Koopman Operator Spectrum for Random Dynamical Systems

  • Nelida Črnjarić-ŽicEmail author
  • Senka Maćešić
  • Igor Mezić


In this paper, we consider the Koopman operator associated with the discrete and the continuous-time random dynamical system (RDS). We provide results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operator associated with different types of linear RDS. Then we consider the RDS for which the associated Koopman operator family is a semigroup, especially those for which the generator can be determined. We define a stochastic Hankel–DMD algorithm for numerical approximations of the spectral objects (eigenvalues, eigenfunctions) of the stochastic Koopman operator and prove its convergence. We apply the methodology to a variety of examples, revealing objects in spectral expansions of the stochastic Koopman operator and enabling model reduction.


Stochastic Koopman operator Random dynamical systems Stochastic differential equations Dynamic mode decomposition 

Mathematics Subject Classification

37H10 47B33 37M99 65P99 



This research has been supported by the DARPA Contract HR0011-16-C-0116 and HR0011-18-9-0033, AFOSR Grants FA9550-08-1-0217 and FA9550-17-C-0012, and support of scientific research of the University of Rijeka, Project No. 18-118-1257. We are thankful to Milan Korda, Allan Avila and anonymous referees for useful comments that helped to substantially improve the manuscript.


  1. Arató, M.: A famous nonlinear stochastic equation: Lotka–Volterra model with diffusion. Math. Comput. Model. 38, 709–726 (2003)MathSciNetzbMATHGoogle Scholar
  2. Arbabi, H., Mezić, 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)MathSciNetzbMATHGoogle Scholar
  3. Arnold, L.: Stochastic Differential Equations. Wiley, New York (1974)zbMATHGoogle Scholar
  4. Arnold, L.: Random Dynamical Systems. Springer, New York (1998)zbMATHGoogle Scholar
  5. Arnold, L., Crauel, H.: Random dynamical systems. Lypunov exponents. In: Proceedings Oberwolfach 1990. Springer Lecture notes in Mathematics, vol. 1486, pp. 1–22 (1991)Google Scholar
  6. Bagheri, S.: Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596–623 (2013)MathSciNetzbMATHGoogle Scholar
  7. Budišić, M., Mohr, R., Mezić, I.: Applied Koopmanism. Chaos 22(4), 596–623 (2012)MathSciNetzbMATHGoogle Scholar
  8. Cohen, S.N., Elliott, R.J.: Stochastic Calculus and Applications, 2nd edn. Birkhäuser, New York (2015)zbMATHGoogle Scholar
  9. Crauel, H.: Markov measures for random dynamical systems. Stoch. Stoch. Rep. 37(3), 153–173 (1991)MathSciNetzbMATHGoogle Scholar
  10. Da Prato, G.J.Z.: Stochastic Equations in Infinite Dimensions, 2nd edn. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  11. Drmač, Z., Mezić, I., Mohr, R.: Data driven modal decompositions: analysis and enhancements. SIAM J. Sci. Comput. 40(4), A2253–A2285 (2018)MathSciNetzbMATHGoogle Scholar
  12. Dynkin, E.B.: Markov Processes, 1, 2, Grundlehren der Mathematischen Wissenschaften, vol. 121, 122. Springer, New York (1965)Google Scholar
  13. Engel, K., Nagel, R.: One-Parameter Semigroups for Linear Evolution Operators. Springer, New York (2001)zbMATHGoogle Scholar
  14. Fantuzzi, G., Goluskin, D., Huang, D., Chernyshenko, S.: Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization. SIAM J. Appl. Dyn. Syst. 15(4), 1962–1988 (2016) MathSciNetzbMATHGoogle Scholar
  15. Gaspard, P., Nicolis, G., Provata, A., Tasaki, S.: Spectral signature of the pitchfork bifurcation: Liouville equation approach. Phys. Rev. E 51(1), 74 (1995)MathSciNetGoogle Scholar
