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Plurifractal signature in the study of resonances of dynamical systems. (English) Zbl 0901.58038

Summary: This paper deals with the numerical investigation of the resonant zones of dynamical equations. Starting from the continuous differential model, the section of trajectories is obtained using resonant principles. Several techniques have been developed for measuring the stability of windings: the Liouville exponent \(\gamma\) of the Kolmogorov-Arnold-Moser theory; the Lipschitz-Hölder exponent \(\alpha\) introduced by Mandelbrot; and a frequency locking exponent \(\beta\) that is related to Fourier spectra. These techniques are implemented on typical models and some original results concerning the fractal structure of resonant regions are emphasized.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
37N99 Applications of dynamical systems
70K50 Bifurcations and instability for nonlinear problems in mechanics
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