Skip to main content Accessibility help

Computing heteroclinic orbits using adjoint-based methods

  • M. Farano (a1) (a2) (a3), S. Cherubini (a1), J.-C. Robinet (a2), P. De Palma (a1) and T. M. Schneider (a3)...


Transitional turbulence in shear flows is supported by a network of unstable exact invariant solutions of the Navier–Stokes equations. The network is interconnected by heteroclinic connections along which the turbulent trajectories evolve between invariant solutions. While many invariant solutions in the form of equilibria, travelling waves and periodic orbits have been identified, computing heteroclinic connections remains a challenge. We propose a variational method for computing orbits dynamically connecting small neighbourhoods around equilibrium solutions. Using local information on the dynamics linearized around these equilibria, we demonstrate that we can choose neighbourhoods such that the connecting orbits shadow heteroclinic connections. The proposed method allows one to approximate heteroclinic connections originating from states with multi-dimensional unstable manifold and thereby provides access to heteroclinic connections that cannot easily be identified using alternative shooting methods. For plane Couette flow, we demonstrate the method by recomputing three known connections and identifying six additional previously unknown orbits.


Corresponding author


Hide All
Budanur, N. B., Short, K. Y., Farazmand, M., Willis, A. P. & Cvitanović, P. 2017 Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech. 833, 274301.
Chantry, M. & Schneider, T. M 2014 Studying edge geometry in transiently turbulent shear flows. J. Fluid Mech. 747, 506517.
Cherubini, S. & De Palma, P. 2014 Minimal perturbations approaching the edge of chaos in a Couette flow. Fluid Dyn. Res. 46 (4), 041403.
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2010 Rapid path to transition via nonlinear localized optimal perturbations in a boundary-layer flow. Phys. Rev. E 82 (6), 066302.
Crommelin, D. T. 2003 Regime transitions and heteroclinic connections in a barotropic atmosphere. J. Atmos. Sci. 60 (2), 229246.
Dong, C. & Lan, Y. 2014 A variational approach to connecting orbits in nonlinear dynamical systems. Phys. Lett. A 378 (9), 705712.
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008 Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255274.
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
Farano, M., Cherubini, S., Robinet, J.-C. & De Palma, P. 2015 Hairpin-like optimal perturbations in plane Poiseuille flow. J. Fluid Mech. 775, R2.
Farano, M, Cherubini, S, Robinet, J.-C. Robinet, De Palma, P. & Schneider, T. M.2018 How hairpin structures emerge from exact solutions of shear flows (in review).
Foures, DPG, Caulfield, CP & Schmid, PJ 2013 Localization of flow structures using -norm optimization. J. Fluid Mech. 729, 672701.
Gibson, J. F.2014 Channelflow: a spectral Navier-Stokes simulator in C++. Tech. Rep. U. New Hampshire,
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107130.
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and traveling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243266.
Halcrow, J., Gibson, J. F., Cvitanović, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365376.
Hof, B. & Budanur, N. B. 2017 Heteroclinic path to spatially localized chaos in pipe flow. J. Fluid Mech. 827, R1.
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305, 15941598.
Hopf, E. 1948 A mathematical example displaying features of turbulence. Commun. Pure Appl. Maths 1, 303322.
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291300.
Kawahara, G., Uhlmann, M. & Van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203225.
Kerswell, R. R. 2018 Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50 (1), 319345.
Krauskopf, B. & Osinga, H. M. 2007 Computing invariant manifolds via the continuation of orbit segments. In Numerical Continuation Methods for Dynamical Systems, pp. 117154. Springer.
Krauskopf, B., Osinga, H. M., Doedel, E. J., Henderson, M. E., Guckenheimer, J., Vladimirsky, A., Dellnitz, M. & Junge, O. 2005 A survey of methods for computing (un) stable manifolds of vector fields. Intl J. Bifurcation Chaos 15 (3), 763791.
Kreilos, T., Veble, G., Schneider, T. M & Eckhardt, B. 2013 Edge states for the turbulence transition in the asymptotic suction boundary layer. J. Fluid Mech. 726, 100122.
Lan, Y. & Cvitanović, P. 2004 Variational method for finding periodic orbits in a general flow. Phys. Rev. E 69 (1), 016217.
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105, 154502.
Schneider, T. M., Gibson, J. F., Lagha, M., De Lillo, F. & Eckhardt, B. 2008 Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E 78, 037301.
Suri, B., Tithof, J., Grigoriev, R. O. & Schatz, M. F. 2017 Forecasting fluid flows using the geometry of turbulence. Phys. Rev. Lett. 118 (11), 114501.
Toh, S. & Itano, T. 2003 A periodic-like solution in channel flow. J. Fluid Mech. 481, 6776.
Van Veen, L. & Kawahara, G. 2011 Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett. 107 (11), 114501.
Van Veen, L., Kawahara, G. & Atsushi, M. 2011 On matrix-free computation of 2D unstable manifolds. SIAM J. Sci. Comput. 33 (1), 2544.
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339358.
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 15171534.
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: traveling wave solutions. J. Fluid Mech. 508, 333371.
Willis, A. P., Cvitanović, P. & Avila, M. 2013 Revealing the state space of turbulent pipe flow by symmetry reduction. J. Fluid Mech. 721, 514540.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO
Type Description Title

Farano et al. supplementary movie
All 9 heteroclinic connections. (Left) State-space projection onto the 3D orthonormal basis. Symbols represent equilibria. (Right) Streamwise averaged velocity field, color represent streamwise velocity (blue negative, red positive) and arrows represent in-plane velocity.

 Video (33.0 MB)
33.0 MB
Supplementary materials

Farano et al. supplementary material
Supplementary data

 PDF (5.2 MB)
5.2 MB

Computing heteroclinic orbits using adjoint-based methods

  • M. Farano (a1) (a2) (a3), S. Cherubini (a1), J.-C. Robinet (a2), P. De Palma (a1) and T. M. Schneider (a3)...


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.