Introduction to Scientific Computing Technologies for Global Analysis of Multidimensional Nonlinear Dynamical Systems

  • Nemanja AndonovskiEmail author
  • Franco Moglie
  • Stefano Lenci
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 69)


To determine global behaviour of a dynamical system, one must find invariant sets (attractors) and their respective basins of attraction. Since this cannot be made extensively with analytical methods, the numerical global analysis is currently the subject of intensive research, especially for strongly nonlinear, multidimensional dynamical systems. Numerical analysis in dimensions higher than four present a challenge, since it requires significant computing resources. Numerical methods used in global analysis that can benefit from high-power computing are those that can parallelize either data or task elaboration on a large scale. Mass parallelization comes with large number of difficulties, restrictions and programming hazards. When not implemented in compliance with hardware organization, data and instruction management can lead to severe loss of parallel algorithm performance. Systematic and methodical approach to design parallel programs is, therefore, critical to get the most from expensive high-power computing systems and to avoid unrealistic speed-up expectation. Considering these difficulties, the goal of this chapter is to introduce readers to the world of high-power computing systems for science and global analysis of strongly nonlinear, multidimensional dynamical systems. Topic covered are classification and performance of hardware and software, classes of computing problems and methodical design of programs. Two major hardware platforms used for scientific computing, clusters and systems with computational GPU are considered. Functionality of widely utilized software solutions (OpenMP, MPI, CUDA and OpenCL) for high-power computing systems is described. Performance of individual computer components are addressed so that the reader can understand advantages, disadvantages, efficiency and limits of each hardware platform. With this knowledge users can judge if their computation problem is suitable for mass parallelization. If this is the case, which hardware and software platforms to use. To avoid many traps of parallel programming, one of the methodical design approaches is covered. Topic is closed with example applications in science and global analysis.



NA and SL would like to thank Radu Serban and Dan Negrut, University of Wisconsin-Madison, USA, for help with HPC and for the kind hospitality during the visit of NA.


