June  2014, 1(2): 391-421. doi: 10.3934/jcd.2014.1.391

On dynamic mode decomposition: Theory and applications

1. 

Dept. of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, United States, United States, United States

2. 

Dept. of Applied Mathematics, University of Washington, Seattle, WA 98195, United States, United States

Received  November 2013 Revised  November 2014 Published  December 2014

Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.
Citation: Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391
References:
[1]

S. Bagheri, Koopman-mode decomposition of the cylinder wake,, J. Fluid Mech., 726 (2013), 596.  doi: 10.1017/jfm.2013.249.  Google Scholar

[2]

B. A. Belson, J. H. Tu and C. W. Rowley, A Parallelized Model Reduction Library,, ACM T. Math. Software, (2013).   Google Scholar

[3]

M. B. Blumenthal, Predictability of a coupled ocean-atmosphere model,, J. Climate, 4 (1991), 766.  doi: 10.1175/1520-0442(1991)004<0766:POACOM>2.0.CO;2.  Google Scholar

[4]

K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses,, J. Nonlinear Sci., 22 (2012), 887.  doi: 10.1007/s00332-012-9130-9.  Google Scholar

[5]

T. Colonius and K. Taira, A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions,, Comput. Method Appl. M., 197 (2008), 2131.  doi: 10.1016/j.cma.2007.08.014.  Google Scholar

[6]

D. Duke, D. Honnery and J. Soria, Experimental investigation of nonlinear instabilities in annular liquid sheets,, J. Fluid Mech., 691 (2012), 594.  doi: 10.1017/jfm.2011.516.  Google Scholar

[7]

D. Duke, J. Soria and D. Honnery, An error analysis of the dynamic mode decomposition,, Exp. Fluids, 52 (2012), 529.  doi: 10.1007/s00348-011-1235-7.  Google Scholar

[8]

P. J. Goulart, A. Wynn and D. Pearson, Optimal mode decomposition for high dimensional systems,, In Proceedings of the 51st IEEE Conference on Decision and Control, 2012 (2012), 4965.  doi: 10.1109/CDC.2012.6426995.  Google Scholar

[9]

M. Grilli, P. J. Schmid, S. Hickel and N. A. Adams, Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction,, J. Fluid Mech., 700 (2012), 16.  doi: 10.1017/jfm.2012.37.  Google Scholar

[10]

K. Hasselmann, PIPs and POPs: The reduction of complex dynamical-systems using Principal Interaction and Oscillation Patterns,, J. Geophys. Res.-Atmos., 93 (1988), 11015.  doi: 10.1029/JD093iD09p11015.  Google Scholar

[11]

B. L. Ho and R. E. Kalman, Effective construction of linear state-variable models from input/output data,, Proceedings of the Third Annual Allerton Conference on Circuit and System Theory, (1965), 449.   Google Scholar

[12]

P. J. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge University Press, (2012).  doi: 10.1017/CBO9780511919701.  Google Scholar

[13]

H. Hotelling, Analysis of a complex of statistical variables into principal components,, J. Educ. Psychol., 24 (1933), 417.  doi: 10.1037/h0071325.  Google Scholar

[14]

H. Hotelling, Analysis of a complex of statistical variables into principal components,, J. Educ. Psychol., 24 (1933), 498.   Google Scholar

[15]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition,, Phys. Fluids, 26 (2014).  doi: 10.1063/1.4863670.  Google Scholar

[16]

J. N. Juang and R. S. Pappa, An eigensystem realization-algorithm for modal parameter-identification and model-reduction,, J. Guid. Control Dynam., 8 (1985), 620.  doi: 10.2514/3.20031.  Google Scholar

[17]

E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction,, Technical report, (1956).   Google Scholar

[18]

Z. Ma, S. Ahuja and C. W. Rowley, Reduced-order models for control of fluids using the eigensystem realization algorithm,, Theor. Comp. Fluid Dyn., 25 (2011), 233.  doi: 10.1007/s00162-010-0184-8.  Google Scholar

[19]

L. Massa, R. Kumar and P. Ravindran, Dynamic mode decomposition analysis of detonation waves,, Phys. Fluids, (2012).   Google Scholar

[20]

I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions,, Nonlin. Dynam., 41 (2005), 309.  doi: 10.1007/s11071-005-2824-x.  Google Scholar

[21]

I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator,, Annu. Rev. Fluid Mech., 45 (2013), 357.  doi: 10.1146/annurev-fluid-011212-140652.  Google Scholar

[22]

T. W. Muld, G. Efraimsson and D. S. Henningson, Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition,, Comput. Fluids, 57 (2012), 87.  doi: 10.1016/j.compfluid.2011.12.012.  Google Scholar

[23]

B. R. Noack, K. Afanasiev, M. Morzyński, G. Tadmor and F. Thiele, A hierarchy of low-dimensional models for the transient and post-transient cylinder wake,, J. Fluid Mech., 497 (2003), 335.  doi: 10.1017/S0022112003006694.  Google Scholar

[24]

B. R. Noack, G. Tadmor and M. Morzyński, Actuation models and dissipative control in empirical Galerkin models of fluid flows,, In Proceedings of the American Control Conference, (2004), 5722.   Google Scholar

[25]

K. Pearson, LIII. on lines and planes of closest fit to systems of points in space,, Philos. Mag., 2 (1901), 559.  doi: 10.1080/14786440109462720.  Google Scholar

[26]

C. Penland, Random forcing and forecasting using Principal Oscillation Pattern analysis,, Mon. Weather Rev., 117 (1989), 2165.   Google Scholar

[27]

C. Penland and T. Magorian, Prediction of Niño 3 sea-surface temperatures using linear inverse modeling,, J. Climate, 6 (1993), 1067.   Google Scholar

[28]

C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition,, Int. J. Bifurcat. Chaos, 15 (2005), 997.  doi: 10.1142/S0218127405012429.  Google Scholar

[29]

C. W. Rowley, I. Mezic, S. Bagheri, P. Schlatter and D. S. Henningson, Spectral analysis of nonlinear flows,, J. Fluid Mech., 641 (2009), 115.  doi: 10.1017/S0022112009992059.  Google Scholar

[30]

P. J. Schmid, Dynamic mode decomposition of numerical and experimental data,, J. Fluid Mech., 656 (2010), 5.  doi: 10.1017/S0022112010001217.  Google Scholar

[31]

P. J. Schmid, Application of the dynamic mode decomposition to experimental data,, Exp. Fluids, 50 (2011), 1123.  doi: 10.1007/s00348-010-0911-3.  Google Scholar

[32]

P. J. Schmid, L. Li, M. P. Juniper and O. Pust, Applications of the dynamic mode decomposition,, Theor. Comp. Fluid Dyn., 25 (2011), 249.  doi: 10.1007/s00162-010-0203-9.  Google Scholar

[33]

P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data,, Journal of Fluid Mechanics, 656 (2010), 5.  doi: 10.1017/S0022112010001217.  Google Scholar

[34]

P. J. Schmid, D. Violato and F. Scarano, Decomposition of time-resolved tomographic PIV,, Exp. Fluids, 52 (2012), 1567.  doi: 10.1007/s00348-012-1266-8.  Google Scholar

[35]

A. Seena and H. J. Sung, Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations,, Int. J. Heat Fluid Fl., 32 (2011), 1098.  doi: 10.1016/j.ijheatfluidflow.2011.09.008.  Google Scholar

[36]

O. Semeraro, G. Bellani and F. Lundell, Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes,, Exp. Fluids, 53 (2012), 1203.  doi: 10.1007/s00348-012-1354-9.  Google Scholar

[37]

J. R. Singler, Optimality of balanced proper orthogonal decomposition for data reconstruction,, Numerical Functional Analysis and Optimization, 31 (2010), 852.  doi: 10.1080/01630563.2010.500022.  Google Scholar

[38]

L. Sirovich, Turbulence and the dynamics of coherent structures. 2. Symmetries and transformations,, Q. Appl. Math., 45 (1987), 573.   Google Scholar

[39]

K. Taira and T. Colonius, The immersed boundary method: A projection approach,, J. Comput. Phys., 225 (2007), 2118.  doi: 10.1016/j.jcp.2007.03.005.  Google Scholar

[40]

L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, SIAM, (1997).  doi: 10.1137/1.9780898719574.  Google Scholar

[41]

J. H. Tu, J. Griffin, A. Hart, C. W. Rowley, L. N. Cattafesta III and L. S. Ukeiley, Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of velocity fields,, Exp. Fluids, 54 (2013).  doi: 10.2514/6.2012-33.  Google Scholar

[42]

J. H. Tu and C. W. Rowley, An improved algorithm for balanced POD through an analytic treatment of impulse response tails,, J. Comput. Phys., 231 (2012), 5317.  doi: 10.1016/j.jcp.2012.04.023.  Google Scholar

[43]

J. H. Tu, C. W. Rowley, E. Aram and R. Mittal, Koopman spectral analysis of separated flow over a finite-thickness flat plate with elliptical leading edge,, AIAA Paper 2011-38, (2011), 2011.  doi: 10.2514/6.2011-38.  Google Scholar

[44]

H. von Storch, G. Bürger, R. Schnur and J. S. von Storch, Principal oscillation patterns: A review,, J. Climate, 8 (1995), 377.   Google Scholar

[45]

M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: extending dynamic mode decomposition,, , (2014).   Google Scholar

[46]

A. Wynn, D. Pearson, B. Ganapathisubramani and P. J. Goulart, Optimal mode decomposition for unsteady flows,, J. Fluid Mech., 733 (2013), 473.  doi: 10.1017/jfm.2013.426.  Google Scholar

show all references

References:
[1]

S. Bagheri, Koopman-mode decomposition of the cylinder wake,, J. Fluid Mech., 726 (2013), 596.  doi: 10.1017/jfm.2013.249.  Google Scholar

[2]

B. A. Belson, J. H. Tu and C. W. Rowley, A Parallelized Model Reduction Library,, ACM T. Math. Software, (2013).   Google Scholar

[3]

M. B. Blumenthal, Predictability of a coupled ocean-atmosphere model,, J. Climate, 4 (1991), 766.  doi: 10.1175/1520-0442(1991)004<0766:POACOM>2.0.CO;2.  Google Scholar

[4]

K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses,, J. Nonlinear Sci., 22 (2012), 887.  doi: 10.1007/s00332-012-9130-9.  Google Scholar

[5]

T. Colonius and K. Taira, A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions,, Comput. Method Appl. M., 197 (2008), 2131.  doi: 10.1016/j.cma.2007.08.014.  Google Scholar

[6]

D. Duke, D. Honnery and J. Soria, Experimental investigation of nonlinear instabilities in annular liquid sheets,, J. Fluid Mech., 691 (2012), 594.  doi: 10.1017/jfm.2011.516.  Google Scholar

[7]

D. Duke, J. Soria and D. Honnery, An error analysis of the dynamic mode decomposition,, Exp. Fluids, 52 (2012), 529.  doi: 10.1007/s00348-011-1235-7.  Google Scholar

[8]

P. J. Goulart, A. Wynn and D. Pearson, Optimal mode decomposition for high dimensional systems,, In Proceedings of the 51st IEEE Conference on Decision and Control, 2012 (2012), 4965.  doi: 10.1109/CDC.2012.6426995.  Google Scholar

[9]

M. Grilli, P. J. Schmid, S. Hickel and N. A. Adams, Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction,, J. Fluid Mech., 700 (2012), 16.  doi: 10.1017/jfm.2012.37.  Google Scholar

[10]

