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Unbounded Growth of Total Variations 43 [6] G. B. Hsu and J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part III: Natural hysteresis memory effect, Int. J. Bifurcation and Chaos 8(3) (1998), 447–470. [7] G. Chen and J. Zhou, Vibration and Damping in Distributed Systems, Vol I: Analysis, Estimation, Attenuation and Design, CRC Press, Boca Raton, FL, 1993. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, New York, 1989.

3), with w0 ∈ C 1 ([0, 1]) and w1 ∈ C 0 ([0, 1]). 10). 13) remains bounded, provided that initially w0 and w1 have bounded total variations. 13) can grow unbounded (when and) only when initially (at least one of) w0 and w1 have unbounded total variations. This is possible, as shown in the following example. 1. Choose x 1 w0 (x) = ξ(ξ − 1) sin dξ, w1 (x) = 0; 0 < x < 1. ξ 0 Then (w0 , w1 ) ∈ H01 (0, 1) × L2 (0, 1). 7)2 are satisfied. 12) is continuous for any (x, t) ∈ [0, 1] × [0, ∞). Here 1 w0 (x) = x(x − 1) sin , x is easily verified to have VI¯(w0 ) = ∞.

3), V[0,T ] (v) ≤ |α|V[0,T ] (u) + |β| · 1 [|u(0+)| + V[0,T ] (u)] + V[0,T ] (f ). 2) holds. M. 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.

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