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Quantifying the Role of Folding in Nonautonomous Flows: The Unsteady Double-Gyre

Department of Mathematics, The Clarkson Center for Complex Systems Science, Clarkson University, 8 Clarkson Ave, Potsdam, New York 13699-5815, USA

E-mail Address: priyankg@clarkson.edu

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Sanjeeva Balasuriya

School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia

E-mail Address: sanjeevabalasuriya@yahoo.com

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Erik Bollt

Department of Mathematics, Electrical and Computer Engineering, The Clarkson Center for Complex Systems Science, Clarkson University, 8 Clarkson Ave, Potsdam, New York 13699-5815, USA

E-mail Address: bolltem@clarkson.edu

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https://doi.org/10.1142/S0218127417501565Cited by:8 (Source: Crossref)

Abstract

We analyze chaos in the well-known nonautonomous Double-Gyre system. A key focus is on folding, which is possibly the less-studied aspect of the “stretching+folding=chaos” mantra of chaotic dynamics. Despite the Double-Gyre not having the classical homoclinic structure for the usage of the Smale–Birkhoff theorem to establish chaos, we use the concept of folding to prove the existence of an embedded horseshoe map. We also show how curvature of manifolds can be used to identify fold points in the Double-Gyre. This method is applicable to general nonautonomous flows in two dimensions, defined for either finite or infinite times.

