December  2020, 7(2): 369-399. doi: 10.3934/jcd.2020015

A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems

School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

* Corresponding author: uqcblach@uq.edu.au

Received  October 2019 Published  July 2020

Fund Project: This work has been partially supported by an Australian Research Council Discovery Early Career Researcher Award (DE160100147) and by an Australian Government Research Training Program Stipend Scholarship (CB)

Coherent structures are spatially varying regions which disperse minimally over time and organise motion in non-autonomous systems. This work develops and implements algorithms providing multilayered descriptions of time-dependent systems which are not only useful for locating coherent structures, but also for detecting time windows within which these structures undergo fundamental structural changes, such as merging and splitting events. These algorithms rely on singular value decompositions associated to Ulam type discretisations of transfer operators induced by dynamical systems, and build on recent developments in multiplicative ergodic theory. Furthermore, they allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. The approach is tested on models of periodically and quasi-periodically driven systems, as well as on a geophysical dataset corresponding to the splitting of the Southern Polar Vortex.

Citation: Chantelle Blachut, Cecilia González-Tokman. A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems. Journal of Computational Dynamics, 2020, 7 (2) : 369-399. doi: 10.3934/jcd.2020015
References:
[1]

M. R. Allshouse and T. Peacock, Lagrangian based methods for coherent structure detection, Chaos, 25 (2015), 13pp. doi: 10.1063/1.4922968.  Google Scholar

[2]

S. BalasuriyaN. T. Ouellette and I. I. Rypina, Generalized Lagrangian coherent structures, Phys. D, 372 (2018), 31-51.  doi: 10.1016/j.physd.2018.01.011.  Google Scholar

[3]

R. Banisch and P. Koltai, Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets, Chaos, 27 (2017), 16pp. doi: 10.1063/1.4971788.  Google Scholar

[4]

M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos, 22 (2012), 33pp. doi: 10.1063/1.4772195.  Google Scholar

[5]

A. J. CharltonA. O'NeillW. A. Lahoz and P. Berrisford, The splitting of the stratospheric polar vortex in the Southern Hemisphere, September 2002: Dynamical evolution, J. Atmospheric Sci., 62 (2005), 590-602.  doi: 10.1175/JAS-3318.1.  Google Scholar

[6]

D. Dee, S. Uppala, A. Simmons, P. Berrisford and P. Poli, et al., The ERA-Interim reanalysis: Configuration and performance of the data assimilation system, Quarterly J. Roy. Meteorological Soc., 137 (2011), 553–597. doi: 10.1002/qj.828.  Google Scholar

[7]

M. DellnitzG. FroylandC. Horenkamp and K. Padberg, On the approximation of transport phenomena—A dynamical systems approach, GAMM-Mitt., 32 (2009), 47-60.  doi: 10.1002/gamm.200910004.  Google Scholar

[8]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, (2001), 145–174,805–807.  Google Scholar

[9]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515.  doi: 10.1137/S0036142996313002.  Google Scholar

[10]

P. Deuflhard, M. Dellnitz, O. Junge and C. Schütte, Computation of essential molecular dynamics by subdivision techniques, in Computational Molecular Dynamics: Challenges, Methods, Ideas, Lecture Notes in Computational Science and Engineering, 4, Springer, Berlin, Heidelberg, 1999, 98–115. doi: 10.1007/978-3-642-58360-5_5.  Google Scholar

[11]

G. FroylandT. HülsG. P. Morriss and T. M. Watson, Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Phys. D, 247 (2013), 18-39.  doi: 10.1016/j.physd.2012.12.005.  Google Scholar

[12]

G. FroylandS. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.  doi: 10.1017/S0143385709000339.  Google Scholar

[13]

G. FroylandS. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860.  doi: 10.3934/dcds.2013.33.3835.  Google Scholar

[14]

G. FroylandS. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541.  doi: 10.1016/j.physd.2010.03.009.  Google Scholar

[15]

G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds—Connecting probabilistic and geometric descriptions of coherent structures in flows, Phys. D, 238 (2009), 1507-1523.  doi: 10.1016/j.physd.2009.03.002.  Google Scholar

[16]

G. Froyland, K. Padberg, M. H. England and A. M. Treguier, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007). doi: 10.1103/PhysRevLett.98.224503.  Google Scholar

[17]

