Systems without equilibria with chaotic flows have been the focus of recent works. Since there are no equilibria to start a local analysis the study of this class of systems is still a challenging task. Some systems have already been found by means of numerical searches that they consider different classes of three-dimensional nonlinear ordinary differential equations, usually with quadratic nonlinearities. Some works present construction approaches for the generation of chaotic attractors via piecewise linear systems (PWL) without equilibria in ℝ³. There are few works on the generation of chaotic attractors through systems without equilibria with differentiable nonlinearities or mechanisms to generate higher dimensional systems with chaotic or hyperchaotic attractors. Here we report a class of systems without equilibria which exhibit a scroll attractor and whose vector field is differentiable. The system construction presents great flexibility for the selection of the number of scrolls exhibited by the attractor. We also report a special coupling for this class of systems which allows the coupling without introducing new equilibria in the system. The coupling is illustrated with a nine-dimensional system which was numerically studied through Lyapunov exponents, Kaplan-Yorke dimension and Poincaré maps. The proposed coupling approach presents flexibility that can be further studied.

中文翻译：

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通过无平衡系统的耦合来实现超混沌吸引子

没有平衡的具有混沌流动的系统一直是最近研究的焦点。由于没有平衡点可以进行局部分析，因此研究此类系统仍然是一项艰巨的任务。通过数值搜索已经找到了一些系统，它们考虑了不同类别的三维非线性常微分方程，通常具有二次非线性。一些适用于混沌吸引的产生通过分段线性系统（PWL），而不在均衡本施工方法ℝ ³。通过没有平衡的具有可区分的非线性或系统来生成具有混沌或超混沌吸引子的高维系统的系统，生成混沌吸引子的工作很少。在这里，我们报告一类没有平衡的系统，它们表现出涡旋吸引子并且其矢量场是可微的。该系统构造为选择吸引器所展示的涡卷数量提供了极大的灵活性。我们还报告了此类系统的特殊耦合，该耦合允许在不引入系统新平衡的情况下进行耦合。通过一个九维系统对耦合进行了说明，该系统通过Lyapunov指数，Kaplan-Yorke维和Poincaré映射进行了数值研究。所提出的耦合方法具有灵活性，可以进一步研究。