Abstract
This paper introduces the trajectory divergence rate, a scalar field which locally gives the instantaneous attraction or repulsion of adjacent trajectories. This scalar field may be used to find highly attracting or repelling invariant manifolds, such as slow manifolds, to rapidly approximate hyperbolic Lagrangian coherent structures, or to provide the local stability of invariant manifolds. This work presents the derivation of the trajectory divergence rate and the related trajectory divergence ratio for two-dimensional systems, investigates their properties, shows their application to several example systems, and presents their extension to higher dimensions.
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Aldridge, B.B., Haller, G., Sorger, P.K., Lauffenburger, D.A.: Direct Lyapunov exponent analysis enables parametric study of transient signalling governing cell behaviour. IEE Proc. Syst. Biol. 153(6), 425–432 (2006)
Ali, F., Menzinger, M.: On the local stability of limit cycles. Chaos Interdiscip. J. Nonlinear Sci. 9(2), 348–356 (1999)
Allshouse, M.R., Thiffeault, J.L.: Detecting coherent structures using braids. Physica D 241(2), 95–105 (2012)
Ameli, S., Desai, Y., Shadden, S.C.: Development of an efficient and flexible pipeline for Lagrangian coherent structure computation. In: Bremer, P.-T., Hotz, I., Pascucci, V., Peikert, R. (eds.) Topological Methods in Data Analysis and Visualization III, pp. 201–215. Springer (2014)
Balasuriya, S., Ouellette, N.T., Rypina, I.I.: Generalized Lagrangian coherent structures. Physica D 372, 31–51 (2018)
Brunton, S.L., Rowley, C.W.: Fast computation of finite-time Lyapunov exponent fields for unsteady flows. Chaos Interdiscip. J. Nonlinear Sci. 20(1), 017–503 (2010)
Budišić, M., Thiffeault, J.L.: Finite-time braiding exponents. Chaos Interdiscip. J. Nonlinear Sci. 25(8), 087,407 (2015)
Chakraborty, P., Balachandar, S., Adrian, R.J.: On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214 (2005)
De Dominicis, M., Falchetti, S., Trotta, F., Pinardi, N., Giacomelli, L., Napolitano, E., Fazioli, L., Sorgente, R., Haley Jr., P.J., Lermusiaux, P.F., et al.: A relocatable ocean model in support of environmental emergencies. Ocean Dyn. 64(5), 667–688 (2014)
Dellnitz, M., Froyland, G., Junge, O.: The algorithms behind GAIO - set oriented numerical methods for dynamical systems. In: Fiedler, B. (ed.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 145–174. Springer (2001)
Dellnitz, M., Junge, O., Lo, M.W., Marsden, J.E., Padberg, K., Preis, R., Ross, S.D., Thiere, B.: Transport of mars-crossing asteroids from the quasi-hilda region. Phys. Rev. Lett. 94, 231,102 (2005)
Desroches, M., Jeffrey, M.R.: Canards and curvature: the smallness of \(\varepsilon \) in slow–fast dynamics. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, p. rspa20110053. The Royal Society (2011)
Froyland, G., Padberg-Gehle, K.: A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data. Chaos Interdiscip. J. Nonlinear Sci. 25(8), 087,406 (2015)
Garth, C., Gerhardt, F., Tricoche, X., Hans, H.: Efficient computation and visualization of coherent structures in fluid flow applications. IEEE Trans. Vis. Comput. Gr. 13(6), 1464–1471 (2007)
Gawlik, E.S., Marsden, J.E., Du Toit, P.C., Campagnola, S.: Lagrangian coherent structures in the planar elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 103(3), 227–249 (2009)
Green, M.A., Rowley, C.W., Smits, A.J.: Using hyperbolic Lagrangian coherent structures to investigate vortices in bioinspired fluid flows. Chaos Interdiscip. J. Nonlinear Sci. 20(1), 017,510 (2010)
Hadjighasem, A., Farazmand, M., Blazevski, D., Froyland, G., Haller, G.: A critical comparison of Lagrangian methods for coherent structure detection. Chaos Interdiscip. J. Nonlinear Sci. 27(5), 053,104 (2017)
Haley, P.J., Lermusiaux, P.F.: Multiscale two-way embedding schemes for free-surface primitive equations in the multidisciplinary simulation, estimation and assimilation system. Ocean Dyn. 60(6), 1497–1537 (2010)
Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13(11), 3365–3385 (2001)
Haller, G.: A variational theory of hyperbolic Lagrangian coherent structures. Physica D 240(7), 574–598 (2011)
Haller, G., Sapsis, T.: Localized instability and attraction along invariant manifolds. SIAM J. Appl. Dyn. Syst. 9(2), 611–633 (2010)
Kai, E.T., Rossi, V., Sudre, J., Weimerskirch, H., Lopez, C., Hernandez-Garcia, E., Marsac, F., Garçon, V.: Top marine predators track Lagrangian coherent structures. Proc. Natl. Acad. Sci. 106(20), 8245–8250 (2009)
Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical Systems, the Three-Body Problem and Space Mission Design. Marsden Books, Pasadena (2011). (ISBN: 978-0-615-24095-4)
Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)
Kuehn, C.: Multiple Time Scale Dynamics, vol. 1. Springer, Berlin (2016)
Lekien, F., Coulliette, C., Mariano, A.J., Ryan, E.H., Shay, L.K., Haller, G., Marsden, J.: Pollution release tied to invariant manifolds: a case study for the coast of Florida. Physica D 210(1–2), 1–20 (2005)
Lekien, F., Ross, S.D.: The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds. Chaos Interdiscip. J. Nonlinear Sci. 20(1), 017,505 (2010)
