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Oscillatory and phase dimensions of solutions of some second-order differential equations. (English) Zbl 1198.34052
Let $x(t)$ be a continuous scalar function defined for $t\geq t_0$ for which there exists a sequence $t_k\to\infty$ such that $x(t_k)=0$ and $x(t)$ changes sign at any point $t_k$. The oscillatory dimension of $x(t)$ near $t=\infty$ is defined as the box dimension of the graph of $x(1/\tau)$ near $\tau=0$. If $x(t)$ is a scalar function of class $C^1$ such that the curve $\Gamma=\{(x(t),x'(t)),t\geq t_0\}$ is a spiral converging to the origin, then the phase dimension of $x(t)$ is the box dimension of $\Gamma$ near the origin. The authors calculate the oscillatory and phase dimensions for some functions of special type. The results are applied to solutions of some Liénard equations and weakly damped oscillators.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
37C45Dimension theory of dynamical systems
Full Text: DOI
[1] Caubergh, M.; Françoise, J. P.: Generalized Liénard equations, cyclicity and Hopf-Takens bifurcations, Qualitative theory dyn. Syst. 5, No. 2, 195-222 (2004) · Zbl 1131.34308 · doi:10.1007/BF02972678
[2] De Maesschalck, P.; Dumortier, F.: The period function of classical Liénard equations, J. differential equations 233, 380-403 (2007) · Zbl 1121.34046 · doi:10.1016/j.jde.2006.09.015
[3] Dumortier, F.; Llibre, J.; Artés, J. C.: Qualitative theory of planar differential systems, (2006) · Zbl 1110.34002
[4] Dumortier, F.; Panazzolo, D.; Roussarie, R.: More limit cycles than expected in Liénard equations, Proc. amer. Math. soc. 135, No. 6, 1895-1904 (2007) · Zbl 1130.34018 · doi:10.1090/S0002-9939-07-08688-1
[5] Elezović, N.; Županović, V.; Žubrinić, D.: Box dimension of trajectories of some discrete dynamical systems, Chaos solitons fract. 34, No. 2, 244-252 (2007) · Zbl 1133.37007 · doi:10.1016/j.chaos.2006.03.060
[6] Falconer, K.: Fractal geometry, (1990) · Zbl 0689.28003
[7] 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
[8] M.K. Kwong, M. Pašić, J.S.W. Wong, Rectifiable oscillations in second-order linear differential equations, preprint · Zbl 1184.34043 · doi:10.1007/s10231-008-0087-0
[9] Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, (1995) · Zbl 0819.28004
[10] Pašić, M.: Minkowski-bouligand dimension of solutions of the one-dimensional p-Laplacian, J. differential equations 190, 268-305 (2003) · Zbl 1054.34034 · doi:10.1016/S0022-0396(02)00149-3
[11] Pašić, M.: Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type, J. math. Anal. appl., 724-738 (2007) · Zbl 1126.34023 · doi:10.1016/j.jmaa.2007.01.099
[12] 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 · doi:10.1016/j.jmaa.2007.09.068
[13] M. Pašić, Rectifiable and unrectifiable oscillations for a generalization of the Riemann-Weber version of Euler differential equations, Georgian Math. J., in press · Zbl 1172.34025 · http://www.heldermann.de/GMJ/GMJ15/GMJ154/gmj15060.htm
[14] M. Pašić, J.S.W. Wong, Rectifiable oscillations in second-order half-linear differential equations, preprint · Zbl 1184.34043
[15] Roussarie, R.: Bifurcations of planar vector fields and Hilbert’s sixteenth problem, (1998) · Zbl 0898.58039
[16] Tricot, C.: Curves and fractal dimension, (1995) · Zbl 0847.28004
[17] Wong, J. S. W.: On rectifiable oscillation of Euler type second-order linear differential equations, E. J. Qualitative theory diff. Equ. 20, 1-12 (2007) · Zbl 1182.34049 · emis:journals/EJQTDE/2007/200720.html
[18] Wong, J. S. W.: On rectifiable oscillation of Emden-Fowler equations, Mem. differential equations math. Phys. 42, 127-144 (2007) · Zbl 1157.34027
[19] Žubrinić, D.: Analysis of Minkowski contents of fractal sets and applications, Real anal. Exchange 31, No. 2, 315-354 (2005/2006) · Zbl 1142.37315
[20] Žubrinić, D.; Županović, V.: Fractal analysis of spiral trajectories of some planar vector fields, Bull. sci. Math. 129, No. 6, 457-485 (2005) · Zbl 1076.37015 · doi:10.1016/j.bulsci.2004.11.007
[21] Žubrinić, D.; Županović, V.: Fractal analysis of spiral trajectories of some vector fields in R3, C. R. Acad. sci. Paris sér. I 342, No. 12, 959-963 (2006) · Zbl 1096.37010 · doi:10.1016/j.crma.2006.04.021
[22] Žubrinić, D.; Županović, V.: Box dimension of spiral trajectories of some vector fields in R3, Qualitative theory dyn. Syst. 6, No. 2, 203-222 (2005) · Zbl 1220.37034 · doi:10.1007/BF02972676
[23] D. Žubrinić, V. Županović, Poincaré map in fractal analysis of spiral trajectories of planar vector fields, Bull. Belg. Math. Soc. Simon Stevin, in press · Zbl 1153.37011 · euclid:bbms/1228486418
[24] Županović, V.; Žubrinić, D.: Fractal dimensions in dynamics, Encyclopedia of mathematical physics, vol. 2 2, 394-402 (2006)