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Koopman Operator Spectrum for Random Dynamical Systems

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Abstract

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.

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References

  • Arató, M.: A famous nonlinear stochastic equation: Lotka–Volterra model with diffusion. Math. Comput. Model. 38, 709–726 (2003)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Arnold, L.: Stochastic Differential Equations. Wiley, New York (1974)

    MATH  Google Scholar 

  • Arnold, L.: Random Dynamical Systems. Springer, New York (1998)

    MATH  Google Scholar 

  • Arnold, L., Crauel, H.: Random dynamical systems. Lypunov exponents. In: Proceedings Oberwolfach 1990. Springer Lecture notes in Mathematics, vol. 1486, pp. 1–22 (1991)

  • Bagheri, S.: Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596–623 (2013)

    MathSciNet  MATH  Google Scholar 

  • Budišić, M., Mohr, R., Mezić, I.: Applied Koopmanism. Chaos 22(4), 596–623 (2012)

    MathSciNet  MATH  Google Scholar 

  • Cohen, S.N., Elliott, R.J.: Stochastic Calculus and Applications, 2nd edn. Birkhäuser, New York (2015)

    MATH  Google Scholar 

  • Crauel, H.: Markov measures for random dynamical systems. Stoch. Stoch. Rep. 37(3), 153–173 (1991)

    MathSciNet  MATH  Google Scholar 

  • Da Prato, G.J.Z.: Stochastic Equations in Infinite Dimensions, 2nd edn. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  • Drmač, Z., Mezić, I., Mohr, R.: Data driven modal decompositions: analysis and enhancements. SIAM J. Sci. Comput. 40(4), A2253–A2285 (2018)

    MathSciNet  MATH  Google Scholar 

  • Dynkin, E.B.: Markov Processes, 1, 2, Grundlehren der Mathematischen Wissenschaften, vol. 121, 122. Springer, New York (1965)

    Google Scholar 

  • Engel, K., Nagel, R.: One-Parameter Semigroups for Linear Evolution Operators. Springer, New York (2001)

    MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Gaspard, P., Nicolis, G., Provata, A., Tasaki, S.: Spectral signature of the pitchfork bifurcation: Liouville equation approach. Phys. Rev. E 51(1), 74 (1995)

    MathSciNet  Google Scholar 

  • Giannakis, D.: Data-driven spectral decomposition and forecasting of ergodic dynamical systems. Appl. Comput. Harmon. Anal. 47(2), 338–396 (2019)

    MathSciNet  MATH  Google Scholar 

  • 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 

  • Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  • 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)

  • Junge, O., Marsden, J.E., Mezić, I.: Uncertainty in the dynamics of conservative maps. In: 43rd IEEE Conference on Decision and Control (2004)

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Koopman, B.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. USA 17(5), 315 (1931)

    MATH  Google Scholar 

  • Korda, M., Mezić, I.: On convergence of extended dynamic mode decomposition to the Koopman operator. J. Nonlinear Sci. 28(2), 687–710 (2018)

    MathSciNet  MATH  Google Scholar 

  • Lan, Y., Mezić, I.: Linearization in the large of nonlinear systems and Koopman operator spectrum. Physica D 242(1), 42–53 (2013). https://doi.org/10.1016/j.physd.2012.08.017

    Article  MathSciNet  MATH  Google Scholar 

  • Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, 2nd edn. Springer, New York (1994)

    MATH  Google Scholar 

  • Leung, H.: Stochastic transient of a noisy van der Pol oscillator. Physica A 221, 340–347 (1995)

    Google Scholar 

  • Maćešić, S., Črnjarić Žic, N., Mezić, I.: Koopman operator family spectrum for nonautonomous systems. SIAM J. Appl. Dyn. Syst. 17(4), 2478–2515 (2018)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Mezić, I.: On the geometrical and statistical properties of dynamical systems: theory and applications. Ph.D. thesis, California Institute of Technology (1994)

  • Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1), 309–325 (2005)

    MathSciNet  MATH  Google Scholar 

  • Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378 (2013)

    MathSciNet  MATH  Google Scholar 

  • 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)

  • Mezić, I.: Koopman operator spectrum and data analysis (2017a). arXiv preprint arXiv:1702.07597

  • Mezić, I.: Koopman Operator Spectrum and Data Analysis. ArXiv e-prints (2017b)

  • 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)

  • Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Physica D 197(1), 101–133 (2004)

    MathSciNet  MATH  Google Scholar 

  • Mezić, I., Surana, A.: Koopman mode decomposition for periodic/quasi-periodic time dependence. IFAC-PapersOnLine 49(18), 690–697 (2016). https://doi.org/10.1016/j.ifacol.2016.10.246

    Article  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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 

  • Proctor, J., Eckhoff, P.A.: Discovering dynamic patterns from infectious disease data using dynamic mode decomposition. Int. Health 7, 139–145 (2015)

    Google Scholar 

  • Rößler, A.: Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Numer. Anal. 48(3), 922–952 (2010)

    MathSciNet  MATH  Google Scholar 

  • Rowley, C., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641(1), 115–127 (2009)

    MathSciNet  MATH  Google Scholar 

  • Schmid, P.: Dynamic mode decomposition of numerical and experimental data. In: Sixty-First Annual Meeting of the APS Division of Fluid Dynamics (2008)

  • Schmid, P.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656(1), 5–28 (2010)

    MathSciNet  MATH  Google Scholar 

  • Schmid, P., Li, L., Juniper, M., Pust, O.: Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn. 25(1), 249–259 (2011)

    MATH  Google Scholar 

  • 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 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Strand, J.: Random ordinary differential equations. J. Differ. Equ. 7, 538–553 (1970)

    MathSciNet  MATH  Google Scholar 

  • Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Perseus Books Publishing, New York (1994)

    MATH  Google Scholar 

  • 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 

  • 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). https://doi.org/10.1109/CDC.2015.7403326

  • 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 

  • Takeishi, N., Kawahara, Y., Yairi, T.: Subspace dynamic mode decomposition for stochastic Koopman analysis. Phys. Rev. E 96, 033,310 (2017)

    Google Scholar 

  • 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)

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Yosida, K.: Functional Analysis, 6th edn. Springer, New York (1980)

    MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Faculty of Engineering, University of Rijeka, Vukovarska 58, 51000, Rijeka, Croatia

    Nelida Črnjarić-Žic & Senka Maćešić

  2. Faculty of Mechanical Engineering and Mathematics, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA

    Igor Mezić

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  1. Nelida Črnjarić-Žic
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  2. Senka Maćešić
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Correspondence to Nelida Črnjarić-Žic.

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Črnjarić-Žic, N., Maćešić, S. & Mezić, I. Koopman Operator Spectrum for Random Dynamical Systems. J Nonlinear Sci 30, 2007–2056 (2020). https://doi.org/10.1007/s00332-019-09582-z

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