Systems of Linear Ordinary Differential Equations with Bounded Coefficients May Have Very Oscillating Solutions

An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system. Bounds for oscillations. In [1] it was shown that a linear ordinary differential equation of order n, with real analytic coefficients bounded in a neighborhood of the interval [−1, 1], admits a uniform upper bound for the number of isolated zeros of a solution defined on this interval. The analyticity condition can be relaxed; only the boundedness of the coefficients matters. Probably, the simplest result in this spirit is the following theorem for the linear ordinary differential equation y(t) + a1(t)y(t) + · · ·+ an(t)y(t) = 0 (1) with continuous coefficients on [α, β] ⊂ R. Theorem 1 ([3, 4]). If the coefficients of the differential equation (1) are uniformly bounded by the constant C ≥ 1 (that is, max{|ai(t)| : i = 1, . . . , n} ≤ C), then a solution defined on [α, β] cannot have more than n−1+ n ln 2C|β−α| isolated zeros. An analog of this result for a system of ordinary differential equations, viewed as a vector field in space, would concern the number of isolated intersections between integral trajectories of the vector field and hyperplanes (or, more generally, hypersurfaces). For polynomial systems of degree d on R of the form ẋi = vi(t, x), i = 1, . . . , n, vi(t, x) = ∑ k+|α|≤d vikαt x, (2) and algebraic hypersurfaces given by {P = 0} where P = P (t, x) is a polynomial of degree d, the following theorem, proved in [3] (see also [2]), gives a bound for the number of isolated intersections in case the magnitude of the domain of the solution and the amplitude of the solution are controlled by the height of the polynomial system, that is, the number max{|vikα| : k + |α| ≤ d, i = 1, . . . , n}. Theorem 2. Suppose that the height of system (2) is bounded by the positive constant C. If γ is an orbit of the system contained in the box BC = {(t, x) ∈ R : |t| < C, |xi| < C}, then the number of isolated intersections of γ and {P = 0} is at most (2 +R) where B = B(n, d) is an explicit elementary function of d and n whose growth rate is smaller than exp exp exp exp(4n ln d+O(1)) as d, n→∞. 1991 Mathematics Subject Classification. Primary 34C10, 34M10; Secondary 34C07.