engleski

The box dimension of a class of degenerate foci

We study a class of polynomial planar systems with singularity of focus type without characteristic directions. A simple and natural transformation of a weak focus has been used to obtain such degenerate focus. We compute the box dimension of a spiral trajectory and show the connection to cyclicity, of the system $$ \begin{;array};{;ccl}; \dot x&=&-y^{;2n-1};\pm x^n y^{;n-1}; (x^{;2n};+y^{;2n};)^k\\ \dot y&=&\phantom{;-};x^{;2n-1};\pm x^{;n-1}; y^n (x^{;2n};+y^{;2n};)^k, \end{;array}; $$ under a perturbation, where $k, n\in\mathbb{;N};$. This is a joint work with Darko \v Zubrini\'c and Vesna \v Zupanovi\'c.

box dimension ; spiral ; weak focus ; degenerate focus

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