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Strange attractors and wandering domains near a homoclinic cycle to a bifocus
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.jde.2020.02.027 Alexandre A. P. Rodrigues
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.jde.2020.02.027 Alexandre A. P. Rodrigues
In this paper, we explore the hyperchaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes $(\Lambda_N)_{N\in \NN}$ arises near the cycle, with one expanding and two contracting directions. We extend previous results on the field and we show that, when the cycle is broken, there are parameters for which the first return map to a given cross section exhibits homoclinic tangencies associated to a dissipative saddle periodic point. These tangencies can be slightly modified in order to satisfy the Tatjer conditions for a generalized tangency of codimension two. This configuration may be seen the organizing center, by which one can obtain Bogdanov-Takens bifurcations and therefore, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. The existence of a homoclinic cycle associated to a bifocus may be considered as a criterion for four-dimensional flows to be $C^1$-approximated by other flows exhibiting strange attractors and non-trivial contracting wandering domains.
中文翻译:
同宿周期附近的奇怪吸引子和游荡域到双焦点
在本文中,我们探索了同宿循环附近的超混沌集到矢量场具有负发散的双曲双焦点。如果双焦点的不变流形满足非简并条件,则在循环附近出现一系列双曲悬挂马蹄形$(\Lambda_N)_{N\in \NN}$,具有一个扩展和两个收缩方向。我们扩展了先前在该领域的结果,我们表明,当循环被打破时,对于给定横截面的第一个返回图显示出与耗散鞍周期点相关的同宿相切的参数。这些相切可以稍作修改,以满足二元广义相切的 Tatjer 条件。这个配置可以看一下组织中心,通过它可以得到 Bogdanov-Takens 分岔,因此,奇怪的吸引子,无限多的汇和非平凡的收缩游荡域。与双焦点相关的同宿循环的存在可以被认为是四维流被 $C^1$ 逼近的标准,其他流表现出奇怪的吸引子和非平凡的收缩游荡域。
更新日期:2020-08-01
中文翻译:
同宿周期附近的奇怪吸引子和游荡域到双焦点
在本文中,我们探索了同宿循环附近的超混沌集到矢量场具有负发散的双曲双焦点。如果双焦点的不变流形满足非简并条件,则在循环附近出现一系列双曲悬挂马蹄形$(\Lambda_N)_{N\in \NN}$,具有一个扩展和两个收缩方向。我们扩展了先前在该领域的结果,我们表明,当循环被打破时,对于给定横截面的第一个返回图显示出与耗散鞍周期点相关的同宿相切的参数。这些相切可以稍作修改,以满足二元广义相切的 Tatjer 条件。这个配置可以看一下组织中心,通过它可以得到 Bogdanov-Takens 分岔,因此,奇怪的吸引子,无限多的汇和非平凡的收缩游荡域。与双焦点相关的同宿循环的存在可以被认为是四维流被 $C^1$ 逼近的标准,其他流表现出奇怪的吸引子和非平凡的收缩游荡域。