Degenerate elliptic resonances

Quasi-periodic solutions for differential equations with an elliptic-type degenerate equilibrium point under small perturbations

This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol's oscillator with zero-Hopf singularity.

Simplified normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps

We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic two-parameter families of area-preserving maps. We also derive a simplified normal form for a generic two-parametric unfolding. The normal forms are used to analyse bifurcations of $n$-periodic orbits. In particular, for $n\ge6$ we find regions of parameters where the normal form has "meandering'' invariant curves.

On bifurcations in degenerate resonance zones

For Hamitonian systems with 3/2 degrees of freedom close to nonlinear integrable and for symplectic maps of the cylinder, bifurcations in degenerate resonance zones are discussed.

非退化紧凸超曲面在R^6中的三个椭圆闭特征

Three elliptic closed characteristics on the non-degenerate compact convex hypersurfaces in R^6

本文证明了在R^{2n}中任何具有有限多个闭特征的紧凸超曲面至少存在两个椭圆闭特征，并进一步证明非退化超曲面中至少存在三个椭圆闭特征，其中至少两个为无理椭圆特征，为该猜想提供了理论支持。