Skip to main content
SpringerLink
Log in
Book cover

Ergodic Theory, Open Dynamics, and Coherent Structures pp 1–30Cite as

Nonautonomous Flows as Open Dynamical Systems: Characterising Escape Rates and Time-Varying Boundaries

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 70))

Abstract

A Lagrangian coherent structure (LCS) in a nonautonomous flow can be viewed as an open dynamical system, from which there is time-dependent escape or entry. A difficulty with this viewpoint is formulating a definition for the time-dependent boundary of the LCS, since it does not correspond to an entity across which there is zero transport. Complementary to this is the question of how to determine the escape rate—the time-dependent fluid flux—across this purported boundary. These questions are addressed within the context of nonautonomously perturbed two-dimensional compressible flow. The LCS boundaries are thought of in terms of time-varying stable and unstable manifolds, whose primary locations are quantified. A definition for the time-varying flux across these is offered, and computationally tractable formulæ with a strong relation to Melnikov functions are provided. Simplifications of these formulæ for frequently considered situations (incompressibility, time-periodic perturbations) are demonstrated to be easily computable using Fourier transforms. Explicit connections to lobe areas and the average flux are also provided.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Indeed, these separating surfaces can be rationalised as being boundaries of LCSs in this highly idealised situation.

  2. 2.

    Transverse intersections are prohibited in autonomous flows, since the presence of such an intersection would violate uniqueness of trajectories—the trajectory passing through the intersection point will not be able to decide which manifold to follow.

  3. 3.

    Chaotic mixing, on the other hand, can be thought of as a purely advective mechanism.

  4. 4.

    \(\frac{dH\left (\boldsymbol{x}(t)\right )} {dt} = DH\left (\boldsymbol{x}(t)\right ) \cdot \dot{\boldsymbol{ x}} = DH\left (\boldsymbol{x}(t)\right ) \cdot \left [-JDH\left (\boldsymbol{x}(t)\right )\right ] = 0\).

  5. 5.

    A concrete example with no lobes is presented in Remark 1.12. Viscosity-induced perturbations also result in no lobes [20, 24, 111].

  6. 6.

    See Corollary 1.5 in Sect. 1.5.

References

  1. Abdullaev, S.: Structure of motion near saddle points and chaotic transport in Hamiltonian systems. Phys. Rev. E 62, 3508–3528 (2000)

    MathSciNet  Google Scholar 

  2. Aharon, R., Rom-Kedar, V., Gildor, H.: When complexity leads to simplicity: ocean surface mixing simplified by vertical convection. Phys. Fluids 24, 056,603 (2012)

    Google Scholar 

  3. Ahn, T., Kim, S.: Separatrix map analysis of chaotic transport in planar periodic vortical flows. Phys. Rev. E 49, 2900–2912 (1994)

    MathSciNet  Google Scholar 

  4. Alligood, K., Sauer, T., Yorke, J.: Chaos: An Introduction to Dynamical Systems. Springer, New York (1996)

    MATH  Google Scholar 

  5. Allshouse, M., Thiffeault, J.L.: Detecting coherent structures using braids. Phys. D 241, 95–105 (2012)

    Google Scholar 

  6. Arnold, V.I.: Sur la topologie des écoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris 261, 17–20 (1965)

    MathSciNet  Google Scholar 

  7. Arnold, V.I.: Sur la géométrie differentielle des groupes de lie de dimension infinie et ses applications á l’hydrodynamique des fluides parfaits. Ann. Inst Fourier 16, 316–361 (1966)

    Google Scholar 

  8. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    MATH  Google Scholar 

  9. Arrowsmith, D., Place, C.: An Introduction to Dynamical Systems. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  10. Balasuriya, S.: Approach for maximizing chaotic mixing in microfluidic devices. Phys. Fluids 17, 118103 (2005)

    MathSciNet  Google Scholar 

  11. Balasuriya, S.: Direct chaotic flux quantification in perturbed planar flows: general time-periodicity. SIAM J. Appl. Dyn. Syst. 4, 282–311 (2005)

    MATH  MathSciNet  Google Scholar 

  12. Balasuriya, S.: Optimal perturbation for enhanced chaotic transport. Phys. D 202, 155–176 (2005)

    MATH  MathSciNet  Google Scholar 

  13. Balasuriya, S.: Cross-separatrix flux in time-aperiodic and time-impulsive flows. Nonlinearity 19, 282–311 (2006)

    MathSciNet  Google Scholar 

  14. Balasuriya, S.: Invasions with density-dependent ecological parameters. J. Theory Biol. 266, 657–666 (2010)

    MathSciNet  Google Scholar 

  15. Balasuriya, S.: Optimal frequency for microfluidic mixing across a fluid interface. Phys. Rev. Lett. 105, 064501 (2010)

    Google Scholar 

  16. Balasuriya, S.: A tangential displacement theory for locating perturbed saddles and their manifolds. SIAM J. Appl. Dyn. Syst. 10, 1100–1126 (2011)

