Geometry of planar quadratic systems

Perturbations of quadratic Hamiltonian two-saddle cycles

We prove that the number of limit cycles, which bifurcate from a two-saddle loop of a planar quadratic Hamiltonian system, under an arbitrary quadratic deformation, is less than or equal to three.

Quadratic vector fields with invariant algebraic curve of large degree

We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.

Planar Semi-quasi Homogeneous Polynomial differential systems with a given degree

This paper study the planar semi-quasi homogeneous polynomial differential systems (short for PSQHPDS), which can be regard as a generalization of semi-homogeneous and of quasi-homogeneous systems. By using the algebraic skills, several important properties of PSQHPDS are derived and are employed to establish an algorithm for obtaining all the explicit expressions of PSQHPDS with a given degree. Afterward, we apply this algorithm to research the center problem of quadratic and cubic PSQHPDS. It is proved that the quadratic one hasn't center, and, that the cubic one has center if and only if it…

Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle

This work studies combinatorics and geometry of the Yoccoz puzzle for quadratic polynomials. It is proven that the moduli of the ``principal nest'' of annuli grow at linear rate. As a corollary we obtain complex a priori bounds and local connectivity of the Julia set for many infinitely renormalizable quadratics.

分段平面二次微分系统中极限环的数量

On the number of limit cycles in piecewise planar quadratic differential systems

本文利用位移映射展开研究直线分区的分段平面二次系统，对 Loud 等时中心及非光滑中心进行一阶、二阶扰动分析，给出至少12个极限环的例子并纠正文献错误。

平面线性二次调节器梯度流中的有限时间边界碰撞

Finite-time boundary collision in planar linear quadratic regulator gradient flows

本文研究平面线性二次调节器策略梯度流能否在有限时间内到达稳定边界，给出充要条件及三角族的显式参数区域，并证明碰撞时稳定裕度和最小特征值线性消失。

On the geometry of bifurcation currents for quadratic rational maps

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive (1, 1)-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

辛平铺台球、平面连杆与双曲几何

Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry

本文结合辛台球与平铺台球提出辛平铺台球，证明其特殊情形下的周期轨道结果，并结合Thurston构造赋予平面等边多边形模空间双曲结构，得出六边形平面连杆位形空间为10尖点双曲3-流形的推论。

The impact of advection on the stability of stripes on lattices near planar Turing instabilities

Striped patterns are known to bifurcate in reaction-diffusion systems with differential isotropic diffusions at a supercritical Turing instability. In this paper we study the impact of weak anisotropy by directional advection on the stability of stripes with respect to various lattice modes, and the role of quadratic terms therein. We focus on the generic form of planar reaction-diffusion systems with two components near such a bifurcation. Using centre manifold reduction we derive a rigorous parameter expansion for the critical eigenvalues for lattice mode perturbations, specifically nearly s…

Width of homoclinic zone for quadratic maps

We study several families of planar quadratic diffeomorphisms near a Bogdanov-Takens bifurcation. For each family, the associate bifurcation diagram can be deduced from the interpolating flow. However, a zone of chaos confined between two lines of homoclinic bifurcation that are exponentially close to one-another is observed. The goal of this paper is to test numerically an accurate asymptotic expansion for the width of this chaotic zone for different families.

Combinatorics, geometry and attractors of quasi-quadratic maps

The Milnor problem on one-dimensional attractors is solved for S-unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set $ω(c)$ of a "non-renormalizable" map. It is proven that the scaling factors characterizing the geometry of this set go down to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynam…

可积连杆的几何

Geometry of Integrable Linkages

本文类比牵引车-挂车问题，定义带无滑动约束的2连杆问题，利用次黎曼几何求测地线，证明其可积并与平面细丝方程建立双重联系。

The impact of advection on large-wavelength stability of stripes near planar Turing instabilities

It is well known that for reaction-diffusion systems with differential isotropic diffusions, a Turing instability yields striped solutions. In this paper we study the impact of weak anisotropy by directional advection on such solutions, and the role of quadratic terms. We focus on the generic form of planar reaction-diffusion systems with two components near such a bifurcation. Using Lyapunov-Schmidt reduction and Floquet-Bloch decomposition we derive a rigorous parameter expansion for existence and stability against large wavelength perturbations. This provides detailed formulae for the loci …