Mem. Differential Equations Math. Phys. 15(1998), 153–156

N. Izobov

ON THE ENUMERABLE SET OF DIFFERENT CHARACTERISTIC SETS OF SOLUTIONS OF A PFAFFIAN LINEAR SYSTEM

(Reported on June 22, 1998)

Consider the Pfaffian linear system

∂x/∂t_{i} = A_{i}(t)t, x ∈ R^{n}, t = (t_{1}, t_{2}) ∈ R_{+}^{2},  (1)

with bounded continuously differentiable matrices A_{1}(t) and A_{2}(t) satisfying the following condition of complete integrability:

∂A_{1}(t)/∂t_{2} + A_{1}(t)A_{2}(t) = ∂A_{2}(t)/∂t_{1} + A_{2}(t)A_{1}(t), t ∈ R_{+}^{2}.

It is well known [1, p. 34] that the ordinary linear system dx/dt = A(t)x, x ∈ R^{n}, t ∈ R_{+}^{1}, with bounded piecewise continuous coefficients has no more than n different characteristic exponents. Let λ[x] = λ ∈ R^{2} be a characteristic vector [2 – 4] of a nontrivial solution x: R_{+}^{2} → R^{n}\{0} of (1) defined by

L_{x}(λ) ≡ \overline{lim}_{t→∞}[ln∥x(t)∥ − (λ, t)]/∥t∥ = 0, L_{x}(λ − εe_{i}) > 0, ∀ε > 0, i = 1, 2.

For the characteristic set Λ_{x} = ⋃ λ[x] of this solution which is the most natural analog of Lyapunov’s characteristic exponent of a one variable vector-function, the essential initial problem about possible number of different ^{∗} characteristic sets Λ_{x} of all nontrivial solutions x of (1) remained open. Note also that the set {P_{x}} of different lower characteristic sets P_{x} = ⋃ p[x] of all nontrivial solutions x of (1) composed of lower characteristic vectors [5, 6] p[x] = p ∈ R^{2} defined by

l_{x}(p) ≡ \underline{lim}_{t→∞}[ln∥x(t)∥ − (p, t)]/∥t∥ = 0, l_{x}(p + εe_{i}) < 0, ∀ε > 0, i = 1, 2,

is nonenumerable and, moreover, the set of the lower characteristic vectors ⋃_{x≠0} P_{x} of (1)

has a positive planar Lebesgue measure [5, 6].

It holds the following

Theorem. For any sequence C = {c_{m}} of pairwise noncollinear vectors there is a complete integrable two-dimensional system (1) with bounded infinitely differentiable coefficients such that all of its solutions x(t, c_{m}), m ∈ N, have pairwise different characteristic sets Λ(m) with a positive linear Lebesgue measure, If x(t) is a solution of (1) linearly independent with any of x(t, c_{m}), c_{m} ∈ C, then its characteristic set Λ_{x} = Lim_{m→∞} Λ(m)

also has a positive measure.

1. Construction of the required system. The preliminary notes. To an enumerable set C ⊂ R^{2}\{0} of the vectors c_{m} = (c_{m}^{1}, c_{m}^{1}) ∈ R^{2} assign the enumerable set α = {α_{m}} ⊂ R of different numbers α_{m} ≡ −c_{m}^{2}/c_{m}^{1} ∈ ∈ (−∞, ∞), the ratios of the

1991 Mathematics Subject Classification. 35F99.

Key words and phrases. Pfaffian linear system, characteristic exponent.

∗The characteristic sets Λ_{x} and Λ_{y} of the solutions x ≠ 0 and y ≠ 0 of (1) are different if Λ_{x} ⋂ Λ_{y} ≠ Λ_{x} ⋃ Λ_{y}. components of the vector c_{m}. Without loss of generality it can be assumed that first components c_{m}^{1} of c_{m} are nonzero.

