∗}(t) by means of replacing of them by the functions f[t; a_{21}(τ_{k}), a_{21}(t_{k})] and f[t; q_{12}(τ_{k}), q_{12}(t_{k})] on all intervals (τ_{k}, t_{k}), one endpoint of which concides with a point of discontinuity of these functions, of such a small length, that the systems (1_{A}) and (1_{A^{∗}}), (1_{A+Q}) and (1_{A^{∗}+Q^{∗}}) are pairwise asymptoticaly equivalent (this is possible according to the Yu. S. Bogdanov–S. A. Mazanik theorem). ■

Problem. Find out whether Theorem 3 is true in the case λ ∈ (2λ_{1} − λ_{2}, λ_{1}).

This work was financed by Byelorussian Fund of Fundamental Researches.

References

1. N. A. Izobov, Linear systems of ordinary differential equations. (Russian) Itogi Nauki i Tekhniki. VINITI. Mat. Anal. 12(1974), 71–146.

2. R. A. Prokhorova, On some propertions of lower exponent under perron perturbations. (Russian) Differentsial’nye Uravnenija 11(1975), No. 6, 997–1004.

Author’s address:

N. Izobov Institute of Mathematics Belorussian Academy of Sciences 11, Surganova St., Minsk 220072 Belarus