Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Previous Article
Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times
JCD Home
This Issue
Next Article
Equation-free computation of coarse-grained center manifolds of microscopic simulators

June 2014, 1(2): 357-375. doi: 10.3934/jcd.2014.1.357

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

José Miguel Pasini ^1, and Tuhin Sahai ^1,

1.	United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06108, United States, United States

Received June 2013 Revised June 2014 Published December 2014

Abstract
Related Papers

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, multiple time scales, or chaotic dynamics. In particular, we prove that the differential equations for the coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of truncation order. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincaré section of the system and that, as the scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

Keywords: polynomial chaos, chaos., multiscale analysis, Hamiltonian systems, Uncertainty quantification.

Mathematics Subject Classification: Primary: 34F05, 65P10, 65P20, 34E13, 60H35; Secondary: 37H9.

Citation: 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

References:

[1]	M. S. Allen and J. A. Camberos, Comparison of uncertainty propagation / response surface techniques for two aeroelastic systems,, in 50th AIAA Structures, (2009), 4. doi: 10.2514/6.2009-2269. Google Scholar
[2]	G. Blatman and B. Sudret, An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis,, Probabilistic Engineering Mechanics, 25 (2010), 183. doi: 10.1016/j.probengmech.2009.10.003. Google Scholar
[3]	G. Blatman and B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression,, Journal of Computational Physics, 230 (2011), 2345. doi: 10.1016/j.jcp.2010.12.021. Google Scholar
[4]	J. Bucklew, Introduction to Rare Event Simulation,, Springer, (2004). doi: 10.1007/978-1-4757-4078-3. Google Scholar
[5]	R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods,, Acta Numerica, 7 (1998), 1. doi: 10.1017/S0962492900002804. Google Scholar
[6]	R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals,, Annals of Mathematics, 48 (1947), 385. doi: 10.2307/1969178. Google Scholar
[7]	C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations,, Prentice Hall, (1971). Google Scholar
[8]	S. E. Geneser, R. M. Kirby and F. B. Sachse, Sensitivity analysis of cardiac electrophysiological models using polynomial chaos,, in Engineering in Medicine and Biology Society, (2006), 4042. doi: 10.1109/IEMBS.2005.1615349. Google Scholar
[9]	R. Ghanem, Probabilistic characterization of transport in heterogeneous media,, Comput. Methods Appl. Mech. Engng., 158 (1998), 199. doi: 10.1016/S0045-7825(97)00250-8. Google Scholar
[10]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42,, Springer-Verlag New York, (1983). doi: 10.1007/978-1-4612-1140-2. Google Scholar
[11]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31,, Springer, (2006). Google Scholar
[12]	T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning,, 2nd edition, (2009). doi: 10.1007/978-0-387-84858-7. Google Scholar
[13]	M. He, S. Murugesan and J. Zhang, Multiple timescale dispatch and scheduling for stochastic reliability in smart grids with wind generation integration,, in Proceedings of the IEEE INFOCOM, (2011), 10. doi: 10.1109/INFCOM.2011.5935204. Google Scholar
[14]	P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511622700. Google Scholar
[15]	B. Huberman and J. P. Crutchfield, Chaotic states of anharmonic systems in periodic fields,, Phys. Rev. Lett., 43 (1979), 1743. doi: 10.1103/PhysRevLett.43.1743. Google Scholar
[16]	S. Klus, T. Sahai, C. Liu and M. Dellnitz, An efficient algorithm for the parallel solution of high-dimensional differential equations,, J. Comput. Appl. Math., 235 (2011), 3053. doi: 10.1016/j.cam.2010.12.026. Google Scholar
[17]	B. Kouchmeshky and N. Zabaras, The effect of multiple sources of uncertainty on the convex hull of material properties of polycrystals,, Computational Materials Science, 47 (2009), 342. doi: 10.1016/j.commatsci.2009.08.010. Google Scholar
[18]	J. Laskar, Large-scale chaos in the solar system,, Astronomy and Astrophysics, 287 (1994). Google Scholar
[19]	R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions,, 2nd edition, (1998). Google Scholar
[20]	E. N. Lorenz, Deterministic nonperiodic flow,, Journal of the atmospheric sciences, 20 (1963), 130. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. Google Scholar
[21]	X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations,, Journal of Computational Physics, 228 (2009), 3084. doi: 10.1016/j.jcp.2009.01.006. Google Scholar
