Fractal Oscillators

Attilio Maccari


We consider a weakly nonlinear oscillator with a fractal forcing, given by the Weierstrass function, and use the asymptotic perturnation (AP) method to study its behavior. Being this function nowhere differentiable we can only use adequate approximations. We find that while in the linear case the resulting motion is a simple superposition between the fractal forcing and the standard oscillation, on the contrary in the nonlinear case the oscillator phase and its frequency also become fractal. We obtain the Poincarè sections in various cases and all theoretical findings are corroborated with numerical simulation .


fractals; nonlinear oscillator; perturbation method; Weierstrass function

Full Text:



M. Schroeder, Fractals, Chaos, Power Laws, W. H. Freeman, New York, (2000)

Lui Lam (editor), Introduction to Nonlinear Physics, Springer, New York , (1997).

B. Mandelbrot, The fractal geometry of Nature, W. H. Freeman, New York, (1982)

D. P. Feldman, Chaos and Fractals: An Elementary Introduction , Oxford University Press, (2012)

K. Falconer, Fractal Geometry:Mathematicall Foundations and Applications, John Wiley, New York (2014)

S.C. Lim and C.H. Eab, Riemann-Liouville and Weyl fractional oscillator processes, Phys. Lett. A 335 87-93, (2006).

R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

B.J. West, M. Bologna and P. Grigolini, Physics of Fractal Operators, Springer-Verlag, New York, (2003).

R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37, R161-R208, (2004).

G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2005

A. Maccari, Modulated motion and infinite-period bifurcation for two non-linearly coupled and parametrically excited van der Pol oscillators, International Journal of Non-Linear Mechanics, 36(3), 335-347, (2001)


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

ISSN: 2394-3688

© Science Front Publishers