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References

Alligood, K., Sauer, T. & Yorke, J. [1996] Chaos: An Introduction to Dynamical Systems (Springer-Verlag, NY). Google Scholar
Allshouse, M. & Peacock, T. [2015] “ Lagrangian based methods for coherent structure detection,” Chaos 25, 097617. Web of Science, Google Scholar
Arrowsmith, D. & Vivaldi, F. [1993] “ Some p-adic representations of the Smale horseshoe,” Phys. Lett. A 176, 292–294. Web of Science, Google Scholar
Balasuriya, S. [2005a] “ Direct chaotic flux quantification in perturbed planar flows: General time-periodicity,” SIAM J. Appl. Dyn. Syst. 4, 282–311. Web of Science, Google Scholar
Balasuriya, S. [2005b] “ Optimal perturbation for enhanced chaotic transport,” Physica D 202, 155–176. Web of Science, Google Scholar
Balasuriya, S. [2006] “ Cross-separatrix flux in time-aperiodic and time-impulsive flows,” Nonlinearity 19, 2775–2795. Web of Science, Google Scholar
Balasuriya, S. [2011] “ A tangential displacement theory for locating perturbed saddles and their manifolds,” SIAM J. Appl. Dyn. Syst. 10, 1100–1126. Web of Science, Google Scholar
Balasuriya, S. [2014] “ Nonautonomous flows as open dynamical systems: Characterising escape rates and time-varying boundaries,” Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Mathematics and Statistics, Vol. 70 (Springer), pp. 1–30. Google Scholar
Balasuriya, S. [2016a] Barriers and Transport in Unsteady Flows: A Melnikov Approach, Mathematical Modeling and Computation (SIAM Press, Philadelphia). Google Scholar
Balasuriya, S. [2016b] “ Local stable and unstable manifolds and their control in nonautonomous finite-time flows,” J. Nonlin. Sci. 26, 895–927. Web of Science, Google Scholar
Balasuriya, S. & Ouellette, N. [2016] “ Hyperbolic neighborhoods as organizers of finite-time exponential stretching,” APS Division of Fluid Dynamics Meeting Abstracts. Google Scholar
Balasuriya, S., Kalampattel, R. & Ouellette, N. [2016] “ Hyperbolic neighbourhoods as organizers of finite-time exponential stretching,” J. Fluid Mech. 807, 509–545. Web of Science, Google Scholar
Balibrea, F. & Snoha, L. [2003] “ Topological entropy of Devaney chaotic maps,” Topol. Appl. 133, 225–239. Web of Science, Google Scholar
Battelli, F. & Lazzari, C. [1986] “ Exponential dichotomies, heteroclinic orbits and Melnikov functions,” J. Diff. Eqs. 86, 342–366. Web of Science, Google Scholar
Bertozzi, A. [1988] “ Heteroclinic orbits and chaotic dynamics in planar fluid flows,” SIAM J. Math. Anal. 19, 1271–1294. Web of Science, Google Scholar
Blazevski, D. & Haller, G. [2014] “ Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows,” Physica D 273, 46–62. Web of Science, Google Scholar
Bollt, E. [2000] “ Controlling chaos and the inverse Frobenius–Perron problem: Global stabilization of arbitrary invariant measures,” Int. J. Bifurcation and Chaos 10, 1033–1050. Link, Web of Science, Google Scholar
Bollt, E., Billings, L. & Schwartz, I. [2002] “ A manifold independent approach to understanding transport in stochastic dynamical systems,” Physica D 173, 153–177. Web of Science, Google Scholar
Bollt, E., Luttman, A., Kramer, S. & Basnayake, R. [2012] “ Measurable dynamics analysis of transport in the Gulf of Mexico during the oil spill,” Int. J. Bifurcation and Chaos 22, 1230012-1–12. Link, Web of Science, Google Scholar
Bollt, E. & Santitissadeekorn, N. [2013] Applied and Computational Measurable Dynamics, Mathematical Modeling and Computation (SIAM Press, Philadelphia). Google Scholar
BozorgMagham, A. & Ross, S. [2015] “ Atmospheric Lagrangian coherent structures considering unresolved turbulence and forecast uncertainty,” Commun. Nonlin. Sci. Numer. Simul. 22, 964–979. Web of Science, Google Scholar
Branicki, M. & Wiggins, S. [2010] “ Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents,” Nonlin. Proc. Geophys. 17, 1–36. Web of Science, Google Scholar
Brunton, S. & Rowley, C. [2010] “ Fast computation of finite-time Lyapunov exponent fields for unsteady flows,” Chaos 20, 017503. Web of Science, Google Scholar
Budisic, M. & Mezic, I. [2012] “ Geometry of the ergodic quotient reveals coherent structures in flows,” Physica D 241, 1255–1269. Web of Science, Google Scholar
Chen, Y. [2006] “ Smale horseshoe via the anti-integrability,” Chaos Solit. Fract. 28, 377–385. Web of Science, Google Scholar
Coppel, W. [1978] Dichotomies in Stability Theory, Lecture Notes in Mathematics (Springer-Verlag, Berlin). Google Scholar
Dellnitz, M. & Junge, O. [2002] “ Set oriented numerical methods for dynamical systems,” Handbook of Dynamical Systems, Vol. 2 (North-Holland, Amsterdam), pp. 221–264. Google Scholar
