Advertisement

Detecting Singularities Using the PowerSeries Library

  • Mahsa KazemiEmail author
  • Marc Moreno Maza
Conference paper
  • 62 Downloads
Part of the Communications in Computer and Information Science book series (CCIS, volume 1125)

Abstract

Local bifurcation analysis of singular smooth maps plays a fundamental role in understanding the dynamics of real world problems. This analysis is accomplished in two steps: first performing the Lyapunov-Schmidt reduction to reduce the dimension of the state variables in the original smooth map and then applying singularity theory techniques to the resulting reduced smooth map. In this paper, we address an important application of the so-called Extended Hensel Construction (EHC) for computing the aforementioned reduced smooth map, which, consequently, leads to detecting the type of singularities of the original smooth map. Our approach is illustrated via two examples displaying pitchfork and winged cusp bifurcations.

Keywords

Singularities Smooth maps Lyapunov-Schmidt reduction Extended Hensel Construction PowerSeries library 

References

  1. 1.
    Alvandi, P., Ataei, M., Kazemi, M., Moreno Maza, M.: On the extended hensel construction and its application to the computation of limit points. J. Symb. Comput. (2019, to appear)Google Scholar
  2. 2.
    Alvandi, P., Kazemi, M., Moreno Maza, M.: Computing limits of real multivariate rational functions. In: Proceedings of ISSAC 2016, pp. 39–46. ACM, New York (2016)Google Scholar
  3. 3.
    Alvandi, P., Ataei, M., Moreno Maza, M.: On the extended Hensel construction and its application to the computation of limit points. In: Proceedings of ISSAC 2017, pp. 13–20. ACM, New York (2017)Google Scholar
  4. 4.
    Gazor, M., Kazemi, M.: Symbolic local bifurcation analysis of scalar smooth maps. ArXiv:1507.06168 (2016)
  5. 5.
    Gazor, M., Kazemi, M.: A user guide for singularity. ArXiv:1601.00268 (2017)
  6. 6.
    Gazor, M., Kazemi, M.: Normal form analysis of \(\mathbb{Z}_2\)-equivariant singularities. Int. J. Bifurcat. Chaos 29(2), 1950015-1–1950015-20 (2019)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Golubitsky, M., Keyfitz, B.L.: A qualitative study of the steady-state solutions for a continuous flow stirred tank chemical reactor. SIAM J. Math. Anal. 11(2), 316–339 (1980)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Golubitsky, M., Stewart, I., Schaeffer, D. G.: Singularities and Groups in Bifurcation Theory, vol. 1–2. Springer, New York (1985 and 1988).  https://doi.org/10.1007/978-1-4612-5034-0CrossRefGoogle Scholar
  9. 9.
    Hamel, P., et al.: Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers. Nat. Photonics 9, 311–315 (2015)CrossRefGoogle Scholar
  10. 10.
    Labouriau, I.: Applications of singularity theory to neurobiology. Ph.D. thesis, Warwick University, (1984)Google Scholar
  11. 11.
    Rossi, J., Carretero-González, R., Kevrekidis, P.G., Haragus, M.: On the spontaneous time-reversal symmetry breaking in synchronously-pumped passive Kerr resonators. J. Phys. A: Math. Theor. 49(45), 455201–455221 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sasaki, T., Kako, F.: Solving multivariate algebraic equation by Hensel construction. Jpn. J. Ind. Appl. Math. 16, 257–285 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Uppal, A., Ray, W.H., Poore, A.B.: The classification of the dynamic behavior of continuous stirred tank reactors-influence of reactor residence time. J. Chem. Eng. Sci. 31(3), 205–214 (1976)CrossRefGoogle Scholar
  14. 14.
    Zeldovich, Y.V., Zisin, U.A.: On the theory of thermal stress. Flow in an exothermic stirred reactor, II. Study of heat loss in a flow reactor. J. Tech. Phys. 11(6), 501–508 (1941). (Russian)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Western OntarioLondonCanada

Personalised recommendations