G. Froyland and K. Padberg-Gehle, Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Math. Stat., Springer, New York, 70 (2014), 171–216. doi: 10.1007/978-1-4939-0419-8_9.  Google Scholar

[18]

G. FroylandC. P. Rock and K. Sakellariou, Sparse eigenbasis approximation: Multiple feature extraction across spatiotemporal scales with application to coherent set identification, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 81-107.  doi: 10.1016/j.cnsns.2019.04.012.  Google Scholar

[19]

G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 10pp. doi: 10.1063/1.3502450.  Google Scholar

[20]

F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi and A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett., 99 (2007). doi: 10.1103/PhysRevLett.99.130601.  Google Scholar

[21]

C. González-Tokman, Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems, in Contributions of Mexican Mathematicians Abroad in Pure and Applied Mathematics, Contemp. Math., Amer. Math. Soc., Providence, RI, 709 (2018), 31–52. doi: 10.1090/conm/709/14290.  Google Scholar

[22]

C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.  doi: 10.1017/etds.2012.189.  Google Scholar

[23]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn., 9 (2015), 237-255.  doi: 10.3934/jmd.2015.9.237.  Google Scholar

[24]

G. Haller, Lagrangian coherent structures, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 47 (2015), 137–162. doi: 10.1146/annurev-fluid-010313-141322.  Google Scholar

[25]

G. HallerD. Karrasch and F. Kogelbauer, Material barriers to diffusive and stochastic transport, Proc. Natl. Acad. Sci. USA, 115 (2018), 9074-9079.  doi: 10.1073/pnas.1720177115.  Google Scholar

[26]

B. Joseph and B. Legras, Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex, J. Atmospheric Sci., 59 (2002), 1198-1212.  doi: 10.1175/1520-0469(2002)059<1198:RBKBSA>2.0.CO; 2.  Google Scholar

[27]

S. KlusP. Koltai and C. Schütte, On the numerical approximation of the Perron-Frobenius and Koopman operator, J. Comput. Dyn., 3 (2016), 51-79.  doi: 10.3934/jcd.2016003.  Google Scholar

[28]

P. Koltai and D. R. M. Renger, From large deviations to semidistances of transport and mixing: Coherence analysis for finite Lagrangian data, J. Nonlinear Sci., 28 (2018), 1915-1957.  doi: 10.1007/s00332-018-9471-0.  Google Scholar

[29]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Applied Mathematical Sciences, 97, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[30]

F. Lekien and S. D. Ross, The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds, Chaos, 20 (2010), 20pp. doi: 10.1063/1.3278516.  Google Scholar

[31]

B. A. Mosovsky and J. D. Meiss, Transport in transitory dynamical systems, SIAM J. Appl. Dyn. Syst., 10 (2011), 35-65.  doi: 10.1137/100794110.  Google Scholar

[32]

M. Ndour and K. Padberg-Gehle, Predicting bifurcations of almost-invariant patterns: A set-oriented approach, preprint, arXiv: 2001.01099. Google Scholar

[33]

P. Newman and E. Nash, The unusual Southern Hemisphere stratosphere winter of 2002, J. Atmospheric Sci., 62 (2005), 614-628.  doi: 10.1175/JAS-3323.1.  Google Scholar

[34]

F. Noethen, A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors, Phys. D, 396 (2019), 18-34.  doi: 10.1016/j.physd.2019.02.012.  Google Scholar

[35]

A. O'NeillC. L. OatleyA. J. Charlton–PerezD. M. Mitchell and T. Jung, Vortex splitting on a planetary scale in the stratosphere by cyclogenesis on a subplanetary scale in the troposphere, Quarterly J. Roy. Meteorological Soc., 143 (2017), 691-705.  doi: 10.1002/qj.2957.  Google Scholar

[36]

Y. J. OrsoliniC. E. RandallG. L. Manney and D. R. Allen, An observational study of the final breakdown of the Southern Hemisphere stratospheric vortex in 2002, J. Atmospheric Sci., 62 (2005), 735-747.  doi: 10.1175/JAS-3315.1.  Google Scholar

[37]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.   Google Scholar

[38]

K. Padberg-Gehle, S. Reuther, S. Praetorius and A. Voigt, Transfer operator-based extraction of coherent features on surfaces, in Topological Methods in Data Analysis and Visualization. IV, Math. Vis., Springer, Cham, (2017), 283–297. doi: 10.1007/978-3-319-44684-4_17.  Google Scholar

