Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs
Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs
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DOI：
10.1063/5.0044731
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发表时间：
2021-04-01
期刊：
影响因子：
2.9
通讯作者：
Belykh, Igor V.
中科院分区：
文献类型：
--
作者：
Belykh, Vladimir N.;Barabash, Nikita V.;Belykh, Igor V.
非平滑系统可以生成与平滑同行截然不同的动力学和分叉。在本文中，我们研究了Belykh等人最近引入的分段平滑分析分析中的同层分叉。 [混乱29，103108（2019）]。通过严格的分析，我们证明了滑动运动的出现导致新的分叉场景，在这种情况下，鞍座不稳定的同层轨道的分叉可以产生稳定的极限周期。这些分叉与它们的平滑类似物形成鲜明对比，这些类似物只能产生不稳定的（马鞍）动力学。我们构建了一张庞科返回图，该图是滑动运动的存在，从而严格地表征了滑动的同层分叉，从而破坏了混乱的洛伦兹型吸引子。特别是，我们得出了一个明确的缩放系数，用于与滑动多环形轨道轨道和准吸引者的形成相关的周期倍增。我们的分析结果为非平滑流动系统中非古典全球分叉理论的发展奠定了基础。
Non-smooth systems can generate dynamics and bifurcations that are drastically different from their smooth counterparts. In this paper, we study such homoclinic bifurcations in a piecewise-smooth analytically tractable Lorenz-type system that was recently introduced by Belykh et al. [Chaos 29, 103108 (2019)]. Through a rigorous analysis, we demonstrate that the emergence of sliding motions leads to novel bifurcation scenarios in which bifurcations of unstable homoclinic orbits of a saddle can yield stable limit cycles. These bifurcations are in sharp contrast with their smooth analogs that can generate only unstable (saddle) dynamics. We construct a Poincare return map that accounts for the presence of sliding motions, thereby rigorously characterizing sliding homoclinic bifurcations that destroy a chaotic Lorenz-type attractor. In particular, we derive an explicit scaling factor for period-doubling bifurcations associated with sliding multi-loop homoclinic orbits and the formation of a quasi-attractor. Our analytical results lay the foundation for the development of non-classical global bifurcation theory in non-smooth flow systems.