This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit. The essential assumptions are hyperbolicity of the vertices and transversality of parameters. Using the result, conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven. Finally, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered.
Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies
Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Graduate School of Business Administration, Hitotsubashi University（一桥大学经营研究科研究生院）；Department of Mathematics, Imperial College London（帝国理工学院数学系）
Homoclinic tangencies to resonant saddles and discrete Lorenz attractors
研究了一类二次同宿切线到鞍不动点的三参数一般开折中周期轨道的分支。我们应用重新标度技术的第一次返回（庞加莱）的地图，并表明，重新标度的地图可以带来一个地图渐近接近的三维Henon地图$\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$，如已知的，表现出野生双曲洛伦兹吸引子在一些开放的参数域。在此基础上，我们证明了类Lorenz吸引子的无穷级联的存在性。
Generic three-parameter unfolding of a completely degenerate germ of a vector field on $\mathbb{R}^3$
University of Kinshasa（金沙萨大学）；Center for Research on Mathematics Education in the Democratic Republic of the Congo（刚果民主共和国数学教育研究中心）；University Institute of Technology, University of Ngaoundere（恩冈代雷大学理工学院）；Institut des Hautes Études Scientifiques, Université Paris-Saclay（法国高等科学研究所，巴黎萨克雷大学）
