Quantification of chaos in a time series generated from a traffic flow model using the extended entropic chaos degree

Abstract

Several Traffic flow models have been proposed to address the underlying causes of traffic congestion and develop effective methods for its suppression. Typically, these models are represented using differential or difference equations. Traffic flow models can exhibit chaotic behaviors. Traffic situations become unstable if chaos prevails in the traffic flow. Therefore, the conditions under which chaos occurs in the traffic flow and the intensity of chaos at that time must be investigated. The Lyapunov exponent is used for quantifying the chaos in the dynamical systems. However, computing the Lyapunov exponent over time series without a dynamical map is challenging. Recently, the extended entropic chaos degree has been introduced as an extension of the entropic chaos degree. The extended entropic chaos degree can be computed directly for any time series. Analytically, the extended entropic chaos degree equals the sum of all the Lyapunov exponents for multidimensional non-periodic maps. Moreover, the extended entropic chaos degree is mathematically shown to equal the sum of one positive and one negative Lyapunov exponents for two-dimensional typical chaotic maps, such as a generalized Baker’s map, and a standard map.In this study, we utilize the extended entropic chaos degree to quantify chaos in a time series with less computing complexity compared with conventional methods. First, we demonstrate that chaos can be quantified in a time series generated from a traffic flow model using an extended entropic chaos degree. Lyapunov exponent for a time series of time traveling between two adjacent traffic lights in the traffic flow model is not directly computable because its dynamics are not given as an explicit differential map. Second, we demonstrate that the computational complexity of the extended entropic chaos degree only corresponds to O(M). However, the computational complexities of the estimated Lyapunov exponents using the Wolf method are O(M2), where M is the number of points on a one-dimensional time series. Thus, we show that the extended chaos degree can quantify the chaos in the traffic flow model, such as one vehicle moving through a sequence of traffic lights in one dimension, with less complexity, where the chaos occurs depending on the timing of signal switching.