[39]

M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.  doi: 10.1007/BF02760464.  Google Scholar

[40]

S. C. ShaddenF. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Phys. D, 212 (2005), 271-304.  doi: 10.1016/j.physd.2005.10.007.  Google Scholar

[41]

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London, 1960.  Google Scholar

[42]

M. O. WilliamsI. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.  doi: 10.1007/s00332-015-9258-5.  Google Scholar

show all references

References:
[1]

M. R. Allshouse and T. Peacock, Lagrangian based methods for coherent structure detection, Chaos, 25 (2015), 13pp. doi: 10.1063/1.4922968.  Google Scholar

[2]

S. BalasuriyaN. T. Ouellette and I. I. Rypina, Generalized Lagrangian coherent structures, Phys. D, 372 (2018), 31-51.  doi: 10.1016/j.physd.2018.01.011.  Google Scholar

[3]

R. Banisch and P. Koltai, Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets, Chaos, 27 (2017), 16pp. doi: 10.1063/1.4971788.  Google Scholar

[4]

M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos, 22 (2012), 33pp. doi: 10.1063/1.4772195.  Google Scholar

[5]

A. J. CharltonA. O'NeillW. A. Lahoz and P. Berrisford, The splitting of the stratospheric polar vortex in the Southern Hemisphere, September 2002: Dynamical evolution, J. Atmospheric Sci., 62 (2005), 590-602.  doi: 10.1175/JAS-3318.1.  Google Scholar

[6]

D. Dee, S. Uppala, A. Simmons, P. Berrisford and P. Poli, et al., The ERA-Interim reanalysis: Configuration and performance of the data assimilation system, Quarterly J. Roy. Meteorological Soc., 137 (2011), 553–597. doi: 10.1002/qj.828.  Google Scholar

[7]

M. DellnitzG. FroylandC. Horenkamp and K. Padberg, On the approximation of transport phenomena—A dynamical systems approach, GAMM-Mitt., 32 (2009), 47-60.  doi: 10.1002/gamm.200910004.  Google Scholar

[8]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, (2001), 145–174,805–807.  Google Scholar

[9]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515.  doi: 10.1137/S0036142996313002.  Google Scholar

[10]

P. Deuflhard, M. Dellnitz, O. Junge and C. Schütte, Computation of essential molecular dynamics by subdivision techniques, in Computational Molecular Dynamics: Challenges, Methods, Ideas, Lecture Notes in Computational Science and Engineering, 4, Springer, Berlin, Heidelberg, 1999, 98–115. doi: 10.1007/978-3-642-58360-5_5.  Google Scholar

[11]

G. FroylandT. HülsG. P. Morriss and T. M. Watson, Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Phys. D, 247 (2013), 18-39.  doi: 10.1016/j.physd.2012.12.005.  Google Scholar

[12]

G. FroylandS. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.  doi: 10.1017/S0143385709000339.  Google Scholar

[13]

G. FroylandS. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860.  doi: 10.3934/dcds.2013.33.3835.  Google Scholar

[14]

G. FroylandS. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541.  doi: 10.1016/j.physd.2010.03.009.  Google Scholar

[15]

G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds—Connecting probabilistic and geometric descriptions of coherent structures in flows, Phys. D, 238 (2009), 1507-1523.  doi: 10.1016/j.physd.2009.03.002.  Google Scholar

[16]

G. Froyland, K. Padberg, M. H. England and A. M. Treguier, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007). doi: 10.1103/PhysRevLett.98.224503.  Google Scholar

[17]

G. Froyland and K. Padberg-Gehle, Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Math. Stat., Springer, New York, 70 (2014), 171–216. doi: 10.1007/978-1-4939-0419-8_9.  Google Scholar

[18]

G. FroylandC. P. Rock and K. Sakellariou, Sparse eigenbasis approximation: Multiple feature extraction across spatiotemporal scales with application to coherent set identification, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 81-107.  doi: 10.1016/j.cnsns.2019.04.012.  Google Scholar

[19]

G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 10pp. doi: 10.1063/1.3502450.  Google Scholar

[20]

F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi and A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett., 99 (2007). doi: 10.1103/PhysRevLett.99.130601.  Google Scholar

[21]

C. González-Tokman, Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems, in Contributions of Mexican Mathematicians Abroad in Pure and Applied Mathematics, Contemp. Math., Amer. Math. Soc., Providence, RI, 709 (2018), 31–52. doi: 10.1090/conm/709/14290.  Google Scholar

