Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations. (28th February 2013)

• Record Type: Journal Article
• Authors: Pašić, Mervan; Tanaka, Satoshi
• Other Names: Yoshida Norio Academic Editor.
• Is Part Of: International journal of differential equations. Volume 2013(2013)
• Publication Date: 2013-02-28
• Subjects: Differential equations -- Periodicals; Differential equations; Periodicals; 515.35
• DOI: 10.1155/2013/857410
• Languages: English
• ISSNs: 1687-9643

Abstract

Abstract : We derive some simple sufficient conditions on the amplitudea ( x ), the phaseφ ( x ), and the instantaneous frequencyω ( x ) such that the so-called chirp functiony ( x ) = a ( x ) S ( φ ( x ) ) is fractal oscillatory near a pointx = x 0, whereφ ′ ( x ) = ω ( x ) andS = S ( t ) is a periodic function onℝ . It means thaty ( x ) oscillates nearx = x 0, and its graphΓ ( y ) is a fractal curve inℝ 2 such that its box-counting dimension equals a prescribed real numbers ∈ [ 1, 2 ) and thes -dimensional upper and lower Minkowski contents ofΓ ( y ) are strictly positive and finite. It numerically determines the order of concentration of oscillations ofy ( x ) nearx = x 0 . 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.