Advertisement

On sample-based computations of invariant sets

  • Shen Zeng
Original Paper
  • 23 Downloads

Abstract

In this paper, the classical problem of uncovering the maximal invariant set of a (discrete-time) dynamical system is illuminated from a novel perspective, which in particular leads to a novel sample-based computational procedure to compute the invariant set. The mathematical description of these new insights can be formulated in strikingly basic set-theoretic terms, and more importantly, be efficiently realized computationally in terms of different sample-based implementations. We illustrate the simplicity and efficiency of the computational method on three examples with a maximal invariant set that is unstable in both time directions, the classical Hénon map, a three-dimensional analogue of the Hénon map, and a Van der Pol oscillator.

Keywords

Discrete-time dynamical systems Invariant set Sample-based techniques 

Notes

Compliance with ethical standards

Conflicts of interest

The author declares that he has no conflict of interest.

References

  1. 1.
    Dellnitz, M., Froyland, G., Junge, O.: The algorithms behind GAIO—set oriented numerical methods for dynamical systems. In: Fiedler, B. (ed.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, Berlin, Heidelberg (2001)Google Scholar
  2. 2.
    Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75(3), 293–317 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dellnitz, M., Junge, O.: On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36(2), 491–515 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Oliver, J., Ioannis, G.K.: On the sighting of unicorns: a variational approach to computing invariant sets in dynamical systems. Chaos Interdiscip. J. Nonlinear Sci. 27(6), 063102 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Arbabi, H., Mezic, I.: Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator. SIAM J. Appl. Dyn. Syst. 16(4), 2096–2126 (2017)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Steven, L.B., Bingni, W.B., Joshua, L.P., Eurika, K., Kutz, J.N.: Chaos as an intermittently forced linear system. Nat. Commun. 8(1), 19 (2017)CrossRefGoogle Scholar
  7. 7.
    Budišić, M., Mohr, R., Mezić, I.: Applied Koopmanism. Chaos Interdiscip. J. Nonlinear Sci. 22(4), 047510 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1–3), 309–325 (2005)MathSciNetMATHGoogle Scholar
  9. 9.
    Samuel, H.R., Steven, L.B., Joshua, L.P., Kutz, J.N.: Data-driven discovery of partial differential equations. Sci. Adv. 3(4), e1602614 (2017)CrossRefGoogle Scholar
  10. 10.
    Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Matthew, O.W., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307–1346 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, vol. 54. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  13. 13.
    Hénon, M.: A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50(1), 69–77 (1976)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Electrical and Systems EngineeringWashington UniversitySt. LouisUSA

Personalised recommendations