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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.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
37C45Dimension theory of dynamical systems
28A80Fractals
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References:
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