The cell mapping methods were originated by Hsu in 1980s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states, periodic motions, and chaotic attractors. The cell mapping methods have been applied to deterministic, stochastic, and fuzzy dynamical systems. Two important extensions of the cell mapping method have been developed to improve the accuracy of the solutions obtained in the cell state space: the interpolated cell mapping (ICM) and the set-oriented method with subdivision technique. For a long time, the cell mapping methods have been applied to dynamical systems with low dimension until now. With the advent of cheap dynamic memory and massively parallel computing technologies, such as the graphical processing units (GPUs), global analysis of moderate- to high-dimensional nonlinear dynamical systems becomes feasible. This paper presents a parallel cell mapping method for global analysis of nonlinear dynamical systems. The simple cell mapping (SCM) and generalized cell mapping (GCM) are implemented in a hybrid manner. The solution process starts with a coarse cell partition to obtain a covering set of the steady-state responses, followed by the subdivision technique to enhance the accuracy of the steady-state responses. When the cells are small enough, no further subdivision is necessary. We propose to treat the solutions obtained by the cell mapping method on a sufficiently fine grid as a database, which provides a basis for the ICM to generate the pointwise approximation of the solutions without additional numerical integrations of differential equations. A modified global analysis of nonlinear systems with transient states is developed by taking advantage of parallel computing without subdivision. To validate the parallelized cell mapping techniques and to demonstrate the effectiveness of the proposed method, a low-dimensional dynamical system governed by implicit mappings is first presented, followed by the global analysis of a three-dimensional plasma model and a six-dimensional Lorenz system. For the six-dimensional example, an error analysis of the ICM is conducted with the Hausdorff distance as a metric.
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November 2015
Research-Article
Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems1
Carlos Hernández,
Carlos Hernández
Computer Science Department,
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: chernandez@computacion.cs.cinvestav.mx
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: chernandez@computacion.cs.cinvestav.mx
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Jesús Fernandez,
Jesús Fernandez
Computer Science Department,
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: jfernandez@computacion.cs.cinvestav.mx
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: jfernandez@computacion.cs.cinvestav.mx
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Jian-Qiao Sun
Jian-Qiao Sun
Professor School of Engineering,
University of California at Merced,
Merced, CA 95343
e-mail: jqsun@ucmerced.edu
University of California at Merced,
Merced, CA 95343
e-mail: jqsun@ucmerced.edu
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Fu-Rui Xiong
Zhi-Chang Qin
Qian Ding
Carlos Hernández
Computer Science Department,
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: chernandez@computacion.cs.cinvestav.mx
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: chernandez@computacion.cs.cinvestav.mx
Jesús Fernandez
Computer Science Department,
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: jfernandez@computacion.cs.cinvestav.mx
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: jfernandez@computacion.cs.cinvestav.mx
Oliver Schütze
Jian-Qiao Sun
Professor School of Engineering,
University of California at Merced,
Merced, CA 95343
e-mail: jqsun@ucmerced.edu
University of California at Merced,
Merced, CA 95343
e-mail: jqsun@ucmerced.edu
2Corresponding author.
3Honorary Professor of Tianjin University, China.
Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 3, 2015; final manuscript received July 18, 2015; published online August 10, 2015. Assoc. Editor: Alexander F. Vakakis.
J. Appl. Mech. Nov 2015, 82(11): 111010 (12 pages)
Published Online: August 10, 2015
Article history
Received:
March 3, 2015
Revision Received:
July 18, 2015
Citation
Xiong, F., Qin, Z., Ding, Q., Hernández, C., Fernandez, J., Schütze, O., and Sun, J. (August 10, 2015). "Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems." ASME. J. Appl. Mech. November 2015; 82(11): 111010. https://doi.org/10.1115/1.4031149
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