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What is the exact condition for fractional integrals and derivatives of Besicovitch functions to have exact box dimension?

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  • He, G.L.
  • Zhou, S.P.

Abstract

Let 10 with λk→∞ satisfy the Hadamard condition λk+1/λk⩾λ>1. For a class of Besicovich functions B(t)=∑k=1∞λks-2sin(λkt), the present paper investigates the intrinsic relationship between box dimension of graphs of their vth fractional integrals g(t) and uth fractional derivatives g˜(t) and the asymptotic behavior of {λk}. We show that: if 01+v, then for sufficiently large λ, dim¯BΓ(g)=dim̲BΓ(g)=s-v holds if and only if limn→∞logλn+1logλn=1; if 0

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  • He, G.L. & Zhou, S.P., 2005. "What is the exact condition for fractional integrals and derivatives of Besicovitch functions to have exact box dimension?," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 867-879.
    • Handle: RePEc:eee:chsofr:v:26:y:2005:i:3:p:867-879
    DOI: 10.1016/j.chaos.2005.01.041
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    Cited by:

    1. Ge, Zheng-Ming & Ou, Chan-Yi, 2008. "Chaos synchronization of fractional order modified duffing systems with parameters excited by a chaotic signal," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 705-717.
    2. Ge, Zheng-Ming & Yi, Chang-Xian, 2007. "Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 42-61.
    3. Xie, T.F. & Zhou, S.P., 2007. "On a class of fractal functions with graph Hausdorff dimension 2," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1625-1630.
    4. Ge, Zheng-Ming & Ou, Chan-Yi, 2007. "Chaos in a fractional order modified Duffing system," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 262-291.

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