[22]

C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.  doi: 10.1017/etds.2012.189.  Google Scholar

[23]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn., 9 (2015), 237-255.  doi: 10.3934/jmd.2015.9.237.  Google Scholar

[24]

G. Haller, Lagrangian coherent structures, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 47 (2015), 137–162. doi: 10.1146/annurev-fluid-010313-141322.  Google Scholar

[25]

G. HallerD. Karrasch and F. Kogelbauer, Material barriers to diffusive and stochastic transport, Proc. Natl. Acad. Sci. USA, 115 (2018), 9074-9079.  doi: 10.1073/pnas.1720177115.  Google Scholar

[26]

B. Joseph and B. Legras, Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex, J. Atmospheric Sci., 59 (2002), 1198-1212.  doi: 10.1175/1520-0469(2002)059<1198:RBKBSA>2.0.CO; 2.  Google Scholar

[27]

S. KlusP. Koltai and C. Schütte, On the numerical approximation of the Perron-Frobenius and Koopman operator, J. Comput. Dyn., 3 (2016), 51-79.  doi: 10.3934/jcd.2016003.  Google Scholar

[28]

P. Koltai and D. R. M. Renger, From large deviations to semidistances of transport and mixing: Coherence analysis for finite Lagrangian data, J. Nonlinear Sci., 28 (2018), 1915-1957.  doi: 10.1007/s00332-018-9471-0.  Google Scholar

[29]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Applied Mathematical Sciences, 97, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[30]

F. Lekien and S. D. Ross, The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds, Chaos, 20 (2010), 20pp. doi: 10.1063/1.3278516.  Google Scholar

[31]

B. A. Mosovsky and J. D. Meiss, Transport in transitory dynamical systems, SIAM J. Appl. Dyn. Syst., 10 (2011), 35-65.  doi: 10.1137/100794110.  Google Scholar

[32]

M. Ndour and K. Padberg-Gehle, Predicting bifurcations of almost-invariant patterns: A set-oriented approach, preprint, arXiv: 2001.01099. Google Scholar

[33]

P. Newman and E. Nash, The unusual Southern Hemisphere stratosphere winter of 2002, J. Atmospheric Sci., 62 (2005), 614-628.  doi: 10.1175/JAS-3323.1.  Google Scholar

[34]

F. Noethen, A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors, Phys. D, 396 (2019), 18-34.  doi: 10.1016/j.physd.2019.02.012.  Google Scholar

[35]

A. O'NeillC. L. OatleyA. J. Charlton–PerezD. M. Mitchell and T. Jung, Vortex splitting on a planetary scale in the stratosphere by cyclogenesis on a subplanetary scale in the troposphere, Quarterly J. Roy. Meteorological Soc., 143 (2017), 691-705.  doi: 10.1002/qj.2957.  Google Scholar

[36]

Y. J. OrsoliniC. E. RandallG. L. Manney and D. R. Allen, An observational study of the final breakdown of the Southern Hemisphere stratospheric vortex in 2002, J. Atmospheric Sci., 62 (2005), 735-747.  doi: 10.1175/JAS-3315.1.  Google Scholar

[37]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.   Google Scholar

[38]

K. Padberg-Gehle, S. Reuther, S. Praetorius and A. Voigt, Transfer operator-based extraction of coherent features on surfaces, in Topological Methods in Data Analysis and Visualization. IV, Math. Vis., Springer, Cham, (2017), 283–297. doi: 10.1007/978-3-319-44684-4_17.  Google Scholar

[39]

M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.  doi: 10.1007/BF02760464.  Google Scholar

[40]

S. C. ShaddenF. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Phys. D, 212 (2005), 271-304.  doi: 10.1016/j.physd.2005.10.007.  Google Scholar

[41]

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London, 1960.  Google Scholar

[42]