In the closed first quarter R_{+}^{2} of the plane R^{2} we will build the required Pfaffian system by constructing its fundamental (lower-triangular and infinitely differentiable) system of solutions X(t) = ((x_{ij}(t))_{1}^{2} with x_{12}(t) ≡ 0 for t ∈ R_{+}^{2}.

On the interval (−∞, ∞) define two infinitely differentiable functions [7, p. 54]

e_{01}(η; η_{1}, η_{2}) = \{ \begin{matrix}0, & if η ∈ (−∞, η_{1}], \\ exp\{−(η − η_{1})^{−2} exp[−(η − η_{2})^{−2}]\}, & if η ∈ (η_{1}, η_{2}), \\ 1, & if η ∈ [η_{2}, ∞), \end{matrix}

e_{11}(η; η_{1}, η_{2}) = \{ \begin{matrix}1, & if η ∉ (η_{1}, η_{2}), \\ 1 − exp[−(η − η_{1})^{−2}(η − η_{2})^{−2}], & if η ∈ (η_{1}, η_{2}), \end{matrix}

where −∞ < η_{1} < η_{2} < +∞ are used for constructing of elements of the matrix X(t).

With the help of the numbers p_{0} = 0, q_{0} = ε ∈ (0, 1/8), and q_{k} = 1 − 2^{−k},p_{k} = q_{k} − 2^{−1−k}, k ∈ N, define the sectors: the closed ones S_{k} = {t ∈ R_{+}^{2} : p_{k} ≤ t_{2}/t_{1} ≤ g_{k}} with k ≥ 0, the open ones s_{k} = {t ∈ R_{+}^{2} : q_{k−1} < < t_{2}/t_{1} < p_{k}} with natural k ≥ 1, and the also sector s_{0} = {t ∈ R_{+}^{2} : 0 ≤ t_{1}/t_{2} ≤ ε}.

2. The construction of the diagonal elements of the fundamental system. In R_{+}^{2} define the positive function x_{2}(t) by

ln x_{2}(t) = \{ \begin{matrix}\sqrt{ε}t_{1} + t_{2}/\sqrt{ε} − ( \sqrt[4]{εt_{1}^{2}} − \sqrt[4]{t_{2}^{2}/ε})^{2}e_{01}(t_{2}/t_{1}; 0, ε), & t ∈ S_{0}, \\ \sqrt{ε}t_{2} + t_{1}/\sqrt{ε} − ( \sqrt[4]{εt_{2}^{2}} − \sqrt[4]{t_{1}^{2}/ε})^{2}e_{01}(t_{1}/t_{2}; 0, ε), & t ∈ s_{0}, \\ 2\sqrt{t_{1}t_{2}}, & t ∈ R_{+}^{2}\(s_{0} ⋃ S_{0}) ≡ S. \end{matrix}

Put the function x_{1} : R_{+}^{2} → [1, +∞) be equal to x_{2}: 1) on a closed sector S̃ ⊂ R_{+}^{2}, which is bounded by the bisectrix t_{2} = t_{1} and the positive coordinate semiaxis t_{1} = 0;

2) on all sectors S_{k}, k ≥ 0. In order to define this function on the remaining sectors s_{k}, k ∈ N, we consider the numbers r_{k} ≥ er_{k−1}, r_{0} = 1, k ∈ N, satisfying

r_{k} > (1 + |α_{k}| + |α_{k+1}|) exp 3(q_{k} − p_{k})^{−2}, k ∈ N; r_{1} > (1 + |α_{1}|) exp 3ε^{−2}.

In the sector s_{k} we will define x_{1}(t) by

ln x_{1}(t) = 2\sqrt{t_{1}t_{2}}\{1 + e_{01}(∥t∥/r_{k}; 1, 3/2)[e_{11}(t_{2}/t_{1}; q_{k−1}, p_{k}) − 1]\}, t ∈ s_{k}, k ∈ N.

