Nonlinear Dynamics

, Volume 96, Issue 1, pp 685–702 | Cite as

Trajectory-free approximation of phase space structures using the trajectory divergence rate

  • Gary K. NaveJr.Email author
  • Peter J. Nolan
  • Shane D. Ross
Original Paper


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.


Vector fields Phase space structure Computational geometry Normally hyperbolic invariant manifolds 



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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Ali, F., Menzinger, M.: On the local stability of limit cycles. Chaos Interdiscip. J. Nonlinear Sci. 9(2), 348–356 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allshouse, M.R., Thiffeault, J.L.: Detecting coherent structures using braids. Physica D 241(2), 95–105 (2012)CrossRefGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Balasuriya, S., Ouellette, N.T., Rypina, I.I.: Generalized Lagrangian coherent structures. Physica D 372, 31–51 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Budišić, M., Thiffeault, J.L.: Finite-time braiding exponents. Chaos Interdiscip. J. Nonlinear Sci. 25(8), 087,407 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chakraborty, P., Balachandar, S., Adrian, R.J.: On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13(11), 3365–3385 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Haller, G.: A variational theory of hyperbolic Lagrangian coherent structures. Physica D 240(7), 574–598 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Haller, G., Sapsis, T.: Localized instability and attraction along invariant manifolds. SIAM J. Appl. Dyn. Syst. 9(2), 611–633 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)CrossRefGoogle Scholar
  23. 23.
    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)zbMATHGoogle Scholar
  24. 24.
    Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kuehn, C.: Multiple Time Scale Dynamics, vol. 1. Springer, Berlin (2016)zbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Madrid, J.J., Mancho, A.M.: Distinguished trajectories in time dependent vector fields. Chaos Interdiscip. J. Nonlinear Sci. 19(1), 013,111 (2009)MathSciNetCrossRefGoogle Scholar
  30. 30.
    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)
  31. 31.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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)CrossRefGoogle Scholar
  33. 33.
    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)Google Scholar
  34. 34.
    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)CrossRefGoogle Scholar
  35. 35.
    Serra, M., Haller, G.: Objective Eulerian coherent structures. Chaos Interdiscip. J. Nonlinear Sci. 26(5), 110 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    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)Google Scholar
  37. 37.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2014)zbMATHGoogle Scholar
  39. 39.
    Tallapragada, P., Sudarsanam, S.: A globally stable attractor that is locally unstable everywhere. AIP Adv. 7(12), 125,012 (2017)CrossRefGoogle Scholar
  40. 40.
    Tanaka, M.L., Ross, S.D., Nussbaum, M.A.: Mathematical modeling and simulation of seated stability. J. Biomech. 43(5), 906–912 (2010)CrossRefGoogle Scholar
  41. 41.
    Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  42. 42.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, Berlin (2003)zbMATHGoogle Scholar
  43. 43.
    Wiggins, S.: The dynamical systems approach to Lagrangian transport in oceanic flows. Ann. Rev. Fluid Mech. 37, 295–328 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Wiggins, S.: Normally Hyperbolic Invariant Manifolds in Dynamical Systems, vol. 105. Springer, Berlin (2013)zbMATHGoogle Scholar
  45. 45.
    Wiggins, S., Wiesenfeld, L., Jaffé, C., Uzer, T.: Impenetrable barriers in phase-space. Phys. Rev. Lett. 86, 5478–5481 (2001)CrossRefGoogle Scholar
  46. 46.
    Xie, X., Nolan, P., Ross, S., Iliescu, T.: Lagrangian data-driven reduced order modeling of finite time Lyapunov exponents. arXiv:1808.05635 (2018)
  47. 47.
    Yeaton, I.J., Socha, J.J., Ross, S.D.: Global dynamics of non-equilibrium gliding in animals. Bioinspir. Biomim. 12(2), 026,013 (2017)CrossRefGoogle Scholar
  48. 48.
    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)CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Engineering Mechanics ProgramVirginia TechBlacksburgUSA

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