Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049-3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vectors expand and so no invariant set can have only negative Lyapunov exponents. Under additional assumptions we characterise an attractor as the closure of the unstable manifold of a fixed point and prove that it satisfies Devaney's definition of chaos.

中文翻译：

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使用不变流形和扩张锥的鲁棒混沌的一种建设性方法

二维边界碰撞法线形式（分段线性映射）的混沌吸引子可以在参数空间的所有开放区域中持续存在。在某些参数机制中已严格建立了这种鲁棒的混乱状态。在这里，我们提供了原始参数体制中鲁棒混沌的正式结果。Banerjee，JA Yorke，C.Grebogi，Robust Chaos，Phys。莱特牧师80（14）：3049-3052，1998]。我们首先在相空间中构造一个陷阱区域，以证明拓扑吸引子的存在。然后，我们在切线空间中构造一个不变的扩张锥，以证明切线向量是扩张的，因此没有任何不变集只能具有负Lyapunov指数。在其他假设下，我们将吸引子的特征定为不动点的不稳定流形的闭合，并证明其满足Devaney对混沌的定义。