Fractal oscillations of self-adjoint and damped linear differential equations of second-order
Fractal oscillations of self-adjoint and damped linear differential equations of second-order
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DOI：
10.1016/j.amc.2011.07.047
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发表时间：
2011-11
期刊：
影响因子：
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通讯作者：
Mervan Pašić;Satoshi Tanaka
中科院分区：
文献类型：
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作者：
Mervan Pašić;Satoshi Tanaka
对于给定的真实的数s∈[1，2），给出了系数p（x）和q（x）的一些充分条件，使得线性微分方程（p（x）y′）′+q（x）y=0在（0，T]上的每一个解y=y（x），y∈C2（（0，T]）在x=0附近有界且分形振动，且分形维数为s.这意味着y在x=0附近振荡，并且y的图Γ（y）的分形（计盒）维数等于s，并且Γ（y）的s维上Minkowski容度（广义长度）是有限的且严格正的。证明了y在x=0附近具有类似于线性调频函数ych（x）=a（x）S（φ（x））的分形几何渐近行为，这种分形几何渐近行为在时频分析及其各种应用中经常出现.进一步，对形式为y″+（μ/x）y′+g（x）y=0，x∈（0，T]的Bessel，chirp等阻尼线性微分方程建立了这类振动性.为了证明主要结果，本文给出了真实的连续函数在x=0附近分形振动的一个新判据，它本质上改进了[1]中的有关判据.
For a prescribed real number s∈[1,2), we give some sufficient conditions on the coefficients p(x) and q(x) such that every solution y=y(x), y∈C2((0,T]) of the linear differential equation (p(x)y′)′+q(x)y=0 on (0,T], is bounded and fractal oscillatory near x=0 with the fractal dimension equal to s. This means that y oscillates near x=0 and the fractal (box-counting) dimension of the graph Γ(y) of y is equal to s as well as the s dimensional upper Minkowski content (generalized length) of Γ(y) is finite and strictly positive. It verifies that y admits similar kind of the fractal geometric asymptotic behaviour near x=0 like the chirp function ych(x)=a(x)S(φ(x)), which often occurs in the time–frequency analysis and its various applications. Furthermore, this kind of oscillations is established for the Bessel, chirp and other types of damped linear differential equations given in the form y″+(μ/x)y′+g(x)y=0, x∈(0,T]. In order to prove the main results, we state a new criterion for fractal oscillations near x=0 of real continuous functions which essentially improves related one presented in [1].