Investigating the possibility of applying techniques from linear systems theory to the setting of non-linear systems has been the focus of many papers. The pseudo-linear (PL) form representation of non-linear dynamical systems has led to the concept of non-linear eigenvalues (NEValues) and non-linear eigenvectors (NEVectors). When the NEVectors do not depend on the state vector of the system, then the NEValues determine the global qualitative behaviour of a non-linear system throughout the state space. The aim of this paper is to use this fact to construct a non-linear dynamical system of which the trajectories of the system show continual stretching and folding. We first prove that the system is globally bounded. Next we analyse the system numerically by studying bifurcations of equilibria and periodic orbits. Chaos arises due to a period doubling cascade of periodic attractors. Chaotic attractors are presumably of Hénon-like type, which means that they are the closure of the unstable manifold of a saddle periodic orbit. We also show how PL forms can be used to control the chaotic system and to synchronize two identical chaotic systems.

中文翻译：

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伪线性系统的混沌动力学

研究将线性系统理论中的技术应用于非线性系统设置的可能性是许多论文的重点。非线性动力系统的伪线性（PL）形式表示法已经提出了非线性特征值（NEValues）和非线性特征向量（NEVectors）的概念。当NEVector不依赖于系统的状态向量时，NEValue会确定整个状态空间中非线性系统的全局定性行为。本文的目的是利用这一事实来构建一个非线性动力系统，该系统的轨迹显示出连续的拉伸和折叠。我们首先证明系统是全球有界的。接下来，我们通过研究平衡点和周期轨道的分歧来对系统进行数值分析。混乱是由于周期性吸引子的级联倍增而引起的。混沌吸引子大概是Hénon类的，这意味着它们是鞍形周期性轨道不稳定流形的闭合。我们还展示了如何使用PL形式来控制混沌系统并使两个相同的混沌系统同步。