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International Journal of Differential Equations
Volume 2013 (2013), Article ID 857410, 11 pages
Research Article

Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations

1Department of Applied Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
2Okayama University of Science, Okayama 700-0005, Japan

Received 7 December 2012; Accepted 8 January 2013

Academic Editor: Norio Yoshida

Copyright © 2013 Mervan Pašić and Satoshi Tanaka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We derive some simple sufficient conditions on the amplitude , the phase and the instantaneous frequency such that the so-called chirp function is fractal oscillatory near a point , where and is a periodic function on . It means that oscillates near , and its graph is a fractal curve in such that its box-counting dimension equals a prescribed real number and the -dimensional upper and lower Minkowski contents of are strictly positive and finite. It numerically determines the order of concentration of oscillations of near . Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions.