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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[17]

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

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[40]

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

[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.

[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.

[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.

[44]

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

[45]

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

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[17]

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

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[40]

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

[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.

[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.

[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.

[44]

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

[45]

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

[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.

[1]

Luigi C. Berselli, Tae-Yeon Kim, Leo G. Rebholz. Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027

[2]

Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Addendum to "Optimal control of multiscale systems using reduced-order models". Journal of Computational Dynamics, 2017, 4 (1&2) : 167-167. doi: 10.3934/jcd.2017006

[3]

Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Optimal control of multiscale systems using reduced-order models. Journal of Computational Dynamics, 2014, 1 (2) : 279-306. doi: 10.3934/jcd.2014.1.279

[4]

Matthew O. Williams, Clarence W. Rowley, Ioannis G. Kevrekidis. A kernel-based method for data-driven koopman spectral analysis. Journal of Computational Dynamics, 2015, 2 (2) : 247-265. doi: 10.3934/jcd.2015005

[5]

Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz. Compressed sensing and dynamic mode decomposition. Journal of Computational Dynamics, 2015, 2 (2) : 165-191. doi: 10.3934/jcd.2015002

[6]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665

[7]

Heikki Haario, Leonid Kalachev, Marko Laine. Reduction and identification of dynamic models. Simple example: Generic receptor model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 417-435. doi: 10.3934/dcdsb.2013.18.417

[8]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817

[9]

Rafael Tiedra De Aldecoa. Spectral analysis of time changes of horocycle flows. Journal of Modern Dynamics, 2012, 6 (2) : 275-285. doi: 10.3934/jmd.2012.6.275

[10]

Rakesh Pilkar, Erik M. Bollt, Charles Robinson. Empirical mode decomposition/Hilbert transform analysis of postural responses to small amplitude anterior-posterior sinusoidal translations of varying frequencies. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1085-1097. doi: 10.3934/mbe.2011.8.1085

[11]

Annalisa Pascarella, Alberto Sorrentino, Cristina Campi, Michele Piana. Particle filtering, beamforming and multiple signal classification for the analysis of magnetoencephalography time series: a comparison of algorithms. Inverse Problems & Imaging, 2010, 4 (1) : 169-190. doi: 10.3934/ipi.2010.4.169

[12]

Zhendong Luo. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1189-1212. doi: 10.3934/dcdsb.2015.20.1189

[13]

Chuang Peng. Minimum degrees of polynomial models on time series. Conference Publications, 2005, 2005 (Special) : 720-729. doi: 10.3934/proc.2005.2005.720

[14]

Siwei Yu, Jianwei Ma, Stanley Osher. Geometric mode decomposition. Inverse Problems & Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035

[15]

Jianquan Li, Zhien Ma, Fred Brauer. Global analysis of discrete-time SI and SIS epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (4) : 699-710. doi: 10.3934/mbe.2007.4.699

[16]

Jiaxu Li, Yang Kuang, Bingtuan Li. Analysis of IVGTT glucose-insulin interaction models with time delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 103-124. doi: 10.3934/dcdsb.2001.1.103

[17]

József Z. Farkas, Gary T. Smith, Glenn F. Webb. A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1203-1224. doi: 10.3934/mbe.2018055

[18]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003

[19]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1

[20]

Vu Hoang Linh, Volker Mehrmann. Spectral analysis for linear differential-algebraic equations. Conference Publications, 2011, 2011 (Special) : 991-1000. doi: 10.3934/proc.2011.2011.991

 Impact Factor: 

Metrics

  • PDF downloads (147)
  • HTML views (0)
  • Cited by (99)

[Back to Top]