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Fractal oscillations of self-adjoint and damped linear differential equations of second-order. (English) Zbl 1244.34052

Authors’ abstract: For a prescribed real number \(s \in \) [1, 2), we give some sufficient conditions on the coefficients \(p(x)\) and \(q(x)\) such that every solution \(y = y(x), ~y \in C^{2}((0, T\)]) of the linear differential equation \[ (p(x)y^{\prime})^{\prime} + q(x)y = 0 \text{ on }(0, T] \] is bounded and fractal oscillatory near \(x = 0\) with the fractal dimension equal to \(s\). This means that \(y\) oscillates near \(x = 0\) and the fractal (box-counting) dimension of the graph \(\Gamma (y)\) of \(y\) is equal to \(s\) as well as the \(s\) dimensional upper Minkowski content (generalized length) of \(\Gamma (y)\) is finite and strictly positive. It verifies that \(y\) admits similar kind of the fractal geometric asymptotic behaviour near \(x = 0\) like the chirp function \(y_{ch}(x) = a(x)S(\varphi (x))\), which often occurs in the time – frequency analysis and its various applications. Furthermore, this kind of oscillations is established for the Bessel, chirp and other types of damped linear differential equations given in the form \[ y''+ (\mu /x)y^{\prime} + g(x)y = 0,~ x \in (0, T]. \] In order to prove the main results, we state a new criterion for fractal oscillations near \(x = 0\) of real continuous functions.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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[1] Pašić, M., Fractal oscillations for a class of second-order linear differential equations of Euler type, J. Math. Anal. Appl., 341, 211-223 (2008) · Zbl 1145.34022
[2] Tricot, C., Curves and Fractal Dimension (1995), Springer-Verlag: Springer-Verlag New York
[3] Borgnat, P.; Flandrin, P., On the chirp decomposition of Weierstrass-Mandelbrot functions, and their time-frequency interpretation, Appl. Comput. Harmon. Anal., 15, 134-146 (2003) · Zbl 1054.94002
[4] Candes, E. J.; Charlton, P. R.; Helgason, H., Detecting highly oscillatory signals by chirplet path pursuit, Appl. Comput. Harmon. Anal., 24, 14-40 (2008) · Zbl 1144.94003
[5] Jaffard, S.; Meyer, Y., Wavelet methods for pointwise regularity and local oscillations of functions, Mem. Amer. Math. Soc., 123, 1-110 (1996) · Zbl 0873.42019
[6] Meyer, Y.; Xu, H., Wavelet analysis and chirp, Appl. Comput. Harmon. Anal., 4, 366-379 (1997) · Zbl 0960.94006
[7] Ren, G.; Chen, Q.; Cerejeiras, P.; Kähler, U., Chirp transforms and chirp series, J. Math. Anal. Appl., 373, 2, 356-369 (2011) · Zbl 1200.42026
[8] Barlow, E.; Mulholland, A. J.; Nordon, A.; Gachagan, A., Theoretical analysis of chirp excitation of contrast agents, Phys. Procedia, 3, 743-747 (2009)
[9] Képesi, M.; Weruaga, L., Adaptive chirp-based time-frequency analysis of speech signals, Speech Commun., 48, 474-492 (2006)
[10] T. Paavle, M. Min, T. Parve, Using of chirp excitation for bioimpedance estimation: theoretical aspects and modeling, Proc. of the Baltic Electronic Conf. BEC2008, 2008, Tallinn, Estonia, pp. 325-328.; T. Paavle, M. Min, T. Parve, Using of chirp excitation for bioimpedance estimation: theoretical aspects and modeling, Proc. of the Baltic Electronic Conf. BEC2008, 2008, Tallinn, Estonia, pp. 325-328.
[11] Pedersen, M. H.; Misaridis, T. X.; Jensen, J. A., Clinical evaluation of chirp-coded excitation in medical ultrasound, Ultrasound Med. Biol., 29, 895-905 (2003)
[12] Weruaga, L.; Képesi, M., The fan-chirp transform for non-stationary harmonic signals, Signal Process., 87, 1504-1522 (2007) · Zbl 1186.94365
[13] Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations (1965), D.C. Heath and Co.: D.C. Heath and Co. Boston · Zbl 0154.09301
[14] Hartman, P., Ordinary Differential Equations (1982), Birkhauser: Birkhauser Boston, Basel, Stuttgart · Zbl 0125.32102
[15] Wong, J. S.W., On rectifiable oscillation of Euler type second order linear differential equations, E.J. Qualitative Theor. Differ. Equat., 20, 1-12 (2007) · Zbl 1182.34049
[16] Pašić, M.; Tanaka, S., Rectifiable oscillations of self-adjoint and damped linear differential equations of second-order, J. Math. Anal. Appl., 381, 27-42 (2011) · Zbl 1223.34047
[17] O’Regan, D., Existence Theory for Nonlinear Ordinary Differential Equations (1997), Kluwer · Zbl 1077.34505
[18] Agarwal, R. P.; Grace, S. R.; O’Regan, D., Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations (2002), Kluwer Academic Publishers.: Kluwer Academic Publishers. London · Zbl 1073.34002
[19] Kwong, M. K.; Pašić, M.; Wong, J. S.W., Rectifiable oscillations in second order linear differential equations, J. Differ. Equat., 245, 2333-2351 (2008) · Zbl 1168.34027
[20] Pašić, M., Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type, J. Math. Anal. Appl., 335, 724-738 (2007) · Zbl 1126.34023
[21] Evans, L. C.; Gariepy, R. F., Measure Theory and Fine Properties of Functions (1999), CRC Press: CRC Press New York · Zbl 0954.49024
[22] Falconer, K., Fractal Geometry. Fractal Geometry, Mathematical Fondations and Applications (1999), John Willey-Sons
[23] Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability (1995), Cambridge · Zbl 0819.28004
[24] Falconer, K. J., On the Minkowski measurability of fractals, Proc. Amer. Math. Soc., 123, 1115-1124 (1995) · Zbl 0838.28006
[25] He, C. Q.; Lapidus, M. L., Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function, Mem. Amer. Math. Soc., 127, 608 (1997)
[26] Lapidus, M.; van Frankenhuijsen, M., Fractal Geometry, Complex Dimensions and Zeta Functions, Geometry and Spectra of Fractal Strings. Fractal Geometry, Complex Dimensions and Zeta Functions, Geometry and Spectra of Fractal Strings, Monographs in Mathematics (2006), Springer: Springer New York · Zbl 1119.28005
[27] Pašić, M., Minkowski-Bouligand dimension of solutions of the one-dimensional p-Laplacian, J. Differ. Equat., 190, 268-305 (2003) · Zbl 1054.34034
[28] Pašić, M., Rectifiable and unrectifiable oscillations for a generalization of the Riemann-Weber version of Euler differential equations, Georgian Math. J., 15, 759-774 (2008) · Zbl 1172.34025
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