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Research Papers

Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems1

[+] Author and Article Information
Fu-Rui Xiong

Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: xfr90311@gmail.com

Zhi-Chang Qin

Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: qinzhichang123@126.com

Qian Ding

Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: qding@tju.edu.cn

Carlos Hernández

Computer Science Department,
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

Oliver Schütze

Computer Science Department,
CINVESTAV-IPN,
Mexico City 07360, Mexico
e-mail: schuetze@cs.cinvestav.mx

Jian-Qiao Sun

Professor School of Engineering,
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 82(11), 111010 (Aug 10, 2015) (12 pages) Paper No: JAM-15-1119; doi: 10.1115/1.4031149 History: Received March 03, 2015

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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Figures

Grahic Jump Location
Fig. 1

Global invariant set was found of the impact model. The subdivision processes in four different cellular space resolutions are presented to show the improvement of solution accuracy. Initial cell space partition is 7 × 7. Final cell space resolution reaches 189 × 189 with 3935 cells found as solutions. Computational time is 64.9844 s for sequential computing and 4.5573 s for parallel computing. (For the interpretation of color in all the figures, the reader is referred to the web version of the paper.)

Grahic Jump Location
Fig. 2

Global properties of the impact model solved by the modified GCM analysis flow. Cell space partition is 189 × 189. Blue cells are the chaotic attractor, black are the unstable manifold, and red are the domain of attraction. Note the attractor and unstable manifold coincide with the invariant set shown in Fig. 1. Sequential computing takes 263.1347 s, while parallel computing takes 29.9928 s.

Grahic Jump Location
Fig. 3

The attractors of the plasma model described in Eq. (31) in the cell space with 30 × 30 × 30 resolution. The four attractors occupy a quarter of the state space each and are symmetric with respect to the xz and yz plane. The CPU time of sequential computing for this example is 1418.9 s, while the parallel GCM analysis on GPU only takes 28.93 s. The parallel computing accelerates the computing by 49 times.

Grahic Jump Location
Fig. 4

The domain of attraction of the first PG of the plasma model. Red cells are the attractor while blue ones are the corresponding domain of attraction. The cells in the domain occupy a quarter of the 3D state space.

Grahic Jump Location
Fig. 5

Number of cells to be processed at each iteration of rolling cut subdivision of one dimension at a time. We chose to stop the subdivision when the number starts to decrease at the 16th iteration.

Grahic Jump Location
Fig. 6

The attractor of the 6D Lorenz system obtained by the SCM–GCM hybrid method and post-processing with interpolation. Blue dots are the central points of the cells representing the attractor projected to the 3D space. Red dots are the interpolated points. The familiar butterfly shape can be seen from the low-dimensional projection.

Grahic Jump Location
Fig. 7

Two-dimensional projections of the attractor of the 6D Lorenz system. Blue dots are the centers of the cells in the invariant set. Red dots showing the fine structure of the attractor are generated with interpolation.

Grahic Jump Location
Fig. 8

The relationship between the number of sampled points and the Hausdorff distance of the invariant set of the 6D Lorenz system is obtained by the interpolation scheme to the reference set. The interpolation scheme appears to be insensitive to the number of sampled points.

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