On the reducibility systems of two linear first-order ordinary differential equations
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On the reducibility systems of two linear first-order ordinary differential equations G. A. Grigorian1 Received: 19 June 2020 / Accepted: 27 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Some global solvability criteria for the scalar Riccati equations are used to establish new reducibility criteria for systems of two linear first-order ordinary differential equations. Some examples are presented. Keywords The Riccati equation · Global solvability · Linear systems of ordinary differential equations · Reducibility Mathematics Subject Classification 34C10 · 34C11 · 34C99
1 Introduction Let a(t), b(t), c(t) and d(t) be real valued continuous and bounded functions on [t0 , +∞). Consider the linear system of ordinary differential equations ⎧ ⎨ φ = a(t)φ + b(t)ψ, ⎩
ψ = c(t)φ + d(t)ψ, t ≥ t0 .
(1.1)
Introduce new unknowns by equalities ⎧ ⎨ φ1 = z 11 (t)φ + z 12 (t)ψ, ⎩
(1.2) ψ1 = z 21 (t)φ + z 22 (t)ψ,
Communicated by Adrian Constantin.
B 1
G. A. Grigorian [email protected] Institute of Mathematics NAS of Armenia, Yerevan, Armenia
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G. A. Grigorian
where φ1 , ψ1 are new unknowns, z jk (t), j, k = 1, 2 are the coefficients of a transformation. For φ1 and φ2 from (1.1) and (1.2) we obtain a new linear system of equations ⎧ ⎨ φ1 = a1 (t)φ1 + b1 (t)ψ1 , ⎩
ψ1 = c1 (t)φ1 + d1 (t)ψ1 , t ≥ t0 .
(1.3)
Definition 1.1 The system (1.1) is called reducible if there exists a bounded matrix function Z (t) ≡ (z jk (t))2j,k=1 of transformation (1.2) for the system (1.1) such that Z (t), Z −1 (t) exist and are bounded with det Z −1 (t) and the coefficients a1 (t), b1 (t), c1 (t) and d1 (t) of the system (1.3) are constants. Study the reducibility behavior of systems of linear ordinary differential equation, in particular of the system (1.1), is an important problem of qualitative theory of differential equations and many works are devoted to it (see [1–6], and cited works therein) The reducible systems (see [7]) play an important role in the study of stability of solutions of nonlinear systems, for which the first approximation contains the time. They play also an important role in the ’.... study of stability of quasi-periodic motion and preservation of invariant tori in Hamiltonian mechanics (where the reducibility of linear equations with quasi-periodic coefficients play an important role)” (see [1]). In this paper some new reducibility criteria for the system (1.1) are obtained.
2 Auxiliary propositions Let f (t), g(t), h(t) be real valued continuous functions on [t0 , +∞). Consider the Riccati equation y + f (t)y 2 + g(t)y + h(t) = 0, t ≥ t0 .
(2.1)
In this section we represent some global existence criteria for Eq. (2.1) proved in [8] and [9]. They will be used in the Sect. 3 to obtain new reducibility criteria for the system (1.1). Theorem 2.1 Let f i (t) and h i (t) be continuously differentiable functions on [t0 , +∞) i = 1, 2. If f 1 (t) ≤ f (t) ≤ such that (−1)i f i (t) > 0, (−1)i h i (t) > 0, t ≥ t0 , √ h i (t) 1 f i (t) f 2 (t), h 1 (t) ≤ h(t) ≤ h 2
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