K. Hasselmann, PIPs and POPs: The reduction of complex dynamical-systems using Principal Interaction and Oscillation Patterns,, J. Geophys. Res.-Atmos., 93 (1988), 11015.  doi: 10.1029/JD093iD09p11015.  Google Scholar

[11]

B. L. Ho and R. E. Kalman, Effective construction of linear state-variable models from input/output data,, Proceedings of the Third Annual Allerton Conference on Circuit and System Theory, (1965), 449.   Google Scholar

[12]

P. J. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge University Press, (2012).  doi: 10.1017/CBO9780511919701.  Google Scholar

[13]

H. Hotelling, Analysis of a complex of statistical variables into principal components,, J. Educ. Psychol., 24 (1933), 417.  doi: 10.1037/h0071325.  Google Scholar

[14]

H. Hotelling, Analysis of a complex of statistical variables into principal components,, J. Educ. Psychol., 24 (1933), 498.   Google Scholar

[15]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition,, Phys. Fluids, 26 (2014).  doi: 10.1063/1.4863670.  Google Scholar

[16]

J. N. Juang and R. S. Pappa, An eigensystem realization-algorithm for modal parameter-identification and model-reduction,, J. Guid. Control Dynam., 8 (1985), 620.  doi: 10.2514/3.20031.  Google Scholar

[17]

E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction,, Technical report, (1956).   Google Scholar

[18]

Z. Ma, S. Ahuja and C. W. Rowley, Reduced-order models for control of fluids using the eigensystem realization algorithm,, Theor. Comp. Fluid Dyn., 25 (2011), 233.  doi: 10.1007/s00162-010-0184-8.  Google Scholar

[19]

L. Massa, R. Kumar and P. Ravindran, Dynamic mode decomposition analysis of detonation waves,, Phys. Fluids, (2012).   Google Scholar

[20]

I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions,, Nonlin. Dynam., 41 (2005), 309.  doi: 10.1007/s11071-005-2824-x.  Google Scholar

[21]

I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator,, Annu. Rev. Fluid Mech., 45 (2013), 357.  doi: 10.1146/annurev-fluid-011212-140652.  Google Scholar

[22]

T. W. Muld, G. Efraimsson and D. S. Henningson, Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition,, Comput. Fluids, 57 (2012), 87.  doi: 10.1016/j.compfluid.2011.12.012.  Google Scholar

[23]

B. R. Noack, K. Afanasiev, M. Morzyński, G. Tadmor and F. Thiele, A hierarchy of low-dimensional models for the transient and post-transient cylinder wake,, J. Fluid Mech., 497 (2003), 335.  doi: 10.1017/S0022112003006694.  Google Scholar

[24]

B. R. Noack, G. Tadmor and M. Morzyński, Actuation models and dissipative control in empirical Galerkin models of fluid flows,, In Proceedings of the American Control Conference, (2004), 5722.   Google Scholar

[25]

K. Pearson, LIII. on lines and planes of closest fit to systems of points in space,, Philos. Mag., 2 (1901), 559.  doi: 10.1080/14786440109462720.  Google Scholar

[26]

C. Penland, Random forcing and forecasting using Principal Oscillation Pattern analysis,, Mon. Weather Rev., 117 (1989), 2165.   Google Scholar

[27]

C. Penland and T. Magorian, Prediction of Niño 3 sea-surface temperatures using linear inverse modeling,, J. Climate, 6 (1993), 1067.   Google Scholar

[28]

C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition,, Int. J. Bifurcat. Chaos, 15 (2005), 997.  doi: 10.1142/S0218127405012429.  Google Scholar

[29]

C. W. Rowley, I. Mezic, S. Bagheri, P. Schlatter and D. S. Henningson, Spectral analysis of nonlinear flows,, J. Fluid Mech., 641 (2009), 115.  doi: 10.1017/S0022112009992059.  Google Scholar

[30]

P. J. Schmid, Dynamic mode decomposition of numerical and experimental data,, J. Fluid Mech., 656 (2010), 5.  doi: 10.1017/S0022112010001217.  Google Scholar

