Koopman Operator Theory for Nonautonomous and Stochastic Systems

  • Senka MaćešićEmail author
  • Nelida Črnjarić-Žic
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 484)


In practice, the dynamics of open systems subject to time-dependent or random forcing is much more present than the dynamics of autonomous systems. Therefore, extension of the Koopman operator theory and applications to such systems is of great importance. At the same time, it brings a new viewpoint to the existing theory of nonautonomous as well as random dynamical systems, particularly, with application of Koopman-based data-driven algorithms. In this chapter, we first review the nonautonomous Koopman operator family based on the two standard nonautonomous dynamical system definitions: skew product and process. Then, we state basic properties of the operator and compare performance of the DMD and Arnoldi-type algorithms in the context of both definitions. In the case of the random dynamical systems (RDS), we introduce the associated stochastic Koopman operator family. We show that when RDS is Markovian, this family satisfies semigroup property and we present some properties for RDS generated by the stochastic differential equations. Finally, we discuss data-driven algorithms in the stochastic framework and illustrate their performance on numerical examples.



This research has been supported by the DARPA Contract HR0011-16-C-0116 “On A Data-Driven, Operator-Theoretic Framework for Space–Time Analysis of Process Dynamics”. S.M. and N.C-Z. are grateful to Prof. Igor Mezić for helpful mathematical discussions and comments on the manuscript.


  1. 1.
    Arbabi, H., Mezić, I.: Ergodic theory, dynamic mode decomposition and computation of spectral properties of the Koopman operator. SIAM J. Appl. Dyn. Syst. 16, 2096–2126 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arnold, L., Kliemann, W.: Qualitative theory of stochastic systems, in Bharucha-Reid, A.T. (ed.) Probabilistic Analysis and Related Topics, vol. 3. Academic Press, New York (1983)Google Scholar
  3. 3.
    Arnold, L.: Stochastic Differential Equations: Theory and Applications. John Wiley Sons, Inc., Hoboken (1974)Google Scholar
  4. 4.
    Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)CrossRefGoogle Scholar
  5. 5.
    Caraballo, T., Han, X.: Applied Nonautonomous and Random Dynamical Systems. BCAM SpringerBriefs. Springer, Cham (2016)Google Scholar
  6. 6.
    Cohen, S.N., Elliot, R.J.: Stochastic Calcukus and Applications. Springer, New York (2015)CrossRefGoogle Scholar
  7. 7.
    Crauel, H.: Markov measures for random dynamical systems. Stoch. Stoch. Rep. 37(3), 153–173 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Črnjarić-Žic, N., Maćešić, S., Mezić, I.: Koopman operator spectrum for random dynamical systems (2017).
  9. 9.
    Drmač, Z., Mezić, I., Mohr, R.: Data driven Koopman spectral analysis in Vandermonde–Cauchy form via the DFT: numerical method and theoretical insights (2018).
  10. 10.
    Drmač, Z., Mezić, I., Mohr, R.: Data driven modal decompositions: analysis and enhancements. SIAM J. Sci. Comput. 40(4), A2253–A2285 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Engel, K.J., Nagel, R.: One-parameter semigroups for linear evolution operators. Springer, New York (2001)Google Scholar
  12. 12.
    Giannakis, D.: Data-driven spectral decomposition and forecasting of ergodic dynamical systems. Appl. Comput. Harmon. Anal. (2017). Scholar
  13. 13.
    Hemati, M.S., Rowley, C.W., Deem, E.A., Cattafesta, L.N.: De-biasing the dynamic mode decomposition for applied Koopman spectral analysis. Theor. Comp. Fluid. Dyn. 31, 349–368 (2017)CrossRefGoogle Scholar
  14. 14.
    Hollingsworth, B.J.: Stochastic Differential Equations: A Dynamical Systems Approach, Dissertation thesis (2008). Auburn University, AuburnGoogle Scholar
  15. 15.
    Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. Mathematical Surveys and Monographs, vol. 176. AMS, Providence (2011)Google Scholar
  16. 16.
    Klus, S., Koltai, P., Schütte, C.: On the numerical approximation of the Perron–Frobenius and Koopman operator. J. Comp. Dyn. 3(1), 51–79 (2016). Scholar
  17. 17.
    Korda, M., Mezić, I.: Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica 93, 149–160 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kutz, J.N., Fu, X., Brunton, S.L.: Multiresolution dynamic mode decomposition. SIAM J. Appl. Dyn. Syst. 15, 713–735 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise. Springer, Berlin (1994)CrossRefGoogle Scholar
  20. 20.
    Maćešić, S., Črnjarić-Žic, N., Mezić, I.: Koopman operator family spectrum for nonauonotomus systems. SIAM J. Appl. Dyn. Syst. 17(4), 2478–2515 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Metafune, G., Pallara, D., Priola, E.: Spectrum of Ornstein–Uhlenbeck operators in \(L^{p}\) spaces with respect to invariant measures. J. Funct. Anal. 196, 40–60 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynam. 41, 309–325 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Physica D 197, 101–133 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mezić, I., Surana, A.: Koopman mode decomposition for periodic/quasi-periodic time dependence. IFAC-PapersOnLine 49, 690–697 (2016). Scholar
  26. 26.
    Proctor, J.L., Brunton, S.L., Kutz, J.N.: Dynamic mode decomposition with control. SIAM J. Appl. Dyn. Syst. 15, 142–161 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Proctor, J.L., Brunton, S.L., Kutz, J.N.: Including inputs and control within equation-free architectures for complex systems. Eur. Phys. J. Special Topics 225, 2413–2434 (2016)CrossRefGoogle Scholar
  28. 28.
    Proctor, J.L., Brunton, S.L., Kutz, J.N.: Generalizing Koopman theory to allow for inputs and control. SIAM J. Appl. Dyn. Syst. 17(1), 909–930 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Shnitzer, T., Talmon, R., Slotine, J.J.: Manifold learning with contracting observers for data-driven time-series analysis. IEEE T. Signal Process. 65, 904–918 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Takeishi, N., Kawahara, Y., Yairi, T.: Subspace dynamic mode decomposition for stochastic Koopman analysis. Phys. Rev. E 96, 033–310 (2017)CrossRefGoogle Scholar
  31. 31.
    Tantet, A., Chekroun, M.D., Dijkstra, H.A., Neelin, J.D.: Mixing spectrum in reduced phase spaces of stochastic differential equations. Part II: Stochastic Hopf bifurcation (2017).
  32. 32.
    Williams, M.O., Hemati, M.S., Dawson T.M., Kevrekidis, I.G., Rowley, C.W.: Extending data-driven Koopman analysis to actuated systems. In: Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems, Monterey (2016)CrossRefGoogle Scholar
  33. 33.
    Williams, M.O., 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)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang, H., Rowley, C.W., Deem, E.A., Cattafesta, L.N.: Online dynamic mode decomposition for time-varying systems (2017).

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Authors and Affiliations

  1. 1.Faculty of EngineeringUniversity of RijekaRijekaCroatia

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