In this paper, we develop the hexagon method and the dodecagon method to estimate the Hausdorff measure of the Sierpiński gasket and show that the Hausdorff measure of the Sierpiński gasket is upper-bounded by a single-variable continuous function. Better upper bounds of the Hausdorff measure of the Sierpiński gasket are also achieved.

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For third-order nilpotent critical points of a planar dynamical system, the analytic center problem is completely solved in this article by using the integrating factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic-order systems under small perturbations, sufficient and necessary conditions for an analytic center are obtained.

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In this paper, an interesting bifurcation phenomenon is investigated -a 3-multiple nilpotent focus of the planar dynamical systems could be broken into two element focuses and an element saddle, and the limit cycles could bifurcate out from two element focuses. As an example, a class of cubic systems with 3-multiple nilpotent focus O(0, 0) is investigated, we prove that nine limit cycles with the scheme 7 ⊃ (1 ∪ 1) could bifurcate out from the origin when the origin is a weak focus of order 8. At the end of this paper, the double bifurcations of a class of Z 2 equivalent cubic system with 3-multiple nilpotent focus or center O(0, 0) are investigated.