稳定性理論中第一临界情形的微分方程与微分差分方程的等价性問題

<正> §1.問題与方法.在1]中提出了等价性問題,并对于一般n的情形作了系統的研究.本文是处理在第一临界情形下的微分方程与微分差分方程的等价性問題. 問題是研究微分方程組

ON THE EQUIVALENCE PROBLEM OF DIFFERENTIAL EQUATIONS AND DIFFERENCE-DIFFERENTIAL EQUATIONS IN THE THEORY OF STABILITY OF THE FIRST CRITICAL CASE

The problem is to investigate the equivalence of stability between the system of differential equations and the system of difference-differential equations where the psσs, qsos, pss and qss are given constants, and δ(t)'s may be non-negative real constants or non-negative real continuous of t. In order to this problem, we consider first the case of Ps= 0, qs=0 (s=1,2,…,n), i.e. to study the eqivalence problem of stability between the system of differential equations and the system of difference-differential equations satisfying the following conditions:(1) X(x_1,…,x_n,x), Y(X_1,…,x_n, x), X_s(x_1, x_2,…, x_n,x) and Y_s(x, x_1, …, x_n) are analytic functions of the variables x_1, …, x_n, x in the neighbourhood of the origin Of coodinates, and the orders of the terms of their expansions are not less than two;(2) All the roots of the characteristic equation D(X)≡|P_(sσ)+q_(sσ)-δ_(sσ)X|=0 (s,σ= 1, 2,…,n) satisfy the conditions(4) m_s≥m.In the article the second type of method mentioned in 1] is used.§1. The equivalence of stabilityTheorem 1. If m is an odd positive integer and g + l < 0, then there exists a positive constant △ = △ (X, Y, p_(sσ), q_(sσ), X_s, Y_s) > 0, such that the trival solution of (2)' is asymptotically stable, provided that the δ(t)'s satisfy the inequality 0 ≤ δ ≤ △.§2. The equivalence of instabilityTheorem 2. If m is an odd positive integer and g + l > 0, then there exists a positive constant △= △(X, Y, p_(sσ), q_(sσ), X_s, Y_s) > 0, such that the trivial solution of (2)' is unstable, provided that the δ(t)'s satisfy the inequality 0 ≤ δ≤△.Theorem 3. If m is an even positive integer and g + l ≠0, then there exists a positive constant △= △(X, Y, p_(sσ), q_(sσ), X, Y) > 0, such that the trivial solution of (2)'is unstable, provided that the δ(t)'s satisfy the inequality 0 ≤ δ ≤ △.§3. The equivalence of stability in the singular caseTheorem 4. The trivial solution of (1)' is stable, then there exists a positive constant △ = △(X, Y, p_(sσ), q_(sσ), X_s, Y_s) > 0, such that the trivial solution of (2)' is stable, provided that the δ(t)'s satisfy the inequality 0 ≤ δ ≤ △.§4. The general caseWe now return to the case between (1) and (2). We can solve this problem on the basis of the equivalence of (1)' and (2)'. Using non-linear transformation x_s = ξ_s + u_s(t) (s = 1, 2, …,n), the system (1) will remain in the same form as the syslem (1)': where u_s(x)'s (s = 1, 2, …, n) satisfy the following system of equations:where x_s = u_s(x) (s = 1,…, n) are analytic functions of x for sufficienfly small |x|.