American Institute of Mathematical Sciences

Advanced
March  2010, 28(1): 199-226. doi: 10.3934/dcds.2010.28.199

Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics

Olivier P. Le Maître 1, , Lionel Mathelin 1, , Omar M. Knio 2, and  M. Yousuff Hussaini 3,

1. 

LIMSI-CNRS, UPR 3251, BP 133, Orsay F-91403, France, France
2. 

Johns Hopkins University, Department of Mechanical Engineering, Baltimore, MD 21218, United States
3. 

Florida State University, Computational Science & Engineering – Department of Mathematics, Tallahassee, FL 32306-4510, United States

Received  November 2009 Revised  February 2010 Published  April 2010

The simulation of dynamical systems involving random coefficients by means of stochastic spectral methods (Polynomial Chaos or other types of orthogonal stochastic expansions) is faced with well known computational difficulties, arising in particular due to the broadening of the solution spectrum as time evolves. The simulation of such systems thus requires increasing the basis dimension and computational resources for long time integration. This paper deals with systems having almost surely a stable limit cycles. It is proposed to reformulate the problem at hand in a rescaled time framework such that the spectrum of the rescaled solution remains narrow-banded. Two variants of this approach are considered and evaluated. The first relies on an explicit expression of a time-dependent, uncertain, time scale related to some distance between the corresponding solution and a reference deterministic system. The time scale is adjusted at each time step so that the distance from the reference system solution remains small, mimicking "in phase'' behavior. The second variant achieves the same objective by borrowing concepts from optimal control theory, and yields more precise time-scale estimates at the price of a higher CPU cost. It is thus more appropriate for uncertain systems exhibiting a stiff behavior and complex limit cycles. The method is applied to the case of a linear oscillator with uncertain properties, and to a stiff nonlinear chemical system involving uncertain reaction constants. The tests demonstrate the effectiveness of the proposed approaches, at least in situations where the topology of the limit cycle does not change when the uncertain system parameters vary.
Keywords: uncertainty, limit-cycle, Dynamical systems, Polynomial Chaos..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Olivier P. Le Maître, Lionel Mathelin, Omar M. Knio, M. Yousuff Hussaini. Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 199-226. doi: 10.3934/dcds.2010.28.199
[1]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447
[2]

José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357
[3]

Andrew J. Majda, Michal Branicki. Lessons in uncertainty quantification for turbulent dynamical systems. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3133-3221. doi: 10.3934/dcds.2012.32.3133
[4]

Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070
[5]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123
[6]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[7]

Sergey Shemyakov, Roman Chernov, Dzmitry Rumiantsau, Dierk Schleicher, Simon Schmitt, Anton Shemyakov. Finding polynomial roots by dynamical systems – A case study. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6945-6965. doi: 10.3934/dcds.2020261
[8]

Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367
[9]

Marta Štefánková. Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3435-3443. doi: 10.3934/dcds.2016.36.3435
[10]

Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065
[11]

Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465
[12]

Armengol Gasull, Hector Giacomini. Upper bounds for the number of limit cycles of some planar polynomial differential systems. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217
[13]

Tao Li, Jaume Llibre. Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3887-3909. doi: 10.3934/cpaa.2021136
[14]

Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161
[15]

Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683
[16]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[17]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5581-5599. doi: 10.3934/dcdsb.2020368
[18]

Elena K. Kostousova. On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty. Conference Publications, 2011, 2011 (Special) : 864-873. doi: 10.3934/proc.2011.2011.864
[19]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[20]

Luca Dieci, Cinzia Elia, Dingheng Pi. Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3091-3112. doi: 10.3934/dcdsb.2017165

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (146)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy