engleski

Fractal properties of a class of polynomial planar systems having degenerate foci

We study a class of polynomial planar systems with singularity of degenerate focus type without characteristic directions. This class is obtained using a natural transformation of a class of systems having weak foci, which is related to the normal form for the Hopf-Takens bifurcation. The class is given by $$ \begin{;array};{;ccl}; \dot x&=&-y^{;2n-1};\pm x^n y^{;n-1}; (x^{;2n};+y^{;2n};)^k\\ \dot y&=&\phantom{;-};x^{;2n-1};\pm x^{;n-1}; y^n (x^{;2n};+y^{;2n};)^k, \end{;array}; $$ where parameters $k, n\in\mathbb{;N};$. For this class we compute the box dimension of any spiral trajectory $\Gamma$, $$ \dim_B\Gamma = 2\left(1-\frac{;1};{;2nk+1};\right) $$ and show the connection to cyclicity under a perturbation. This work is a continuation of the previous work done by Darko \v Zubrini\'c and Vesna \v Zupanovi\'c, regarding fractal analysis of spiral trajectories of planar vector fields.

polynomial planar systems ; degenerate focus ; box dimension ; spiral

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