丁 錦 紅

(河海大學(xué) 理學(xué)院數(shù)學(xué)系, 南京 210098)

0 引 言

對時間周期非線性微分方程平衡點的Lyapunov穩(wěn)定性研究是微分方程與動力系統(tǒng)的一個重要問題. 然而, 研究保守系統(tǒng)的穩(wěn)定性很困難, 因為Lyapunov直接方法不適用. 文獻[1-8]在Birkhoff標(biāo)準(zhǔn)型和Moser扭轉(zhuǎn)定理的基礎(chǔ)上, 發(fā)展了研究二階拉格朗日方程穩(wěn)定性的解析方法, 即三階近似方法.

文獻[8-9]將上述方法推廣到帶有阻尼的二階微分方程中, 即計算了二階非線性微分方程

x″+h(t)x′+f(t,x)=0

(1)

的三階近似方程

x″+h(t)x′+a(t)x+b(t)x2+c(t)x3+…=0

(2)

的第一扭轉(zhuǎn)系數(shù)β, 其中:f: R×R→ R是關(guān)于t的2π-周期函數(shù), 且充分光滑;

若ψ(t)是方程(1)的2π-周期解, 則方程(2)中的系數(shù)a,b,c∈C(R/2πZ), 且

若β≠0, 則稱解ψ(t)是扭轉(zhuǎn)的. 由Moser扭轉(zhuǎn)定理可知, 此扭轉(zhuǎn)周期解在Lyapunov意義下是穩(wěn)定的.

方程(2)的一階線性部分是

x″+h(t)x′+a(t)x=0.

(3)

文獻[9]給出了關(guān)于式(3)的穩(wěn)定性準(zhǔn)則. 但式(3)的穩(wěn)定性并不能保證方程(1)的穩(wěn)定性.

本文考慮阻尼奇異微分方程

(4)

1 預(yù)備知識

引理1[8-9]方程(2)的扭轉(zhuǎn)系數(shù)為

其中:

θ=2πρ,ρ是Hill方程

(6)

的旋轉(zhuǎn)數(shù);r是奇異方程

(7)

的唯一正2π-周期解, 且滿足

這里

引理2[10]假設(shè)

(8)

則存在0

1) 方程(7)唯一的正2π-周期解滿足

(9)

2) 方程(6)的旋轉(zhuǎn)度ρ滿足

(10)

2 主要結(jié)果

由文獻[11]可知, 當(dāng)滿足條件(8)時, 對于任意的α>0, 方程(4)有一個正的2π-周期解.

定理1存在常數(shù)L(α)>0, 使得如果a,h滿足條件:

(11)

則存在一個正常數(shù)α*, 對于所有的α>0且α≠α*, 方程(4)的正周期解都是穩(wěn)定的.

證明: 設(shè)ψ(t)是奇異方程(4)的正2π-周期解, 則其三階近似可寫為

(12)

其中:

(13)

將式(12)兩邊同乘σ(h)(t), 可得

(14)

將方程(4)兩邊同乘σ(h)(t), 有

(15)

再將方程(15)兩邊同時積分可得

(16)

(17)

由文獻[10]可知, 0<γ(1-S(A1))≤ψ(t)≤γ(1+S(A1)), 且當(dāng)A1→ 0+時,ψmax/ψmin=1+O(A1). 故當(dāng)式(11)成立時, 若A1→ 0+, 則有

又由式(9)與(10)可知

故有

根據(jù)式(5),(13)和(17), 奇異方程(4)的扭轉(zhuǎn)系數(shù)β為

(23)

第二項為

(24)

又由式(23)和(24), 當(dāng)A1→ 0+時,

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