The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.

中文翻译：

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一种受李雅普诺夫指数启发的用于表征状态转移网络动力学的新方法

网络科学、非线性动力学和时间序列分析的结合为复杂系统的不同方法之间提供了新颖的见解和类比。通过结合动力系统的李雅普诺夫指数和马尔可夫链的转移概率的平均熵背后的考虑，我们引入了一种网络度量来表征状态转移网络的动力学，特别关注区分混沌和循环模式。这个李雅普诺夫测度的一个重要特性是它对动力学循环性的非单调依赖。通过为研究新措施提供适当的用例，我们还提出了一种通过相空间粗粒度将时间序列映射到状态转换网络的方法。使用离散时间和连续时间动态系统，从相应的状态转移网络中提取的李雅普诺夫测度表现出与李雅普诺夫指数相似的行为。此外，它对边界危机表现出很强的敏感性，这表明它适用于预测混乱的崩溃。