[31]

P. J. Schmid, Application of the dynamic mode decomposition to experimental data,, Exp. Fluids, 50 (2011), 1123.  doi: 10.1007/s00348-010-0911-3.  Google Scholar

[32]

P. J. Schmid, L. Li, M. P. Juniper and O. Pust, Applications of the dynamic mode decomposition,, Theor. Comp. Fluid Dyn., 25 (2011), 249.  doi: 10.1007/s00162-010-0203-9.  Google Scholar

[33]

P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data,, Journal of Fluid Mechanics, 656 (2010), 5.  doi: 10.1017/S0022112010001217.  Google Scholar

[34]

P. J. Schmid, D. Violato and F. Scarano, Decomposition of time-resolved tomographic PIV,, Exp. Fluids, 52 (2012), 1567.  doi: 10.1007/s00348-012-1266-8.  Google Scholar

[35]

A. Seena and H. J. Sung, Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations,, Int. J. Heat Fluid Fl., 32 (2011), 1098.  doi: 10.1016/j.ijheatfluidflow.2011.09.008.  Google Scholar

[36]

O. Semeraro, G. Bellani and F. Lundell, Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes,, Exp. Fluids, 53 (2012), 1203.  doi: 10.1007/s00348-012-1354-9.  Google Scholar

[37]

J. R. Singler, Optimality of balanced proper orthogonal decomposition for data reconstruction,, Numerical Functional Analysis and Optimization, 31 (2010), 852.  doi: 10.1080/01630563.2010.500022.  Google Scholar

[38]

L. Sirovich, Turbulence and the dynamics of coherent structures. 2. Symmetries and transformations,, Q. Appl. Math., 45 (1987), 573.   Google Scholar

[39]

K. Taira and T. Colonius, The immersed boundary method: A projection approach,, J. Comput. Phys., 225 (2007), 2118.  doi: 10.1016/j.jcp.2007.03.005.  Google Scholar

[40]

L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, SIAM, (1997).  doi: 10.1137/1.9780898719574.  Google Scholar

[41]

J. H. Tu, J. Griffin, A. Hart, C. W. Rowley, L. N. Cattafesta III and L. S. Ukeiley, Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of velocity fields,, Exp. Fluids, 54 (2013).  doi: 10.2514/6.2012-33.  Google Scholar

[42]

J. H. Tu and C. W. Rowley, An improved algorithm for balanced POD through an analytic treatment of impulse response tails,, J. Comput. Phys., 231 (2012), 5317.  doi: 10.1016/j.jcp.2012.04.023.  Google Scholar

[43]

J. H. Tu, C. W. Rowley, E. Aram and R. Mittal, Koopman spectral analysis of separated flow over a finite-thickness flat plate with elliptical leading edge,, AIAA Paper 2011-38, (2011), 2011.  doi: 10.2514/6.2011-38.  Google Scholar

[44]

H. von Storch, G. Bürger, R. Schnur and J. S. von Storch, Principal oscillation patterns: A review,, J. Climate, 8 (1995), 377.   Google Scholar

[45]

M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: extending dynamic mode decomposition,, , (2014).   Google Scholar

[46]

A. Wynn, D. Pearson, B. Ganapathisubramani and P. J. Goulart, Optimal mode decomposition for unsteady flows,, J. Fluid Mech., 733 (2013), 473.  doi: 10.1017/jfm.2013.426.  Google Scholar

[1]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, Stock price fluctuation prediction method based on time series analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-915. doi: 10.3934/dcdss.2019061

[2]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[3]

P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178

[4]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[5]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[6]

Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021003

[7]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

[8]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[9]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[10]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[11]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281

[12]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354

[13]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156

[14]

Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085

[15]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[16]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[17]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331

[18]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[19]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111

[20]

Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097

 Impact Factor: 

Metrics

  • PDF downloads (2495)
  • HTML views (0)
  • Cited by (308)

[Back to Top]