Naslov
Towards bifurcations of complex dimensions

Autori
Lapidus, Michel L. ; Radunović, Goran ; Žubrinić, Darko

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Equadiff 2019: Book of Abstracts / - , 2019, 89-89

Skup
Equadiff 2019

Mjesto i datum
Liblice, Češka Republika, 08.07.2019. - 12.07.2019

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
bifurcations ; complex dimensions, fractal zeta function, fracctal set, Minkowski content

Sažetak
It is known that at the moment when a limit cycle is born from a weak focus in a Hopf-Takens bifurcation, the Minkowski dimension of any associated spiral trajectory jumps from trivial, i.e., 1 to nontrivial, i.e. a rational number of the form 4k/(2k+1) where the integer k is the multiplicity of the weak focus. For a given set, its complex dimensions are defined as the poles of the associated fractal distance zeta function and provide a far-reaching generalization of the classical notion of the Minkowski dimension. The higher-dimensional theory of complex dimensions has been developed in the recent extensive research monograph by the co-authors. One defines the order of a given complex dimension as the order of the pole of the associated fractal zeta function. We show on a geometric example of a fractal nest the effect of merging of two simple complex dimensions of order one into a single complex dimension of order 2. This is interesting since the fractal nest can be considered as a geometric simplification of a focus trajectory of a dynamical system. We conjecture that this effect of merging of several complex dimensions into a single one can give new insights into bifurcations of dynamical systems.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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