Lopesino, C., Balibrea-Iniesta, F., García-Garrido, V.J., Wiggins, S., Mancho, A.M.: A theoretical framework for Lagrangian descriptors. Int. J. Bifurc. Chaos 27(01), 1730,001 (2017)
Madrid, J.J., Mancho, A.M.: Distinguished trajectories in time dependent vector fields. Chaos Interdiscip. J. Nonlinear Sci. 19(1), 013,111 (2009)
Nave Jr., G.K., Ross, S.D.: Global phase space structures in a model of passive descent. Commun. Nonlinear Sci. Numer. Simul., Under Review arXiv:1804.05099 (2019)
Norris, J.A., Marsh, A.P., Granata, K.P., Ross, S.D.: Revisiting the stability of 2D passive biped walking: local behavior. Physica D 237(23), 3038–3045 (2008)
Peng, J., Dabiri, J.O.: The ‘upstream wake’ of swimming and flying animals and its correlation with propulsive efficiency. J. Exp. Biol. 211(16), 2669–2677 (2008)
Schindler, B., Peikert, R., Fuchs, R., Theisel, H.: Ridge concepts for the visualization of Lagrangian coherent structures. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds.) Topological Methods in Data Analysis and Visualization II, pp. 221–235. Springer (2012)
Schmale III, D.G., Ross, S.D.: Highways in the sky: Scales of atmospheric transport of plant pathogens. Ann. Rev. Phytopathol. 53, 591–611 (2015)
Serra, M., Haller, G.: Objective Eulerian coherent structures. Chaos Interdiscip. J. Nonlinear Sci. 26(5), 110 (2016)
Shadden, S.C.: Lagrangian coherent structures. In: Grigoriev, R. (ed.) Transport and Mixing in Laminar Flows: From Microfluidics to Oceanic Currents, pp. 59–89. John Wiley & Sons, Ltd. (2011)
Shadden, S.C., Lekien, F., Marsden, J.E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212(3), 271–304 (2005)
Strogatz, S.H.: Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2014)
Tallapragada, P., Sudarsanam, S.: A globally stable attractor that is locally unstable everywhere. AIP Adv. 7(12), 125,012 (2017)
Tanaka, M.L., Ross, S.D., Nussbaum, M.A.: Mathematical modeling and simulation of seated stability. J. Biomech. 43(5), 906–912 (2010)
Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics. Springer, Berlin (2004)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, Berlin (2003)
Wiggins, S.: The dynamical systems approach to Lagrangian transport in oceanic flows. Ann. Rev. Fluid Mech. 37, 295–328 (2005)
Wiggins, S.: Normally Hyperbolic Invariant Manifolds in Dynamical Systems, vol. 105. Springer, Berlin (2013)
Wiggins, S., Wiesenfeld, L., Jaffé, C., Uzer, T.: Impenetrable barriers in phase-space. Phys. Rev. Lett. 86, 5478–5481 (2001)
Xie, X., Nolan, P., Ross, S., Iliescu, T.: Lagrangian data-driven reduced order modeling of finite time Lyapunov exponents. arXiv:1808.05635 (2018)
Yeaton, I.J., Socha, J.J., Ross, S.D.: Global dynamics of non-equilibrium gliding in animals. Bioinspir. Biomim. 12(2), 026,013 (2017)
Zhong, J., Virgin, L.N., Ross, S.D.: A tube dynamics perspective governing stability transitions: an example based on snap-through buckling. Int. J. Mech. Sci. 149, 413–428 (2018)
Acknowledgements
This work was supported by National Science Foundations Grants Division of Atmospheric and Geospace Sciences (Grant No. 1520825) Division of Civil, Mechanical and Manufacturing Innovation (Grant No. 1537349) and Division of Mathematical Sciences (Grant No. 1821145) and by the Biological Transport (BioTrans) Interdisciplinary Graduate Education Program at Virginia Tech. We thank Pierre Lermusiaux, P.J. Haley, and the MIT-MSEAS team for providing the MSEAS model data.
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Appendix: Derivation of Eq. (17)
Appendix: Derivation of Eq. (17)
Starting with (15), the trajectory-normal repulsion rate, \(\rho _T\) can be written, to leading order in T, as,
For a 2-tensor, \({\mathbf {A}}\), the relation \(\text {tr}({\mathbf {A}}){\mathbf {I}}-{\mathbf {A}}\) in index notation may be written as \(A_{ii}\delta _{jk} - A_{jk}\).
where \(\varepsilon _{ij}\) is the two-dimensional Levi-Civita symbol which, for a 2x2 matrix, is the index representation of the negative of the \(90^\circ \) counter-clockwise rotation matrix, \(\varepsilon _{ij} = -{\mathbf {R}}\). Therefore, for small time T, \(\rho _T\) may be written as,
which can alternatively be written in terms of the unit normal field, \({\mathbf {n}} = {\mathbf {R}}{\mathbf {v}}/\left| {\mathbf {v}}\right| \), as in (3), yielding
which gives the leading order behavior defined by the instantaneous rate,
Note that the rate of length change for an infinitesimal material element vector \(\ell \) based at \({\mathbf {x}}_0\) and advected under the flow is
Thus, the leading order behavior of the trajectory-normal repulsion rate for short time \(T\) can be thought of as the rate of stretching of unit normal vectors, normal to the invariant manifold passing through \({\mathbf {x}}_0\). This value is locally maximized along the most repulsive (or attractive) manifolds, which provide the most influential core of phase space deformation patterns.
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Nave, G.K., Nolan, P.J. & Ross, S.D. Trajectory-free approximation of phase space structures using the trajectory divergence rate. Nonlinear Dyn 96, 685–702 (2019). https://doi.org/10.1007/s11071-019-04814-z
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DOI: https://doi.org/10.1007/s11071-019-04814-z