Duc, L. & Siegmund, S. [2008] “ Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals,” Int. J. Bifurcation and Chaos 18, 641–674. Link, Web of Science, Google Scholar
Farazmand, M. & Haller, G. [2013] “ Attracting and repelling Lagrangian coherent structures from a single computation,” Chaos 23, 023101. Web of Science, Google Scholar
Froyland, G., Padberg-Gehle, K., England, M. & Treguier, A. [2007] “ Detection of coherent oceanic structures via transfer operators,” Phys. Rev. Lett. 98, 224503. Web of Science, Google Scholar
Froyland, G. & Padberg-Gehle, K. [2009] “ Almost-invariant sets and invariant manifolds connecting probabilistic and geometric descriptions of coherent structures in flows,” Physica D 238, 1507–1523. Web of Science, Google Scholar
Froyland, G., Santitissadeekorn, N. & Monahan, A. [2010] “ Transport in time-dependent dynamical systems: Finite-time coherent sets,” Chaos 20, 043116. Web of Science, Google Scholar
Froyland, G. & Padberg-Gehle, K. [2012] “ Finite-time entropy: A probabilistic approach for measuring nonlinear stretching,” Physica D 241, 1612–1628. Web of Science, Google Scholar
Froyland, G. [2013] “ An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems,” Physica D 250, 1–19. Web of Science, Google Scholar
Froyland, G. & Padberg-Gehle, K. [2015] “ A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data,” Chaos 25, 087406. Web of Science, Google Scholar
Gajamannage, K. & Bollt, E. [2016] “ Detecting phase transitions in collective motion using manifold’s curvature,” Math. Biosci. Engin. 14, 437–453. Web of Science, Google Scholar
Garaboa-Paz, D. & Perez-Munuzuri, V. [2015] “ A method to calculate finite-time Lyapunov exponents for intertial particles in incompressible flows,” Nonlin. Proc. Geophys. 22, 571–577. Web of Science, Google Scholar
Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, NY). Google Scholar
Hadjighasem, A., Karrasch, D., Teramoto, H. & Haller, G. [2016] “ Spectral-clustering approach to Lagrangian vortex detection,” Phys. Rev. E 93, 063107. Web of Science, Google Scholar
Haller, G. & Yuan, G. [2000] “ Lagrangian coherent structures and mixing in two-dimensional turbulence,” Physica D 147, 352–370. Web of Science, Google Scholar
Haller, G. [2015] “ Lagrangian coherent structures,” Ann. Rev. Fluid Mech. 47, 137–162. Web of Science, Google Scholar
He, G., Pan, C., Feng, L., Gao, Q. & Wang, J. [2016] “ Evolution of Lagrangian coherent structures in a cylinder-wake disturbed flat plate boundary layer,” J. Fluid Mech. 792, 274–306. Web of Science, Google Scholar
Holmes, P. [1986] “ Knotted periodic orbits in suspensions of Smale’s horseshoe: Period multiplying and cabled knots,” Physica D 21, 7–41. Web of Science, Google Scholar
Holmes, P. [1990] “ Poincaré, celestial mechanics, dynamical-systems theory and chaos,” Phys. Rep. 193, 137–163. Web of Science, Google Scholar
Huntley, H., Lipphardt, B., Jacobs, G. & Kirwan, A. [2015] “ Clusters, deformation, and dilation: Diagnostics for material accumulation regions,” J. Geophys. Res.: Oceans 120, 6622–6636. Web of Science, Google Scholar
Johnson, P. & Meneveau, C. [2015] “ Large-deviation joint statistics of the finite-time Lyapunov spectrum in isotropic turbulence,” Phys. Fluids 27, 085110. Web of Science, Google Scholar
Ju, N., Small, D. & Wiggins, S. [2003] “ Existence and computation of hyperbolic trajectories of aperiodically time dependent vector fields and their approximations,” Int. J. Bifurcation and Chaos 13, 1449–1457. Link, Web of Science, Google Scholar
Karrasch, D., Farazmand, M. & Haller, G. [2015] “ Attraction-based computation of hyperbolic Lagrangian coherent structures,” J. Comput. Dyn. 2, 83–93. Google Scholar
Kelley, D., Allshouse, M. & Ouellette, N. [2013] “ Lagrangian coherent structures separate dynamically distinct regions in fluid flows,” Phys. Rev. E 88, 013017. Web of Science, Google Scholar
Lipinski, D. & Mohseni, K. [2010] “ A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures,” Chaos 20, 017504. Web of Science, Google Scholar
Ma, T. & Bollt, E. [2013] “ Relatively coherent sets as a hierarchical partition method,” Int. J. Bifurcation and Chaos 23, 1330026-1–17. Link, Web of Science, Google Scholar
Ma, T. & Bollt, E. [2014] “ Differential geometry perspective of shape coherence and curvature evolution by finite-time nonhyperbolic splitting,” SIAM J. Appl. Dyn. Syst. 13, 1106–1136. Web of Science, Google Scholar
Ma, T. & Bollt, E. [2015] “ Shape coherence and finite-time curvature evolution,” Int. J. Bifurcation and Chaos 25, 1550076-1–10. Link, Web of Science, Google Scholar
Ma, T., Ouellette, N. & Bollt, E. [2016] “ Stretching and folding in finite time,” Chaos 26, 023112. Web of Science, Google Scholar