  16. Giannakis, D.: Data-driven spectral decomposition and forecasting of ergodic dynamical systems. Appl. Comput. Harmon. Anal. 47(2), 338–396 (2019)MathSciNetGoogle Scholar
  17. Hemati, M.S., Rowley, C.W., Deem, E.A., Cattafesta, L.N.: De-biasing the dynamic mode decomposition for applied koopman spectral analysis. Theor. Comput. Fluid. Dyn. 31(4), 349–368 (2017)Google Scholar
  18. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  19. Hutzenthaler, M., Jentzen, A.: Numerical Approximations of Stochastic Differential Equations with Non-globally Lipschitz Continuous Coefficients. Memoirs of the American Mathematical Society, vol. 236, no. 1112 (2015)Google Scholar
  20. Junge, O., Marsden, J.E., Mezić, I.: Uncertainty in the dynamics of conservative maps. In: 43rd IEEE Conference on Decision and Control (2004)Google Scholar
  21. Klus, S., Koltai, P., Schütte, C.: On the numerical approximation of the Perron–Frobenius and Koopman operator. J. Comput. Dyn. 3(1), 51–79 (2016)MathSciNetzbMATHGoogle Scholar
  22. Klus, S., Nüske, F., Koltai, P., Wu, H., Kevrekidis, I., Schütte, C., Noé, F.: Data-driven model reduction and transfer operator approximation. J. Nonlinear Sci. 28(3), 985–1010 (2018)MathSciNetzbMATHGoogle Scholar
  23. Klus, S., Schütte, C.: Towards tensor-based methods for the numerical approximation of the Perron–Frobenius and Koopman operator. J. Comput. Dyn. 3(2), 139–161 (2016)MathSciNetzbMATHGoogle Scholar
  24. Koopman, B.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. USA 17(5), 315 (1931)zbMATHGoogle Scholar
  25. Korda, M., Mezić, I.: On convergence of extended dynamic mode decomposition to the Koopman operator. J. Nonlinear Sci. 28(2), 687–710 (2018)MathSciNetzbMATHGoogle Scholar
  26. Lan, Y., Mezić, I.: Linearization in the large of nonlinear systems and Koopman operator spectrum. Physica D 242(1), 42–53 (2013). MathSciNetzbMATHGoogle Scholar
  27. Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, 2nd edn. Springer, New York (1994)zbMATHGoogle Scholar
  28. Leung, H.: Stochastic transient of a noisy van der Pol oscillator. Physica A 221, 340–347 (1995)Google Scholar
  29. Maćešić, S., Črnjarić Žic, N., Mezić, I.: Koopman operator family spectrum for nonautonomous systems. SIAM J. Appl. Dyn. Syst. 17(4), 2478–2515 (2018)MathSciNetzbMATHGoogle Scholar
  30. Mauroy, A., Mezić, I., Moehlis, J.: Isostables, isochrons, and koopman spectrum for the action-angle representation of stable fixed point dynamics. Physica D Nonlinear Phenom. 261, 19–30 (2013)MathSciNetzbMATHGoogle Scholar
  31. Mezić, I.: On the geometrical and statistical properties of dynamical systems: theory and applications. Ph.D. thesis, California Institute of Technology (1994)Google Scholar
  32. Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1), 309–325 (2005)MathSciNetzbMATHGoogle Scholar
  33. Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378 (2013)MathSciNetzbMATHGoogle Scholar
  34. Mezić, I.: On applications of the spectral theory of the Koopman operator in dynamical systems and control theory. In: 2015 IEEE 54th Annual Conference on Decision and Control (CDC), pp. 7034–7041. IEEE (2015)Google Scholar
  35. Mezić, I.: Koopman operator spectrum and data analysis (2017a). arXiv preprint arXiv:1702.07597
  36. Mezić, I.: Koopman Operator Spectrum and Data Analysis. ArXiv e-prints (2017b)Google Scholar
  37. Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior: Spectral methods. In: Proceedings of the 39th IEEE Conference on Decision and Control, 2000, vol. 2, pp. 1224–1231. IEEE (2000)Google Scholar
