Abstract

Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modeling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte Carlo methods or by direct numerical simulation techniques using polynomial chaos (PC), i.e., by a series expansion of the random parameters in combination with forward integration. Here we take a dynamical systems viewpoint and focus on the invariant sets of differential equations such as steady states, stable/unstable manifolds, periodic orbits, and heteroclinic orbits. We employ PC to compute representations of all these different types of invariant sets for RODEs. This allows us to obtain fast sampling, geometric visualization of distributional properties of invariants sets, and uncertainty quantification of dynamical output such as periods or locations of orbits. We apply our techniques to a predator-prey model, where we compute steady states and stable/unstable manifolds. We also include several benchmarks to illustrate the numerical efficiency of adaptively chosen PC depending upon the random input. Then we employ the methods for the Lorenz system, obtaining computational PC representations of periodic orbits, stable/unstable manifolds, and heteroclinic orbits.

Keywords

  1. invariant manifold
  2. periodic orbit
  3. heteroclinic orbit
  4. Lorenz system
  5. polynomial chaos
  6. random differential equation

MSC codes

  1. 37H10
  2. 34F05
  3. 60H35
  4. 41A58
  5. 65C50

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References

1.
U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Classics in Appl. Math. 13, SIAM, Philadelphia, 1987.
2.
M. Breden and C. Kuehn, MATLAB Code for “Computing Invariant Sets of Random Differential Equations Using Polynomial Chaos,'' https://sites.google.com/site/maximebreden/research.
3.
M. Breden, J.-P. Lessard, and J. D. Mireles James, Computation of maximal local (un)stable manifold patches by the parameterization method, Indag. Math., 27 (2016), pp. 340--367.
4.
X. Cabré, E. Fontich, and R. de la Llave, The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), pp. 283--328.
5.
X. Cabré, E. Fontich, and R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), pp. 329--360.
6.
X. Cabré, E. Fontich, and R. de la Llave, The parameterization method for invariant manifolds. III. Overview and applications, J. Differential Equations, 218 (2005), pp. 444--515.
7.
B. J. Debusschere, H. N. Najm, P. P. Pébay, O. M. Knio, R. G. Ghanem, and O. Le Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes, SIAM J. Sci. Comput., 26 (2004), pp. 698--719.
8.
M. Dellnitz, S. Klus, and A. Ziessler, A set-oriented numerical approach for dynamical systems with parameter uncertainty, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 120--138.
9.
A. Desai, J. A. Witteveen, and S. Sarkar, Uncertainty quantification of a nonlinear aeroelastic system using polynomial chaos expansion with constant phase interpolation, J. Vibration Acoustics, 135 (2013), 051034.
10.
T. A. Driscoll, N. Hale, and L. N. Trefethen, Chebfun Guide, https://www.chebfun.org/, 2014.
11.
T. A. Driscoll and J. Weideman, Optimal domain splitting for interpolation by Chebyshev polynomials, SIAM J. Numer. Anal., 52 (2014), pp. 1913--1927.
12.
O. G. Ernst, A. Mugler, H.-J. Starkloff, and E. Ullmann, On the convergence of generalized polynomial chaos expansions, Math. Model. Numer. Anal., 46 (2012), pp. 317--339.
13.
J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 1070--1088.
14.
G. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, Springer, New York, 2013.
15.
R. Ghanem and D. Ghosh, Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition, Internat. J. Numer. Methods Engrg., 72 (2007), pp. 486--504.
16.
R. G. Ghanem and P. D. Spanos, Stochastic finite element method: Response statistics, in Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991, pp. 101--119.
17.
M. Grigoriu, Stochastic Systems: Uncertainty Quantification and Propagation, Springer, New York, 2012.
18.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
19.
A. Haro, M. Canadell, J.-L. Figueras, A. Luque, and J.-M. Mondelo, The parameterization method for invariant manifolds, Appl. Math. Sci, 195 (2016).
20.
T. Hurth and C. Kuehn, Random switching near bifurcations, Stoch. Dyn., to appear.