Mancho, M., Small, D., Wiggins, S. & Ide, K. [2003] “ Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields,” Physica D 182, 188–222. Web of Science, Google Scholar
Mancho, A., Wiggins, S., Curbelo, J. & Mendoza, C. [2013] “ Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems,” Commun. Nonlin. Sci. Numer. Simul. 18, 3530–3557. Web of Science, Google Scholar
McIlhany, K. & Wiggins, S. [2012] “ Eulerian indicators under continuously varying conditions,” Phys. Fluids 24, 073601. Web of Science, Google Scholar
Melnikov, V. [1963] “ On the stability of the centre for time-periodic perturbations,” Trans. Moscow Math. Soc. 12, 1–56. Google Scholar
Mezic, I., Loire, S., Fonoberov, V. & Hogan, P. [2010] “ A new mixing diagnostic and Gulf oil spill movement,” Science 330, 486–489. Web of Science, Google Scholar
Mosovsky, B. & Meiss, J. [2011] “ Transport in transitory dynamical systems,” SIAM J. Appl. Dyn. Syst. 10, 35–65. Web of Science, Google Scholar
Muller-Karger, C., Mirena, A. & Lopez, J. [2000] “ Hyperbolic trajectories for pick-and-place operations to elude obstacles,” IEEE Trans. Robot. Automat. 16, 294–300. Google Scholar
Namikawa, J. & Hashimoto, T. [2004] “ Dynamics and computation in functional shifts,” Nonlinearity 17, 1317. Web of Science, Google Scholar
Nelson, D. & Jacobs, G. [2015] “ DG-FTLE: Lagrangian coherent structures with high-order discontinuous-Galerkin methods,” J. Comput. Phys. 295, 65–86. Web of Science, Google Scholar
Nevins, T. & Kelley, D. [2016] “ Optimal stretching in advection–reaction–diffusion systems,” Phys. Rev. Lett. 117, 164502. Web of Science, Google Scholar
Onu, K., Huhn, F. & Haller, G. [2015] “ {LCS} Tool: A computational platform for Lagrangian coherent structures,” J. Comput. Sci. 7, 26–36. Web of Science, Google Scholar
Palmer, K. [1984] “ Exponential dichotomies and transversal homoclinic points,” J. Diff. Eqs. 55, 225–256. Web of Science, Google Scholar
Poje, A., Haller, G. & Mezic, I. [1999] “ The geometry and statistics of mixing in aperiodic flows,” Phys. Fluids 11, 2963–2968. Web of Science, Google Scholar
Pratt, K., Meiss, J. & Crimaldi, J. [2015] “ Reaction enhancement of initially distant scalars by Lagrangian coherent structures,” Phys. Fluids 27, 035106. Web of Science, Google Scholar
Robinson, C. [1999] Dynamical Systems: Stabiliy, Symbolic Dynamics and Chaos, Studies in Advanced Mathematics (CRC Press). Google Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. [1990] “ An analytical study of transport, mixing and chaos in an unsteady vortical flow,” J. Fluid Mech. 214, 347–394. Web of Science, Google Scholar
Rom-Kedar, V. & Poje, A. [1999] “ Universal properties of chaotic transport in the presence of diffusion,” Phys. Fluids 11, 2044–2057. Web of Science, Google Scholar
Rosi, G., Walker, A. & Rival, D. [2015] “ Lagrangian coherent structure identification using a Voronoi tessellation-based networking algorithm,” Exp. Fluids 56, 189. Web of Science, Google Scholar
Rössler, O. [1977] “ Horseshoe-map chaos in the Lorenz equation,” Phys. Lett. A 60, 392–394. Web of Science, Google Scholar
Sattari, S., Chen, Q. & Mitchell, K. [2016] “ Using heteroclinic orbits to quantify topological entropy in fluid flows,” Chaos 26, 033112. Web of Science, Google Scholar
Shadden, S., Lekien, F. & Marsden, J. [2005] “ Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,” Physica D 212, 271–304. Web of Science, Google Scholar
Sudharsan, M., Brunton, S. & Riley, J. [2016] “ Lagrangian coherent structures and inertial particle dynamics,” Phys. Rev. E 93, 033108. Web of Science, Google Scholar
Tallapragada, P. & Ross, S. [2013] “ A set-oriented defintion of finite-time Lyapunov exponents and coherent sets,” Commun. Nonlin. Sci. Numer. Simul. 18, 1106–1126. Web of Science, Google Scholar
Teramoto, H., Haller, G. & Komatsuzaki, T. [2013] “ Detecting invariant manifolds as stationary LCSs in autonomous dynamical systems,” Chaos 23, 043107. Web of Science, Google Scholar
Tumasz, S. & Thiffeault, J. [2013] “ Estimating topological entropy from the motion of stirring rods,” Procedia IUTAM 7, 117–126. Google Scholar
Vleck, E. V. [1994] “ Numerical shadowing near hyperbolic trajectories,” SIAM J. Sci. Comput. 16, 1177–1189. Web of Science, Google Scholar
Wiggins, S. [1992] Chaotic Transport in Dynamical Systems (Springer-Verlag, NY). Google Scholar
Wiggins, S. & Mancho, A. M. [2014] “ Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev’s theorem and “Nearly Invariant” tori,” Nonlin. Proc. Geophys. 21, 165–185. Web of Science, Google Scholar
Williams, M., Rypina, I. & Rowley, C. [2015] “ Identifying finite-time coherent sets from limited quantities of Lagrangian data,” Chaos 25, 087408. Web of Science, Google Scholar