M. O. WilliamsI. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.  doi: 10.1007/s00332-015-9258-5.  Google Scholar

Figure 1.  Figures 1a and 1b show almost invariant structures as described by the (evolved) subdominant eigenvector of an Ulam matrix approximation to the transfer operator in the periodically driven double gyre flow, with parameters as in [40]. Figures 1c and 1d show finite-time coherent structures as described by the (evolved) subdominant initial time singular vector of a composition of $ 10 $ Ulam matrices describing the evolution of the transitory double gyre flow introduced by [31]. See [17] for a thorough discussion of both models
Figure 2.  Evolution in non-autonomous dynamical systems: driving system (above the arrow), particle evolution (2nd row), transfer operators (3rd row) and Ulam's method (bottom row)
Figure 3.  Selected vector field instances for the periodically forced double well potential
Figure 10.  An illustration of the behaviour of $ \alpha(t) $ and $ \tilde{\alpha}(t) $ over $ 5 $ periods
Figure 4.  Tracking modes over rolling windows for the periodically forced double well potential
Figure 5.  Tracking modes for time windows of length $ n = 50 $, evolved using Algorithm 4
Figure 6.  Crossing introduced by shifting from $ n = 54 $ to $ n = 51 $ for the periodically forced double well potential
Figure 7.  Equivariance mismatch for the periodically forced double well potential when $ n = 50 $
Figure 8.  Leading $ 6 $ of $ \mathcal{N} = 6 $ modes for the periodically forced double well potential when $ n = 50 $
Figure 9.  Leading $ 4 $ of $ \mathcal{N} = 4 $ modes for the periodically forced double well potential when $ n = 100 $
Figure 11.  Mean equivariance mismatch, as per Algorithm 5, for the leading $ 4 $ of $ \mathcal{N} $ modes using the two pairing methods given by Algorithms 2 ($ \bar{\varsigma}_{S} $) and 3 ($ \bar{\varsigma}_{U} $) for $ n = 50 $
Figure 12.  Leading $ 4 $ from a total $ \mathcal{N} = 5 $ tracked modes for $ n = 50 $ using Algorithms 3 (top) and 5 (bottom)
Figure 13.  Leading $ 4 $ from a total $ \mathcal{N} = 7 $ tracked modes for $ n = 50 $ using Algorithms 3 (top) and 5 (bottom)
Figure 14.  Consecutive windows corresponding to reasonable equivariance for $ S_{U}^{(4)} $ of Figure 12
Figure 15.  Initial time singular vectors corresponding to rolling windows initialised at the various $ t_{0} $ indicated by colour coded bars and column headings. These are paired according to the paths illustrated in Figure 12
Figure 16.  Evolved $ u^{(50)}_{75,4} $ (top) and $ u^{(50)}_{274,4} $ (bottom) of mode $ S_{U}^{(4)} $ in Figure 15, evolved as per Algorithm 4
Figure 17.  Southern hemisphere wind speed (easterly and northerly) on the $ 850 $ K isentropic surface
Figure 18.  Mean equivariance mismatch, as per Algorithm 5, for the leading $ 3 $ of $ \mathcal{N} $ modes using the two pairing methods given in Algorithms 2 and 3 with $ n = 56 $ and $ t_0 \in[0000 \: 1 \: \text{August}, 1800 \: 30 \: \text{September}] $. Here the Ulam matrices, describing transitions for the area south of $ 30^{\circ} $S, are of dimension $ m \times m $ for $ m = 2^{14} $
Figure 19.  Leading $ 3 $ of $ \mathcal{N} = 3 $ tracked paths of singular values of rolling windows paired using Algorithm 2 for $ n = 56 $
Figure 20.  Leading singular vectors, for various $ t_{0} $, of matrix compositions associated with Figure 19b where time windows are of length $ n = 56 $. The area illustrated is south of $ 50^{\circ} $S and the time given in the label is the relevant $ t_{0} $ for that window
Figure 21.  Evolved leading mode associated with Figure 19a for a time window centred on the peak at $ 1800 $ on $ 23 $ Sep. This is illustrated on the area south of $ 15^{\circ} $S
Figure 22.  Evolved subdominant mode associated with Figure 19b for a time window centred on the peak at $ 0600 $ on $ 24 $ Sep. This is illustrated on the area south of $ 15^{\circ} $S
Figure 23.  Evolved leading singular vectors for time windows centred at $ 0000 $ on $ 24 $ Sep. for $ m = 12,800 $ initially seeded bins whose centres are south of $ 20^{\circ} $S. This is illustrated on the full southern hemisphere
Figure 24.  Evolved subdominant mode normalised as in [19] for time windows centred at $ 0000 $ on $ 24 $ Sep. for $ m = 12,800 $ initially seeded bins whose centres are south of $ 20^{\circ} $S. This is illustrated on the full southern hemisphere
[1]

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