  1. 1.
    Hsu, C.S.: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. Springer, New York (1987)CrossRefGoogle Scholar
  2. 2.
    Sun, J.-Q., Luo, A.C.J. (eds.): Global Analysis of Nonlinear Dynamics. Springer, New York (2012)Google Scholar
  3. 3.
    Category: History of computing hardware—Wikipedia, the Free Encyclopedia. Accessed 15 Nov 2017Google Scholar
  4. 4.
    Disrupting the datacenter: Qualcomm CentriqTM 2400 processor. Accessed 15 Jan 2018Google Scholar
  5. 5.
    Null, L., Lobur, Julia: The Essentials of Computer Organization and Architecture. Jones and Bartlett Publishers, Sudbury, Mass (2003)Google Scholar
  6. 6.
    Hilborn, R.C.: Chaos and nonlinear dynamics: an introduction for scientists and engineers, 2nd edn. Oxford University Press, Oxford (2000)Google Scholar
  7. 7.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering. Mass, Wokingham, Addison-Wesley Pub, Reading (1994)Google Scholar
  8. 8.
    Jain, M.K., Iyengar, S.R.K., Jain, R.K.: Numerical Methods for Scientific and Engineering Computation. Wiley Eastern Ltd., New Delhi etc. (1986)Google Scholar
  9. 9.
    Aguirre, J., Viana, R.L., Sanjuán, M.A.F.: Fractal structures in nonlinear dynamics. Rev. Mod. Phys. 81, 333–386 (2009)Google Scholar
  10. 10.
    Engelina Nusse, H., Hunt, B.R., John Kostelich, E., Yorke, J.A.: Dynamics: numerical explorations. In: Applied Mathematical Sciences, 2nd edn. Springer, New York (1998)Google Scholar
  11. 11.
    Comer, D.: Essentials of Computer Architecture, 2nd edn. Chapman and Hall CRC (2017)Google Scholar
  12. 12.
    Elahi, A.: Computer Systems: Digital Design. Fundamentals of Computer Architecture and Assembly Language. Springer, Cham (2018)CrossRefGoogle Scholar
  13. 13.
    Czarnul, Pawel: Parallel Programming for Modern High Performance Computing Systems. CRC Press, Taylor and Francis Group (2018)CrossRefGoogle Scholar
  14. 14.
    Aubanel, E.: Elements of Parallel Computing. Chapman and Hall CRC (2016)Google Scholar
  15. 15.
    Hwang, K.: Advanced Computer Architecture: Parallelism, Scalability, Programmability. McGraw-Hill Series in Computer Engineering. McGraw-Hill, New York (1993)Google Scholar
  16. 16.
    Clements, A.: Principles of Computer Hardware, 4th edn. Oxford University Press, Oxford, New York (2006)Google Scholar
  17. 17.
    Padua, D. (ed.): Encyclopedia of Parallel Computing. Springer US (2011)Google Scholar
  18. 18.
    Rauber, T., Rnger, G.: Parallel Programming for Multicore and Cluster Systems, 2nd edn. Springer, Berlin, Heidelberg (2013)Google Scholar
  19. 19.
    Jiao, Y., Lin, H., Balaji, P., Feng, W.: Power and performance characterization of computational kernels on the gpu. In: Green Computing and Communications (GreenCom), 2010 IEEE/ACM International Conference on Cyber, Physical and Social Computing (CPSCom), pp. 221–228, Dec 2010Google Scholar
  20. 20.
    Hennessy, J.L., Patterson, D.A., Asanovi, K.: Computer architecture: a quantitative approach, 5th edn. Morgan Kaufmann/Elsevier, Amsterdam, Boston (2012)Google Scholar
  21. 21.
    The Top500 List—November 2017. Accessed 11 Feb 2018Google Scholar
  22. 22.
    The Green500 List—November 2017. Accessed 11 Feb 2018Google Scholar
  23. 23.
    CUDA zone. Accessed 15 Jan 2018Google Scholar
  24. 24.
    Sourouri, M., Langguth, J., Spiga, F., Baden, S.B., Cai, X.: Cpu+gpu programming of stencil computations for resource-efficient use of gpu clusters. In: 2015 IEEE 18th International Conference on Computational Science and Engineering, pp. 17–26, Oct 2015Google Scholar
  25. 25.
    Chakrabarti, S., Demmel, J., Yelick, K.: Modeling the benefits of mixed data and task parallelism. In: Proceedings of the Seventh Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA ’95, pp. 74–83. ACM, New York, NY, USA (1995)Google Scholar
  26. 26.
    Raicu, I., Foster, I.T., Zhao, Y.: Many-task computing for grids and supercomputers. In: 2008 Workshop on Many-Task Computing on Grids and Supercomputers, pp. 1–11, Nov 2008Google Scholar
  27. 27.
    Foster, Ian: Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering. Addison-Wesley Longman Publishing Co., Inc, Boston, MA, USA (1995)zbMATHGoogle Scholar
  28. 28.
    Grama, A.: Introduction to Parallel Computing, 2nd edn. Addison-Wesley, Harlow, England, New York (2003)Google Scholar
  29. 29.
    Solihin, Y.: Fundamentals of Parallel Multicore Architecture, 1st edn. Chapman-Hall, CRC (2015)CrossRefGoogle Scholar
  30. 30.
    Shi, Y.: Reevaluating Amdahl’s law and Gustafson’s law (1996)Google Scholar
  31. 31.
    Sun, X.-H., Ni, L.M.: Institute for computer applications in science, and engineering. In: Scalable Problems and Memory-Bounded Speedup. ICASE Report. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Va. (1992)Google Scholar
  32. 32.
    Pllana, S, Xhafa, F. (eds.) Programming Multi-core and Many-core Computing Systems, 1st edn. Wiley Publishing (2014)Google Scholar
  33. 33.
    Stallings, W., Paul, G.: Operating Systems: Internals and Design Principles, 7th international edn. Pearson, Boston, Mass, London (2012)Google Scholar
  34. 34.
    The OpenMP API specification for parallel programming. Accessed 15 Jan 2018Google Scholar
  35. 35.
    Message Passing Interface (MPI) Forum. Accessed 15 Jan 2018Google Scholar
  36. 36.
    The OpenCLTM specification. Accessed 15 Jan 2018Google Scholar
  37. 37.
    Cineca SCAI application software for science. Accessed 01 June 2018Google Scholar
  38. 38.
    Medio, A., Lines, M.: Nonlinear Dynamics: A Primer. Cambridge University Press (2001)Google Scholar
  39. 39.
    Tongue, B.-H., Gu, K.: Interpolated cell mapping of dynamical systems. J. Appl. Mech 55(2), 461–466 (1988)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Ge, Z.-M., Lee, S.-C.: A modified interpolated cell mapping method. J. Sound Vib. 199(2), 189–206 (1997)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Spek, J.A.W., van der, D.H. Campen, V., Kraker de, A.: Cell mapping for multi degrees of freedom systems. In: Bajaj, A.K. (ed.) Nonlinear and Stochastic Dynamics: Presented at 1994 International Mechanical Engineering Congress and Exhibition, Nov 6–11, 1994, pp. 151–159. Chicago, Illinois, AMD, ASME (1994)Google Scholar
  42. 42.
    Eason, R.P., Dick, A.J.: A parallelized multi-degrees-of-freedom cell mapping method. Nonlinear Dyn. 77(3), 467–479 (2014)CrossRefGoogle Scholar
  43. 43.
    Belardinelli, P., Lenci, S.: A first parallel programming approach in basins of attraction computation. Int. J. Non-Linear Mech. Dyn. Stab. Control Flexible Struct. 80, 76–81 (2016)Google Scholar
  44. 44.
    Belardinelli, P., Lenci, S.: An efficient parallel implementation of cell mapping methods for mdof systems. Nonlinear Dyn. 86(4), 2279–2290 (2016)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Belardinelli, P., Lenci, S.: Improving the global analysis of mechanical systems via parallel computation of basins of attraction. In: Procedia IUTAM, IUTAM Symposium on Nonlinear and Delayed Dynamics of Mechatronic Systems, vol. 22, pp. 192–199 (2017)Google Scholar
  46. 46.
    Fernndez, J., Schtze, O., Hernndez, C., Sun, J.-Q., Xiong, Fu-Rui: Parallel simple cell mapping for multi-objective optimization. Eng. Opt. 48(11), 1845–1868 (2016)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Xiong, F., Qin, Z.-C., Ding, Q., Castellanos, C.H., Fernandez, J., Schetze, O., Sun, J.Q.: Parallel cell mapping method for global analysis of high-dimensional nonlinear dynamical systems, vol. 82 (2015)Google Scholar
  48. 48.
    Battelino, P.M., Grebogi, C., Ott, E., Yorke, J.A., Yorke, E.D.: Multiple coexisting attractors, basin boundaries and basic sets. Phys. D 32(2), 296–305 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Nemanja Andonovski
    • 1
    Email author
  • Franco Moglie
    • 1
  • Stefano Lenci
    • 1
  1. 1.Polytechnic University of MarcheAnconaItaly

Personalised recommendations