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Zbl 1198.34052
Pašić, Mervan; Žubrinić, Darko; Županović, Vesna
Oscillatory and phase dimensions of solutions of some second-order differential equations.
(English)
[J] Bull. Sci. Math. 133, No. 8, 859-874 (2009). ISSN 0007-4497

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.
[Sergei Yu. Pilyugin (St. Petersburg)]
MSC 2000:
*34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
37C45 Dimension theory of dynamical systems
28A80 Fractals

Keywords: nonlinear differential equation; nonlinear oscillations; box dimension; chirp; spiral; Liénard equation; weakly damped oscillator

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