An Integral Equality and its Applications by Hille E.

By Hille E.

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Apostal, Mathematical Analysis, Addison Wesley, Reading, MA, 1957. [2] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99 (1992), 332–334. [3] G. B. Hsu and J. Zhou, Linear superposition of chaotic and orderly vibrations on two serially connected strings with a van der Pol joint, Int. J. Bifurcation and Chaos 6 (1996), 1509–1527. [4] G. B. Hsu and J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part I: Controlled hysteresis, Trans.

M} is an orbit of g ≡ f 2 with prime period m. 5) are satisfied. Therefore, we have O2 = {xj | j = 1, 2, . . , m} and for some integer j1 : 0 < j1 ≤ m, k k x1 = f j1·2 (ξ), k x2 = f (j1 +1)·2 (ξ), . . , xm = f (j1 +m)·2 (ξ). 16) ) j→∞ =∞ for any = 0, 1, 2, . . , m · 2k − 1, for some subinterval I0 ⊆ J (where I0 depends on given ). Given any such ∈ {0, 1, 2, . . , m · 2k − 1}, we can find a positive integer ˆ > 0 such that + ˆ = j1 · 2k (modm · 2k ). 17) y2 = f ˆ+2k (ξ), I0 = [y1 , y2 ], if y1 < y2 , [y2 , y1 ], if y1 > y2 .

36) is degenerate) and, therefore, the 38 Chen et al. 1 again apply. Nevertheless, we could not locate a precise reference to this effect. 35) are not periodic points. ) 4 Miscellaneous Remarks In this paper, we have successfully shown that when chaotic vibration occurs for the wave equation caused by the nonlinear boundary condition specified here, the total variations of snapshots tend to infinity as t → ∞ for a large class of initial data, even though the total variation of any such initial data is finite at time t = 0.

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