Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method

Abstract

When a dynamical system has a complex structure of fixed points, periodic cycles or even chaotic attractors, Cell Mapping methods are excellent tools to discover and thoroughly analyse all features in the state space. These methods discretize a region of the state space into cells and examine the dynamics in the cell state space. By determining one or more image cells for each cell, the global behaviour within the region can be quickly determined. In the simplest case -Simple Cell Mapping (SCM) method -only one image corresponds to a cell and usually a rectangular grid of cells is used. In typical applications the grid of cells is refined at specific locations. This paper, however, introduces a different approach, which is useful to expand the analysed state space region to include all features which properly characterize the global dynamics of the system. Instead of refining the initial cell state space, we start with a small initial state space region, analyse other interesting regions of the state space and incorporate them into a cluster of cell mapping solutions. By this approach, trajectories escaping the original state space region can be followed automatically and additional objects in the state space can be discovered. To illustrate the benefits of the method, we present the exploration of the phase-space of the micro-chaos map -a simple model of digitally controlled systems.Keywords: discretized state space, adaptive global analysis, algorithm, micro-chaos Highlights:• Extension of the Simple Cell Mapping method by adaptively expanding cell state space.• New state space regions can be attached to the cluster of SCM solutions.• Clustering makes parallel computation trivial.

The Hybrid Micro-chaos Map: Digitally Controlled Inverted Pendulum with Dry Friction

“…We present simple analytical formulas to determine if chaotic attractors are affected by the sticking introduced by the friction or not. Finally, we present numerical results obtained using Clustered Simple Cell Mapping [11] to support our theoretical results.…”

“…Chaotic attractors are usually covered with one or more high-period orbits. We have applied Clustered Simple Cell Mapping (C-SCM) [11] to Eq. (12) to determine whether the chaotic attractors disappear due to the dry friction or not.…”

Twofold quantization in digital control: deadzone crisis and switching line collision

“…Utilizing Clustered Simple Cell Mapping [9], it is possible to automatically extend the analysed state space region and also execute cell mapping in a parallel computing environment.…”

Nonlinear effects of saturation in the car-following model

“…Isola is a special, isolated and closed solution to nonlinear problems, which is detailed in the scientific literature [38][39][40] and appears in a variety of disciplines [41,42]. Detecting and finding an isola in a system is not straightforward and requires advanced analytical [43,44] or numerical techniques [45][46][47][48]. On this basis, it is substantial for engineers in practice to be aware of this phenomenon and to have knowledge about isola, and understand its nonlinear dynamics.…”