  38. Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Physica D 197(1), 101–133 (2004)MathSciNetzbMATHGoogle Scholar
  39. Mezić, I., Surana, A.: Koopman mode decomposition for periodic/quasi-periodic time dependence. IFAC-PapersOnLine 49(18), 690–697 (2016). Google Scholar
  40. Mezić, I., Wiggins, S.: A method for visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos Interdiscip. J. Nonlinear Sci. 9(1), 213–218 (1999)MathSciNetzbMATHGoogle Scholar
  41. Pavliotis, G.A.: Stochastic Processes and Applications: Diffusion Processes, the Fokker–Planck and Langevin Equations, Volume 60 of Texts in Applied Mathematics. Springer, New York (2014)Google Scholar
  42. Proctor, J., Eckhoff, P.A.: Discovering dynamic patterns from infectious disease data using dynamic mode decomposition. Int. Health 7, 139–145 (2015)Google Scholar
  43. Rößler, A.: Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Numer. Anal. 48(3), 922–952 (2010)MathSciNetzbMATHGoogle Scholar
  44. Rowley, C., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641(1), 115–127 (2009)MathSciNetzbMATHGoogle Scholar
  45. Schmid, P.: Dynamic mode decomposition of numerical and experimental data. In: Sixty-First Annual Meeting of the APS Division of Fluid Dynamics (2008)Google Scholar
  46. Schmid, P.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656(1), 5–28 (2010)MathSciNetzbMATHGoogle Scholar
  47. Schmid, P., Li, L., Juniper, M., Pust, O.: Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn. 25(1), 249–259 (2011)zbMATHGoogle Scholar
  48. Sharma, A.S., Mezić, I., McKeon, B.J.: Correspondence between koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the navier-stokes equations. Phys. Rev. Fluids 1(3), 032,402 (2016)Google Scholar
  49. Shnitzer, T., Talmon, R., Slotine, J.: Manifold learning with contracting observers for data-driven time-series analysis. IEEE Trans. Signal Proces. 65, 904–918 (2017)MathSciNetzbMATHGoogle Scholar
  50. Strand, J.: Random ordinary differential equations. J. Differ. Equ. 7, 538–553 (1970)MathSciNetzbMATHGoogle Scholar
  51. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Perseus Books Publishing, New York (1994)zbMATHGoogle Scholar
  52. Susuki, Y., Mezić, I.: Nonlinear Koopman modes and a precursor to power system swing instabilities. IEEE Trans. Power Syst. 27(3), 1182–1191 (2012)Google Scholar
  53. Susuki, Y., Mezić, I.: A Prony approximation of Koopman mode decomposition. In: Decision and Control (CDC), 2015 IEEE 54th Annual Conference on Decision and Control (2015).
  54. Susuki, Y., Mezić, I., Raak, F.T.H.: Applied Koopman operator theory for power system technology. Nonlinear Theory Appl. 7(4), 430–459 (2016)Google Scholar
  55. Takeishi, N., Kawahara, Y., Yairi, T.: Subspace dynamic mode decomposition for stochastic Koopman analysis. Phys. Rev. E 96, 033,310 (2017)Google Scholar
  56. 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. ArXiv e-prints (2017)Google Scholar
  57. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014)MathSciNetzbMATHGoogle Scholar
  58. Williams, M., Kevrekidis, I., Rowley, C.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307–1346 (2015)MathSciNetzbMATHGoogle Scholar
  59. Yosida, K.: Functional Analysis, 6th edn. Springer, New York (1980)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Nelida Črnjarić-Žic
    • 1
    Email author
  • Senka Maćešić
    • 1
  • Igor Mezić
    • 2
  1. 1.Faculty of EngineeringUniversity of RijekaRijekaCroatia
  2. 2.Faculty of Mechanical Engineering and MathematicsUniversity of California, Santa BarbaraSanta BarbaraUSA

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