×

zbMATH — the first resource for mathematics

Clustered simple cell mapping: an extension to the simple cell mapping method. (English) Zbl 07257115
Summary: 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.
MSC:
68 Computer science
92 Biology and other natural sciences
Software:
Dynamics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hsu, C., Cell-to-cell mapping: a method of global analysis for nonlinear systems. Cell-to-cell mapping: a method of global analysis for nonlinear systems, Applied mathematical sciences, 64 (1987), Springer: Springer Singapore
[2] Xiong, F.-R.; Qin, Z.-C.; Xue, Y.; Schtze, O.; Ding, Q.; Sun, J.-Q., Multi-objective optimal design of feedback controls for dynamical systems with hybrid simple cell mapping algorithm, Commun Nonlinear Sci Numer Simul, 19, 5, 1465-1473 (2014)
[3] Zou, H.; Xu, J., Improved generalized cell mapping for global analysis of dynamical systems, Sci China Ser E, 52, 3, 787-800 (2009) · Zbl 1186.37097
[4] de Kraker, B.; van der Spek, J. A.W.; van Campen, D. H., Extensions of cell mapping for discontinuous systems, (Wiercigroch, M.; de Kraker, B., Applied nonlinear dynamics and chaos of mechanical systems with discontinuities (2000), World Scientific), 61-102
[5] Klages, R., Deterministic diffusion in one-dimensional chaotic dynamical systems (1996), TU Berlin
[6] Xiong, F.-R.; Schtze, O.; Ding, Q.; Sun, J.-Q., Finding zeros of nonlinear functions using the hybrid parallel cell mapping method, Commun Nonlinear Sci Numer Simul, 34, 23-37 (2016)
[7] Csernák, G.; Stépán, G., Digital control as source of chaotic behavior, Int J Bifurcations Chaos, 5, 20, 1365-1378 (2010) · Zbl 1193.37041
[8] Csernák, G.; Gyebrószki, G.; Stépán, G., Multi-baker map as a model of digital PD control, Int J Bifurcations Chaos, 26, 2, 1650023-1-1650023-11 (2016) · Zbl 1334.34142
[9] Nusse, H. E.; Yorke, J. A., Basins of attraction, Science, 271, 1376-1380 (1996) · Zbl 1226.37009
[10] Aguirre, J.; Viana, R. L.; Sanjun, M. A.F., Fractal structures in nonlinear dynamics, Rev Mod Phys, 81, 333-386 (2009)
[11] Nusse, H. E.; Yorke, J. A., Dynamics: numerical explorations. Dynamics: numerical explorations, Applied Mathematical Sciences, 101 (1998), Springer-Verlag: Springer-Verlag New York
[12] van der Spek, J. A.W., Cell mapping methods: modifications and extensions (1994), Eindhoven university of technology, Eindhoven
[13] Haller, G.; Stépán, G., Micro-chaos in digital control, J Nonlinear Sci, 6, 415-448 (1996) · Zbl 0863.93050
[14] Csernák, G.; Stépán, G., Sampling and round-off, as sources of chaos in PD-controlled systems, Proc 19th Mediterr Conf Control Autom (2011)
[15] Gyebrószki, G.; Csernák, G., Methods for the quick analysis of micro-chaos, (Awrejcewicz, J., Applied Non-Linear Dynamical Systems (2014), Springer International Publishing), 383-395
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.