[22]	X. Ma and N. Zabaras, Kernel principal component analysis for stochastic input model generation,, Journal of Computational Physics, 230 (2011), 7311. doi: 10.1016/j.jcp.2011.05.037. Google Scholar
[23]	Y. M. Marzouk and H. N. Najm, Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems,, Journal of Computational Physics, 228 (2009), 1862. doi: 10.1016/j.jcp.2008.11.024. Google Scholar
[24]	R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook, Response Surface Methodology,, 3rd edition, (2009). Google Scholar
[25]	H. N. Najm, B. J. Debusschere, Y. M. Marzouk, S. Widmer and O. P. Le Maìtre, Uncertainty quantification in chemical systems,, Int. J. Numer. Meth. Engng., 80 (2009), 789. doi: 10.1002/nme.2551. Google Scholar
[26]	H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers,, Bulletin of the American Mathematical Society, 84 (1978), 957. doi: 10.1090/S0002-9904-1978-14532-7. Google Scholar
[27]	F. Nobile, R. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with randon input data,, SIAM J. Numer. Anal., 46 (2008), 2309. doi: 10.1137/060663660. Google Scholar
[28]	H. Ogura, Orthogonal functions of the Poisson processes,, IEEE Transactions on Information Theory, 18 (1972), 473. doi: 10.1109/TIT.1972.1054856. Google Scholar
[29]	P. Parpas and M. Webster, A stochastic multiscale model for electricity generation capacity expansion,, Eur. J. Oper. Res., 232 (2014), 359. doi: 10.1016/j.ejor.2013.07.022. Google Scholar
[30]	R. H. Rand, Lecture Notes on Nonlinear Vibrations,, Internet-First University Press, (2012). Google Scholar
[31]	T. Sahai, V. Fonoberov and S. Loire, Uncertainty as a stabilizer of the head-tail ordered phase in carbon-monoxide monolayers on graphite,, Physical Review B, 80 (2009). doi: 10.1103/PhysRevB.80.115413. Google Scholar
[32]	T. Sahai, Backbone transitions and invariant tori in forced micromechanical oscillators with optical detection,, Nonlinear Dynamics, 62 (2010), 273. doi: 10.1007/s11071-010-9716-4. Google Scholar
[33]	T. Sahai, R. B. Bhiladvala and A. T. Zehnder, Thermomechanical transitions in doubly-clamped micro-oscillators,, International Journal of Non-Linear Mechanics, 42 (2007), 596. doi: 10.1016/j.ijnonlinmec.2006.12.009. Google Scholar
[34]	T. Sahai and J. M. Pasini, Uncertainty quantification in hybrid dynamical systems,, J. Comput. Phys., 237 (2013), 411. doi: 10.1016/j.jcp.2012.10.030. Google Scholar
[35]	T. Sahai, A. Speranzon and A. Banaszuk, Hearing the clusters in a graph: A dristributed algorithm,, Automatica, 48 (2012), 15. doi: 10.1016/j.automatica.2011.09.019. Google Scholar
[36]	T. Sahai and A. T. Zehnder, Modeling of coupled dome-shaped microoscillators,, Microelectromechanical Systems, 17 (2008), 777. doi: 10.1109/JMEMS.2008.924844. Google Scholar
[37]	S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering,, Perseus Books Group, (2001). Google Scholar
[38]	A. Surana, T. Sahai and A. Banaszuk, Iterative methods for scalable uncertainty quantification in complex networks,, International Journal for Uncertainty Quantification, 2 (2012), 413. doi: 10.1615/Int.J.UncertaintyQuantification.2012004138. Google Scholar
[39]	Y. Susuki, I. Mezić and T. Hikihara, Coherent swing instability of power grids,, J. Nonlinear Sci., 21 (2011), 403. doi: 10.1007/s00332-010-9087-5. Google Scholar
[40]	G. Szegö, Orthogonal Polynomials, vol. 23,, Amer Mathematical Society, (1967). Google Scholar
[41]	Y. Ueda, Explosion of strange attractors exhibited by Duffing's equation,, in Nonlinear Dynamics* (ed. R. H. G. Hellerman), 357* (1980), 422. doi: 10.1111/j.1749-6632.1980.tb29708.x. Google Scholar
[42]	X. Wan and G. E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs,, Journal of Scientific Computing, 27 (2006), 455. doi: 10.1007/s10915-005-9038-8. Google Scholar
[43]	X. Wan and G. E. Karniadakis, Recent advances in polynomial chaos methods and extensions,, in Computational Uncertainty in Military Vehicle Design Meeting Proceedings, (2008). Google Scholar
[44]	X. Wan and G. E. Karniadakis, Long-term behavior of polynomial chaos in stochastic flow simulations,, Computer methods in applied mechanics and engineering, 195 (2006), 5582. doi: 10.1016/j.cma.2005.10.016. Google Scholar
[45]	C. G. Webster, Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data,, PhD thesis, (2007). Google Scholar
[46]	N. Wiener, The homogeneous chaos,, American Journal of Mathematics, 60 (1938), 897. doi: 10.2307/2371268. Google Scholar
[47]	D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos,, J. Comp. Phys., 187 (2003), 137. doi: 10.1016/S0021-9991(03)00092-5. Google Scholar
[48]	N. Zabaras and B. Ganapathysubramanian, A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach,, J. Comput. Phys., 227 (2008), 4697. doi: 10.1016/j.jcp.2008.01.019. Google Scholar

