In some maps the existence of an attractor with a positive Lyapunov exponent can be proved by constructing a trapping region in phase space and an invariant expanding cone in tangent space. If this approach fails it may be possible to adapt the strategy by considering an induced map (a first return map for a well-chosen subset of phase space). In this paper we show that such a construction can be applied to the two-dimensional border-collision normal form (a continuous piecewise-linear map) if a certain set of conditions are satisfied and develop an algorithm for checking these conditions. The algorithm requires relatively few computations, so it is a more efficient method than, for example, estimating the Lyapunov exponent from a single orbit in terms of speed, numerical accuracy, and rigor. The algorithm is used to prove the existence of an attractor with a positive Lyapunov exponent numerically in an area of parameter space where the map has strong rotational characteristics and the consideration of an induced map is critical for the proof of robust chaos.

在一些映射中，具有正 Lyapunov 指数的吸引子的存在可以通过在相空间中构造一个俘获区域和在切空间中构造一个不变的扩展锥来证明。如果这种方法失败，则可以通过考虑诱导图（精心选择的相空间子集的第一个返回图）来调整策略。在本文中，我们表明，如果满足特定条件集，则可以将这种构造应用于二维边界碰撞法线形式（连续分段线性映射），并开发一种检查这些条件的算法。该算法需要相对较少的计算，因此它是一种比从单个轨道估计 Lyapunov 指数在速度、数值精度和严谨性方面更有效的方法。