### Abstract

The stabilization of nonlinear systems depend strongly on the initial state and the parameters of the systems. The initial state and the parameters with which the system is stabilized can be distinguished by the geometrical structure. It is, however, difficult and sometimes impossible to analyze the structure analytically. Therefore it comes important to show and analyze the structure of the parameters and initial states numerically and visually. In this paper, we present a method to draw and visualize such region and structure in the three dimensional space. In general, the projection of the original high-dimensional space to the lower dimension one is required for using visual analysis. Thus, it is convenient that the viewpoint can be moved, without time loss, in the direction where analyst would like to see. As often as the viewpoint moves, the recomputation as quick as possible is required to realize the quick motion of viewpoint. It is, however, obvious that lots of computation and time are taken to draw the region. Therefore, high performance calculators are needed to realize the real-time drawing. In order to overcome this problem, FPGA and cluster-computing is used in this paper. Then it is demonstrated by illustrative examples that FPGA and cluster-computing shows high performance to draw the region of the parameters and initial state in 3D with which z
_{n+1} = z
^{2}
_{n} + C can be stabilized, that is Mandelbrot and Julia sets, respectively.

Original language | English |
---|---|

Title of host publication | Proceedings of the Third Workshop - 2005 IEEE Intelligent Data Acquisition and Advanced Computing Systems |

Subtitle of host publication | Technology and Applications, IDAACS 2005 |

Pages | 588-592 |

Number of pages | 5 |

DOIs | |

Publication status | Published - Dec 1 2007 |

Externally published | Yes |

Event | 3rd IEEE Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, IDAACS 2005 - Sofia, Bulgaria Duration: Sep 5 2005 → Sep 7 2005 |

### Other

Other | 3rd IEEE Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, IDAACS 2005 |
---|---|

Country | Bulgaria |

City | Sofia |

Period | 9/5/05 → 9/7/05 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Computer Vision and Pattern Recognition
- Theoretical Computer Science

### Cite this

*Proceedings of the Third Workshop - 2005 IEEE Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, IDAACS 2005*(pp. 588-592). [4062203] https://doi.org/10.1109/IDAACS.2005.283052

**Visualization of stability of dynamical systems by 3D graphics supported by cluster computing.** / Funasaka, Takashi; Iwase, Masami; Fujisawa, Katsuki; Hatakeyama, Shoshiro.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Third Workshop - 2005 IEEE Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, IDAACS 2005.*, 4062203, pp. 588-592, 3rd IEEE Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, IDAACS 2005, Sofia, Bulgaria, 9/5/05. https://doi.org/10.1109/IDAACS.2005.283052

}

TY - GEN

T1 - Visualization of stability of dynamical systems by 3D graphics supported by cluster computing

AU - Funasaka, Takashi

AU - Iwase, Masami

AU - Fujisawa, Katsuki

AU - Hatakeyama, Shoshiro

PY - 2007/12/1

Y1 - 2007/12/1

N2 - The stabilization of nonlinear systems depend strongly on the initial state and the parameters of the systems. The initial state and the parameters with which the system is stabilized can be distinguished by the geometrical structure. It is, however, difficult and sometimes impossible to analyze the structure analytically. Therefore it comes important to show and analyze the structure of the parameters and initial states numerically and visually. In this paper, we present a method to draw and visualize such region and structure in the three dimensional space. In general, the projection of the original high-dimensional space to the lower dimension one is required for using visual analysis. Thus, it is convenient that the viewpoint can be moved, without time loss, in the direction where analyst would like to see. As often as the viewpoint moves, the recomputation as quick as possible is required to realize the quick motion of viewpoint. It is, however, obvious that lots of computation and time are taken to draw the region. Therefore, high performance calculators are needed to realize the real-time drawing. In order to overcome this problem, FPGA and cluster-computing is used in this paper. Then it is demonstrated by illustrative examples that FPGA and cluster-computing shows high performance to draw the region of the parameters and initial state in 3D with which z n+1 = z 2 n + C can be stabilized, that is Mandelbrot and Julia sets, respectively.

AB - The stabilization of nonlinear systems depend strongly on the initial state and the parameters of the systems. The initial state and the parameters with which the system is stabilized can be distinguished by the geometrical structure. It is, however, difficult and sometimes impossible to analyze the structure analytically. Therefore it comes important to show and analyze the structure of the parameters and initial states numerically and visually. In this paper, we present a method to draw and visualize such region and structure in the three dimensional space. In general, the projection of the original high-dimensional space to the lower dimension one is required for using visual analysis. Thus, it is convenient that the viewpoint can be moved, without time loss, in the direction where analyst would like to see. As often as the viewpoint moves, the recomputation as quick as possible is required to realize the quick motion of viewpoint. It is, however, obvious that lots of computation and time are taken to draw the region. Therefore, high performance calculators are needed to realize the real-time drawing. In order to overcome this problem, FPGA and cluster-computing is used in this paper. Then it is demonstrated by illustrative examples that FPGA and cluster-computing shows high performance to draw the region of the parameters and initial state in 3D with which z n+1 = z 2 n + C can be stabilized, that is Mandelbrot and Julia sets, respectively.

UR - http://www.scopus.com/inward/record.url?scp=43549111401&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43549111401&partnerID=8YFLogxK

U2 - 10.1109/IDAACS.2005.283052

DO - 10.1109/IDAACS.2005.283052

M3 - Conference contribution

SN - 0780394461

SN - 9780780394469

SP - 588

EP - 592

BT - Proceedings of the Third Workshop - 2005 IEEE Intelligent Data Acquisition and Advanced Computing Systems

ER -