show all references

References:

[1]	M. S. Allen and J. A. Camberos, Comparison of uncertainty propagation / response surface techniques for two aeroelastic systems,, in 50th AIAA Structures, (2009), 4. doi: 10.2514/6.2009-2269. Google Scholar
[2]	G. Blatman and B. Sudret, An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis,, Probabilistic Engineering Mechanics, 25 (2010), 183. doi: 10.1016/j.probengmech.2009.10.003. Google Scholar
[3]	G. Blatman and B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression,, Journal of Computational Physics, 230 (2011), 2345. doi: 10.1016/j.jcp.2010.12.021. Google Scholar
[4]	J. Bucklew, Introduction to Rare Event Simulation,, Springer, (2004). doi: 10.1007/978-1-4757-4078-3. Google Scholar
[5]	R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods,, Acta Numerica, 7 (1998), 1. doi: 10.1017/S0962492900002804. Google Scholar
[6]	R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals,, Annals of Mathematics, 48 (1947), 385. doi: 10.2307/1969178. Google Scholar
[7]	C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations,, Prentice Hall, (1971). Google Scholar
[8]	S. E. Geneser, R. M. Kirby and F. B. Sachse, Sensitivity analysis of cardiac electrophysiological models using polynomial chaos,, in Engineering in Medicine and Biology Society, (2006), 4042. doi: 10.1109/IEMBS.2005.1615349. Google Scholar
[9]	R. Ghanem, Probabilistic characterization of transport in heterogeneous media,, Comput. Methods Appl. Mech. Engng., 158 (1998), 199. doi: 10.1016/S0045-7825(97)00250-8. Google Scholar
[10]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42,, Springer-Verlag New York, (1983). doi: 10.1007/978-1-4612-1140-2. Google Scholar
[11]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31,, Springer, (2006). Google Scholar
[12]	T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning,, 2nd edition, (2009). doi: 10.1007/978-0-387-84858-7. Google Scholar
[13]	M. He, S. Murugesan and J. Zhang, Multiple timescale dispatch and scheduling for stochastic reliability in smart grids with wind generation integration,, in Proceedings of the IEEE INFOCOM, (2011), 10. doi: 10.1109/INFCOM.2011.5935204. Google Scholar
[14]	P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511622700. Google Scholar
[15]	B. Huberman and J. P. Crutchfield, Chaotic states of anharmonic systems in periodic fields,, Phys. Rev. Lett., 43 (1979), 1743. doi: 10.1103/PhysRevLett.43.1743. Google Scholar
[16]	S. Klus, T. Sahai, C. Liu and M. Dellnitz, An efficient algorithm for the parallel solution of high-dimensional differential equations,, J. Comput. Appl. Math., 235 (2011), 3053. doi: 10.1016/j.cam.2010.12.026. Google Scholar
[17]	B. Kouchmeshky and N. Zabaras, The effect of multiple sources of uncertainty on the convex hull of material properties of polycrystals,, Computational Materials Science, 47 (2009), 342. doi: 10.1016/j.commatsci.2009.08.010. Google Scholar
[18]	J. Laskar, Large-scale chaos in the solar system,, Astronomy and Astrophysics, 287 (1994). Google Scholar
[19]	R. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions,, 2nd edition, (1998). Google Scholar
[20]	E. N. Lorenz, Deterministic nonperiodic flow,, Journal of the atmospheric sciences, 20 (1963), 130. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. Google Scholar
[21]	X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations,, Journal of Computational Physics, 228 (2009), 3084. doi: 10.1016/j.jcp.2009.01.006. Google Scholar
[22]	X. Ma and N. Zabaras, Kernel principal component analysis for stochastic input model generation,, Journal of Computational Physics, 230 (2011), 7311. doi: 10.1016/j.jcp.2011.05.037. Google Scholar
[23]	Y. M. Marzouk and H. N. Najm, Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems,, Journal of Computational Physics, 228 (2009), 1862. doi: 10.1016/j.jcp.2008.11.024. Google Scholar
[24]	R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook, Response Surface Methodology,, 3rd edition, (2009). Google Scholar
[25]	H. N. Najm, B. J. Debusschere, Y. M. Marzouk, S. Widmer and O. P. Le Maìtre, Uncertainty quantification in chemical systems,, Int. J. Numer. Meth. Engng., 80 (2009), 789. doi: 10.1002/nme.2551. Google Scholar
