The transition to turbulence in many shear flows proceeds along two competing routes, one linked with finite-amplitude disturbances and the other one originating from a linear instability, as in, e.g., boundary layer flows. The dynamical systems concept of an edge manifold has been suggested in the subcritical case to explain the partition of the state space of the system. This investigation is devoted to the evolution of the edge manifold when linear stability is added in such subcritical systems, a situation poorly studied despite its prevalence in realistic fluid flows. In particular, the fate of the edge state as a mediator of transition is unclear. A deterministic three-dimensional model is suggested, parametrized by the linear instability growth rate. The edge manifold evolves topologically, via a global saddle-loop bifurcation of the underlying invariant sets, from the separatrix between two attraction basins to the mediator between two transition routes. For larger instability rates, the stable manifold of the saddle point increases in codimension from 1 to 2 after an additional local pitchfork node bifurcation, causing the collapse of the edge manifold. As the growth rate is increased, three different regimes of this model are identified, each one associated with a flow case from the recent hydrodynamic literature. A simple nonautonomous generalization of the model is also suggested in order to capture the complexity of spatially developing flows.

中文翻译：

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当两条过渡路线竞争时，对边缘的塌陷进行建模

在许多剪切流中，湍流的过渡沿两条相互竞争的路径进行，一条与有限幅值扰动相关，另一条源自线性不稳定性，例如边界层流。在次临界情况下，提出了边缘歧管的动力学系统概念，以解释系统状态空间的划分。这项研究致力于在这种亚临界系统中增加线性稳定性时边缘歧管的演变，尽管这种情况在现实的流体流动中普遍存在，但研究情况却很少。尤其是，边缘状态作为过渡媒介的命运还不清楚。提出了确定性的三维模型，其参数由线性不稳定性增长率决定。边缘流形在拓扑上演化，通过基础不变集的全局鞍环分叉，从两个吸引盆之间的分界线到两个过渡路径之间的介体。对于较大的不稳定性率，在附加局部干草叉节点分叉之后，鞍点的稳定歧管的维数从1增加到2，从而导致边缘歧管崩溃。随着增长率的增加，确定了该模型的三种不同状态，每种状态都与最近的水动力文献中的流动情况有关。还建议对模型进行简单的非自治概括，以捕获空间上发展的流程的复杂性。附加的局部干草叉节点分叉后，鞍点的稳定歧管的维数从1增加到2，从而导致边缘歧管崩溃。随着增长率的提高，确定了该模型的三种不同状态，每种状态都与最近的水动力文献中的流动情况有关。还建议对模型进行简单的非自治概括，以捕获空间上发展的流程的复杂性。附加局部干草叉节点分叉后，鞍点的稳定歧管的维数从1增加到2，从而导致边缘歧管崩溃。随着增长率的提高，确定了该模型的三种不同状态，每种状态都与最近的水动力文献中的流动情况有关。还建议对模型进行简单的非自治概括，以捕获空间上发展的流程的复杂性。