21.
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Int. J. Bifur. Chaos, 15 (2005), pp. 763--791.
22.
B. Krauskopf, H. M. Osinga, and J. Galán-Vique, eds., Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Springer, New York, 2007.
23.
C. Kuehn, Deterministic continuation of stochastic metastable equilibria via Lyapunov equations and ellipsoids, SIAM J. Sci. Comput., 34 (2012), pp. A1635--A1658.
24.
C. Kuehn, Quenched noise and nonlinear oscillations in bistable multiscale systems, Europhys. Lett., 120 (2017), 10001.
25.
C. Kuehn, Uncertainty transformation via Hopf bifurcation in fast-slow systems, Proc. A, 473 (2017), 20160346.
26.
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci. 112, Springer, New York, 2013.
27.
O. Le Maître and O. M. Knio, Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, Springer, New York, 2010.
28.
O. Le Maître, L. Mathelin, O. M. Knio, and M. Y. Hussaini, Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics, Discrete Contin. Dyn. Syst., 28 (2010), pp. 199--226.
29.
D. Lucor and G. E. Karniadakis, Adaptive generalized polynomial chaos for nonlinear random oscillators, SIAM J. Sci. Comput., 26 (2004), pp. 720--735.
30.
D. R. Millman, P. I. King, R. C. Maple, P. S. Beran, and L. K. Chilton, Airfoil pitch-and-plunge bifurcation behavior with Fourier chaos expansions, J. Aircraft, 42 (2005), pp. 376--384.
31.
I. Molchanov, Theory of Random Sets, Springer, New York, 2017.
32.
H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer Ser. Inform. Sci. 2, Springer, New York, 2012.
33.
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, UK, 2010.
34.
D. Potts, G. Steidl, and M. Tasche, Fast algorithms for discrete polynomial transforms, Math. Comp., 67 (1998), pp. 1577--1590.
35.
K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press, New York, 2014.
36.
M. Schick, V. Heuveline, and O. Le Maître, A Newton--Galerkin method for fluid flow exhibiting uncertain periodic dynamics, SIAM/ASA J. Uncertain. Quantif., 2 (2014), pp. 153--173.
37.
R. Szwarc, Orthogonal polynomials and banach algebras, in Inzell Lectures on Orthogonal Polynomials, Advances in the Theory of Special Functions and Orthogonal Polynomials, Vol. 2, Nova Science Publishers, Hauppauge, NY, 2005, pp. 103--139.
38.
L. N. Trefethen, Approximation Theory and Approximation Practice, Appl. Math. 128, SIAM, Philadelphia, 2013.
39.
W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011.
40.
J. B. van den Berg, J. D. Mireles James, and C. Reinhardt, Computing (un)stable manifolds with validated error bounds: Non-resonant and resonant spectra, J. Nonlinear Sci., 26 (2016), pp. 1055--1095.
41.
J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015).
42.
J. B. van den Berg and R. Sheombarsing, Rigorous Numerics for ODEs Using Chebyshev Series and Domain Decomposition, preprint, 2015.
43.
A. J. Veraart, E. J. Faassen, V. Dakos, E. H. van Nes, M. Lurling, and M. Scheffer, Stochastic bifurcation analysis of Rayleigh-Bénard convection, J. Fluid Mech., 650 (2010), pp. 391--413.
44.
X. Wan and G. E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28 (2006), pp. 901--928.
45.
N. Wiener, The homogeneous chaos, Amer. J. Math., 60 (1938), pp. 897--936.
46.
D. Xiu, Generalized (Wiener-Askey) Polynomial Chaos, Ph.D. thesis, Brown University, Providence, RI, 2004.
47.
D. Xiu, Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), pp. 293--309.
48.
D. Xiu, Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys., 5 (2009), pp. 242--272.
49.
D. Xiu and G. E. Karniadakis, The Wiener--Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619--644.
50.
D. Xiu and G. E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 4927--4948.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 577 - 618
ISSN (online): 1536-0040

History

Submitted: 28 December 2018
Accepted: 22 January 2020
Published online: 17 March 2020

Keywords

  1. invariant manifold
  2. periodic orbit
  3. heteroclinic orbit
  4. Lorenz system
  5. polynomial chaos
  6. random differential equation

MSC codes

  1. 37H10
  2. 34F05
  3. 60H35
  4. 41A58
  5. 65C50

Authors

Affiliations

Funding Information

Volkswagen Foundation https://doi.org/10.13039/501100001663

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