We study the number of limit cycles of the discontinuous piecewise linear differential systems in ℝ2n with two zones separated by a hyperplane. Our main result shows that at most (8n - 6)n-1 limit cycles can bifurcate up to first-order expansion of the displacement function with respect to a small parameter. For proving this result, we use the averaging theory in a form where the differentiability of the system is not necessary.
In [Rong, F., Quasi-parabolic analytic transformations of C n , J. Math. Anal. Appl. 343 (2008), 99-109], we showed the existence of "parabolic curves" for certain quasiparabolic analytic transformations of C n . Under some extra assumptions, we show the existence of "parabolic manifolds" for such transformations.
We study holomorphic maps of C 2 tangent to the identity at a fixed point which have degenerate characteristic directions. With the help of some new invariants, we give sufficient conditions for the existence of attracting domains in these degenerate characteristic directions.
Abstract. We construct a class of bounded domains, on which the squeezing function is not uniformly bounded from below near a smooth and pseudoconvex boundary point.
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