[26]	H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers,, Bulletin of the American Mathematical Society, 84 (1978), 957. doi: 10.1090/S0002-9904-1978-14532-7. Google Scholar
[27]	F. Nobile, R. Tempone and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with randon input data,, SIAM J. Numer. Anal., 46 (2008), 2309. doi: 10.1137/060663660. Google Scholar
[28]	H. Ogura, Orthogonal functions of the Poisson processes,, IEEE Transactions on Information Theory, 18 (1972), 473. doi: 10.1109/TIT.1972.1054856. Google Scholar
[29]	P. Parpas and M. Webster, A stochastic multiscale model for electricity generation capacity expansion,, Eur. J. Oper. Res., 232 (2014), 359. doi: 10.1016/j.ejor.2013.07.022. Google Scholar
[30]	R. H. Rand, Lecture Notes on Nonlinear Vibrations,, Internet-First University Press, (2012). Google Scholar
[31]	T. Sahai, V. Fonoberov and S. Loire, Uncertainty as a stabilizer of the head-tail ordered phase in carbon-monoxide monolayers on graphite,, Physical Review B, 80 (2009). doi: 10.1103/PhysRevB.80.115413. Google Scholar
[32]	T. Sahai, Backbone transitions and invariant tori in forced micromechanical oscillators with optical detection,, Nonlinear Dynamics, 62 (2010), 273. doi: 10.1007/s11071-010-9716-4. Google Scholar
[33]	T. Sahai, R. B. Bhiladvala and A. T. Zehnder, Thermomechanical transitions in doubly-clamped micro-oscillators,, International Journal of Non-Linear Mechanics, 42 (2007), 596. doi: 10.1016/j.ijnonlinmec.2006.12.009. Google Scholar
[34]	T. Sahai and J. M. Pasini, Uncertainty quantification in hybrid dynamical systems,, J. Comput. Phys., 237 (2013), 411. doi: 10.1016/j.jcp.2012.10.030. Google Scholar
[35]	T. Sahai, A. Speranzon and A. Banaszuk, Hearing the clusters in a graph: A dristributed algorithm,, Automatica, 48 (2012), 15. doi: 10.1016/j.automatica.2011.09.019. Google Scholar
[36]	T. Sahai and A. T. Zehnder, Modeling of coupled dome-shaped microoscillators,, Microelectromechanical Systems, 17 (2008), 777. doi: 10.1109/JMEMS.2008.924844. Google Scholar
[37]	S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering,, Perseus Books Group, (2001). Google Scholar
[38]	A. Surana, T. Sahai and A. Banaszuk, Iterative methods for scalable uncertainty quantification in complex networks,, International Journal for Uncertainty Quantification, 2 (2012), 413. doi: 10.1615/Int.J.UncertaintyQuantification.2012004138. Google Scholar
[39]	Y. Susuki, I. Mezić and T. Hikihara, Coherent swing instability of power grids,, J. Nonlinear Sci., 21 (2011), 403. doi: 10.1007/s00332-010-9087-5. Google Scholar
[40]	G. Szegö, Orthogonal Polynomials, vol. 23,, Amer Mathematical Society, (1967). Google Scholar
[41]	Y. Ueda, Explosion of strange attractors exhibited by Duffing's equation,, in Nonlinear Dynamics* (ed. R. H. G. Hellerman), 357* (1980), 422. doi: 10.1111/j.1749-6632.1980.tb29708.x. Google Scholar
[42]	X. Wan and G. E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs,, Journal of Scientific Computing, 27 (2006), 455. doi: 10.1007/s10915-005-9038-8. Google Scholar
[43]	X. Wan and G. E. Karniadakis, Recent advances in polynomial chaos methods and extensions,, in Computational Uncertainty in Military Vehicle Design Meeting Proceedings, (2008). Google Scholar
[44]	X. Wan and G. E. Karniadakis, Long-term behavior of polynomial chaos in stochastic flow simulations,, Computer methods in applied mechanics and engineering, 195 (2006), 5582. doi: 10.1016/j.cma.2005.10.016. Google Scholar
[45]	C. G. Webster, Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data,, PhD thesis, (2007). Google Scholar
[46]	N. Wiener, The homogeneous chaos,, American Journal of Mathematics, 60 (1938), 897. doi: 10.2307/2371268. Google Scholar
[47]	D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos,, J. Comp. Phys., 187 (2003), 137. doi: 10.1016/S0021-9991(03)00092-5. Google Scholar
[48]	N. Zabaras and B. Ganapathysubramanian, A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach,, J. Comput. Phys., 227 (2008), 4697. doi: 10.1016/j.jcp.2008.01.019. Google Scholar

