Poincaré map in fractal analysis of spiral trajectories of planar vector fields. (English) Zbl 1153.37011

Žubrinić, Darko; Županović, Vesna

Bull. Belg. Math. Soc. - Simon Stevin 15, No. 5, 947-960 (2008).

Summary: We study the box dimension and Minkowski content of spiral trajectories of planar vector fields, using information about the asymptotic behaviour of iterates of the Poincaré map. An auxilliary tool is a flow-sector theorem near the weak focus, which is of a similar nature as the well known flow-box theorem. Applications include Hopf bifurcation and Liénard systems. We obtain all possible values of box dimensions of spiral trajectories around weak focus, associated with polynomial vector fields.

MSC:

37C45	Dimension theory of smooth dynamical systems
37G10	Bifurcations of singular points in dynamical systems
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations

Keywords:

Poincaré map; spiral; box dimension; flow-sector; Hopf bifurcation; Liénard system