Note that by definition of the function e_{01}(η; η_{1}, η_{2}) on the whole axis (−∞, +∞) we have

ln x_{1}(t) = 2\sqrt{t_{1}t_{2}}e_{11}(t_{2}/t_{1}; q_{k−1}, p_{k}), t ∈ s_{k}, and ∥t∥ ≥ 3r_{k}/2.

3. The construction of the off-diagonal elements of the fundamental system. Due to [5, 6], define the off-diagonal element x_{21}(t) of a constructed two-di-mesional linear Pfaffian system with bounded infinitely differentiable coefficients and two-dimensional time by the equality x_{21}(t) = x_{2}(t)F (t), t ∈ R_{+}^{2}, where the infinitely differentiable function F (t) is defined by

F (t) = \{ \begin{matrix}0, & if t ∈ S̃, \\ α_{k}e_{01}(∥t∥/r_{k}; 1/2, 1), & if t ∈ s_{k}, k ∈ N, \\ α_{k}e_{01}(∥t∥/r_{k}; 1/2, 1) + e_{01}(t_{2}/t_{1}; p_{k}, q_{k})[α_{k+1}e_{01}(∥t∥× \\ ×r_{k+1}^{−1}; 1/2, 1) − α_{k}e_{01}(∥t∥/r_{k}; 1/2, 1), α_{0} = 0, & if t ∈ S_{k}, k ≥ 0. \end{matrix}

The infinite differentiability of the functions x_{1}(t) ≥ 1, x_{2}(t) ≥ 1, and F (t) on R_{+}^{2} follows from the same property of the functions e_{01}(t_{2}/t_{1}; p_{k}, q_{k}) for k ≥ 0, e_{01}(∥t∥/r_{k}; 1/2, 1) for k ≥ 1, and e_{11}(t_{2}/t_{1}; q_{k−1}, p_{k}) for k ∈ N.

4. The boundedness of coefficient matrices

A_{i}(t) = \frac{∂X(t)}{∂t_{i}} X^{−1}(t) = ( \begin{matrix}x_{1}^{−1}(t) \frac{∂x_{1}(t)}{∂t_{i}} & 0 \\ \frac{x_{2}(t)}{x_{1}(t)} \frac{∂F (t)}{∂t_{i}} & x_{2}^{−1}(t) \frac{∂x_{2}(t)}{∂t_{i}} \end{matrix} ) , i = 1, 2

of the constructed two-dimensional system (1) is proved by the following statement:

Lemma. For all m ∈ N and any (η_{1}, η_{2}) with the lengths ≤ 1/2 there are the estimates

(η − η_{1})^{−m}e_{01}(η; η_{1}, η_{2}) ≤ [\sqrt{m/2e} exp(η_{2} − η_{1})^{−2}]^{m}, η ∈ (η_{1}, η_{2}),
(η_{2} − η)^{−m} exp[−(η_{2} − η)^{−2}] ≤ (\sqrt{m/2e})^{m}, η ∈ (η_{1}, η_{2}).

It is evident, that the infinite differentiability of the matrices A_{i}(t) in R_{+}^{2} follows from the same property of the nonsingular lower-triangular matrix X(t). Similarly, the infinite differentiability of the fundamental solutions system X(t) ensures the feasibility of the complete integrability conditions (2) for the constructed two-dimensional system (1).

5. The construction of the characteristic set of solutions. First for the characteristic set Λ_{x_{2}} of the solution x(t, l_{2}) = (0, x_{2}(t)) of system (1) we obtain the representation Λ_{x_{2}} = Λ = \{(λ_{1}, 1/λ_{1}) ∈ R_{+}^{2} : λ_{1} ∈ [\sqrt{ε}, 1/\sqrt{ε}]\}. Then for the solution x(t, c_{m}) we establish the relations