[1]	Carles Simó. Measuring the total amount of chaos in some Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5135-5164. doi: 10.3934/dcds.2014.34.5135
[2]	Flaviano Battelli, Michal Fe?kan. Chaos in forced impact systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861
[3]	Andrew J. Majda, Michal Branicki. Lessons in uncertainty quantification for turbulent dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3133-3221. doi: 10.3934/dcds.2012.32.3133
[4]	Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[5]	Y. Charles Li. Existence of chaos in weakly quasilinear systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1331-1344. doi: 10.3934/cpaa.2011.10.1331
[6]	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 - A, 2010, 28 (1) : 199-226. doi: 10.3934/dcds.2010.28.199
[7]	Marta Štefánková. Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3435-3443. doi: 10.3934/dcds.2016.36.3435
[8]	Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231
[9]	Tijana Levajković, Hermann Mena, Amjad Tuffaha. The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach. Evolution Equations & Control Theory, 2016, 5 (1) : 105-134. doi: 10.3934/eect.2016.5.105
[10]	H. T. Banks, Robert Baraldi, Karissa Cross, Kevin Flores, Christina McChesney, Laura Poag, Emma Thorpe. Uncertainty quantification in modeling HIV viral mechanics. Mathematical Biosciences & Engineering, 2015, 12 (5) : 937-964. doi: 10.3934/mbe.2015.12.937
[11]	Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553
[12]	Jing Li, Panos Stinis. Mori-Zwanzig reduced models for uncertainty quantification. Journal of Computational Dynamics, 2018, 0 (0) : 1-30. doi: 10.3934/jcd.2019002
[13]	Ryan Bennink, Ajay Jasra, Kody J. H. Law, Pavel Lougovski. Estimation and uncertainty quantification for the output from quantum simulators. Foundations of Data Science, 2019, 1 (2) : 157-176. doi: 10.3934/fods.2019007
[14]	Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971
[15]	Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[16]	Arsen R. Dzhanoev, Alexander Loskutov, Hongjun Cao, Miguel A.F. Sanjuán. A new mechanism of the chaos suppression. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 275-284. doi: 10.3934/dcdsb.2007.7.275
[17]	Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic & Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85
[18]	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
[19]	Y. Charles Li. Chaos phenotypes discovered in fluids. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1383-1398. doi: 10.3934/dcds.2010.26.1383
[20]	Kaijen Cheng, Kenneth Palmer. Chaos in a model for masting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1917-1932. doi: 10.3934/dcdsb.2015.20.1917

Impact Factor:

American Institute of Mathematical Sciences