Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations
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...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2013-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2013/857410 |
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| _version_ | 1850159504094658560 |
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| author | Mervan Pašić Satoshi Tanaka |
| author_facet | Mervan Pašić Satoshi Tanaka |
| author_sort | Mervan Pašić |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b53a1bcdaec94d69b1c88b60892a4d67 |
| institution | OA Journals |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-b53a1bcdaec94d69b1c88b60892a4d672025-08-20T02:23:31ZengWileyInternational Journal of Differential Equations1687-96431687-96512013-01-01201310.1155/2013/857410857410Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential EquationsMervan Pašić0Satoshi Tanaka1Department of Applied Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, CroatiaOkayama University of Science, Okayama 700-0005, JapanWe 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.http://dx.doi.org/10.1155/2013/857410 |
| spellingShingle | Mervan Pašić Satoshi Tanaka Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations International Journal of Differential Equations |
| title | Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations |
| title_full | Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations |
| title_fullStr | Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations |
| title_full_unstemmed | Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations |
| title_short | Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations |
| title_sort | fractal oscillations of chirp functions and applications to second order linear differential equations |
| url | http://dx.doi.org/10.1155/2013/857410 |
| work_keys_str_mv | AT mervanpasic fractaloscillationsofchirpfunctionsandapplicationstosecondorderlineardifferentialequations AT satoshitanaka fractaloscillationsofchirpfunctionsandapplicationstosecondorderlineardifferentialequations |