∥x(t, c_{m})∥ = x_{1}(t) = |x_{2}(t)|^{e_{11}(t_{2}/t_{1};q_{m−1},p_{m})} ≡ ρ_{m}(t), t ∈ s_{m}, ∥t∥ ≥ 3r_{m}/2;

max{x_{1}(t), |α_{k} − α_{m}|x_{2}(t)} ≤ ∥x(t, c_{m})∥ ≤
≤ (1 + |α_{k} − α_{m}|)x_{2}(t), t ∈ s_{k}, ∥t∥ ≥ 3r_{k}/2, k ≠ m;

1 ≤ ∥x(t, c_{m})∥/x_{2}(t) ≤ 1 + |α_{k} − α_{m}| + |α_{k+1} − α_{k}|, t ∈ S_{k}, ∥t∥ ≥ r_{k+1}, k ≥ 0;

∥x(t, c_{m})∥ = \sqrt{1 + α_{m}^{2}}x_{2}(t), t ∈ S̃.

Hence in view of the equality lim_{k→∞} r_{k}^{−1} ln(1 + |α_{k}| + |α_{k+1}|) = 0, true by the choice of

the numbers r_{k}, and the uniform in t ∈ s_{k} tending of e_{11}(t_{2}/t_{1}; q_{k−1}, p_{k}) as k → ∞, it follows that the characteristic set Λ(m) of x(t, c_{m}) coincides with the characteristic set of the function ρ_{m}(t), which is equal to x_{2}(t) outside the sector S_{m}, m ∈ N. By nontrivial reasonings it established then, that the vector λ_{2}(η) ∈ R^{2} with the components λ_{2}(η) = = φ′_{m}(η), λ_{1}(η) = φ_{m}(η) − ηφ′_{m}(η) for any η ∈ [ε, 1/ε] is a characteristic vector of the function ρ_{m}(t), where the function φ_{m}(η) = 2\sqrt{η}e_{11}(η; q_{m−1}, p_{m}) is infinitely differentiable and convex up.

Thus we have the representation Λ(m) = {λ(η) ∈ R^{2} : η ∈ [ε, 1/ε]}. The curve Λ(m) coincides with the hyperbola Λ at λ_{1} ∈ [\sqrt{ε}, \sqrt{q_{m−1}}] ⋃[\sqrt{p_{m}}, 1/\sqrt{ε}] and is located below this hiperbola at λ_{1} ∈ (\sqrt{q_{m−1}}, \sqrt{p_{m}}). In particular, for η = η_{m} ≡ (q_{m−1} + p_{m})/2 we obtain the point λ(η_{m}) ∈ Λ(m) with the coordinates λ_{1}(η_{m}) = \sqrt{η_{m}}(1 − e^{−γ_{m}}), λ_{2}(η_{m}) = (1 − e^{−γ_{m}})/\sqrt{η_{m}}, where γ_{m} ≡ 16(p_{m} − q_{m−1})^{−4} and their product λ_{1}(η_{m})λ_{2}(η_{m}) < 1. Obviously, Λ(l) ≠ Λ(m) ≠ Λ for any l, m ∈ N, l ≠ m, and Lim_{m→∞} Λ(m) = Λ. It is not dificult to prove also the equality Λ_{x} = Λ for a solution x(t)

linearly independent with any of x(t, c_{m}), m ∈ N, of the system (1).

The construction of the characteristic sets of all solutions of (1) is completed.

Remark. Obviously, from the constructed two-dimensional system (1) it may be possible to obtain an n-dimensional completely integrable system (1) with bounded infinitely differentiable coefficients in R_{+}^{2}, which have enumerable number of dofferent characteristic sets of the solutions.

Problem. It ought be to clarified, whether the set {Λ_{x}} of different characteristic sets Λ_{x} of solutions x : R_{+}^{2} → R^{n} of a Pfaffian system (1) is finite or enumerable.

References

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Author’s address: Institute of Mathematics Belorussian National Academy of Sciences 11, Surganova St., Minsk 220072 Belarus