    MATH  MathSciNet  Google Scholar 

  17. Balasuriya, S.: Explicit invariant manifolds and specialised trajectories in a class of unsteady flows. Phys. Fluids 24, 127101 (2012)

    Google Scholar 

  18. Balasuriya, S., Finn, M.: Energy constrained transport maximization across a fluid interface. Phys. Rev. Lett. 108, 244503 (2012)

    Google Scholar 

  19. Balasuriya, S., Gottwald, G.: Wavespeed in reaction-diffusion systems, with applications to chemotaxis and population pressure. J. Math. Biol. 61, 377–399 (2010)

    MATH  MathSciNet  Google Scholar 

  20. Balasuriya, S., Jones, C.K.R.T.: Diffusive draining and growth of eddies. Nonlin. Proc. Geophys. 8, 241–251 (2001)

    Google Scholar 

  21. Balasuriya, S., Padberg-Gehle, K.: Controlling the unsteady analogue of saddle stagnation points. SIAM J. Appl. Math. 73, 1038–1057 (2013)

    MATH  MathSciNet  Google Scholar 

  22. Balasuriya, S., Padberg-Gehle, K.: Nonautonomous control of stable and unstable manifolds in two-dimensional flows (2014, submitted)

    Google Scholar 

  23. Balasuriya, S., Volpert, V.: Wavespeed analysis: approximating Arrhenius kinetics with step-function kinetics. Combust. Theor. Model. 12, 643–670 (2008)

    MATH  MathSciNet  Google Scholar 

  24. Balasuriya, S., Jones, C.K.R.T., Sandstede, B.: Viscous perturbations of vorticity-conserving flows and separatrix splitting. Nonlinearity 11(1), 47–77 (1998)

    MATH  MathSciNet  Google Scholar 

  25. Balasuriya, S., Mezić, I., Jones, C.K.R.T.: Weak finite-time Melnikov theory and 3D viscous perturbations of Euler flows. Phys. D 176(1–2), 82–106 (2003)

    MATH  MathSciNet  Google Scholar 

  26. Balasuriya, S., Gottwald, G., Hornibrook, J., Lafortune, S.: High Lewis number combustion wavefronts: a perturbative Melnikov analysis. SIAM J. Appl. Math. 67, 464–486 (2007)

    MATH  MathSciNet  Google Scholar 

  27. Battelli, F., Lazzari, C.: Exponential dichotomies, heteroclinic orbits, and Melnikov functions. J. Differ. Equ. 86, 342–366 (1990)

    MATH  MathSciNet  Google Scholar 

  28. Beigie, D., Leonard, A., Wiggins, S.: Invariant manifold templates for chaotic advection. Chaos Solitons Fractals 4, 749–868 (1994)

    MATH  Google Scholar 

  29. Beron-Vera, F., Olascoaga, M., Brown, M., Kocak, H., Rypina, I.: Invariant-tori-like Lagrangian coherent structures in geophysical flows. Chaos 20, 017,514 (2010)

    MathSciNet  Google Scholar 

  30. Beron-Vera, F., Olascoaga, M., Brown, M., Kocak, H.: Zonal jets as meridional transport barriers in the subtropical and polar lower stratosphere. J. Atmos. Sci 69, 753–767 (2012)

    Google Scholar 

  31. Bettencourt, J., López, C., Hernández-García, E.: Oceanic three-dimensional Lagrangian coherent structures: a study of a mesoscale eddy in the Benguela upwelling region. Ocean Model. 51, 73–83 (2012)

    Google Scholar 

  32. Boffetta, G., Lacorata, G., Radaelli, G., Vulpiani, A.: Detecting barriers to transport: a review of different techniques. Phys. D 159, 58–70 (2001)

    MATH  Google Scholar 

  33. Borgogno, D., Grasso, D., Pegoraro, F., Schep, T.: Barriers in the transition to global chaos in collisionless magnetic reconnection. I. Ridges of the finite time Lyapunov exponent field. Phys. Plasmas 18, 102307 (2011)

    Google Scholar 

  34. Branicki, M., Mancho, A., Wiggins, S.: A Lagrangian description of transport associated with front-eddy interaction: application to data from the North-Western mediterranean sea. Phys. D 240, 282–304 (2011)

    MATH  MathSciNet  Google Scholar 

  35. Brunton, S., Rowley, C.: Fast computation of finite-time Lyapunov exponent fields for unsteady flows. Chaos 20, 017,503 (2010)

    MathSciNet  Google Scholar 

  36. Budis̆ić, M., Mezić, I.: Geometry of ergodic quotient reveals coherent structures in flows. Phys. D 241, 1255–1269 (2012)

    Google Scholar 

  37. de la Camara, A., Mechoso, C., Ide, K., Walterscheid, R., Schubert, G.: Polar night vortex breakdown and large-scale stirring in the southern stratosphere. Climate Dyn. 35, 965–975 (2010)

    Google Scholar 

  38. del Castillo Negrete, D., Morrison, P.: Chaotic transport by Rossby waves in shear flow. Phys. Fluids A 5, 948–965 (1993)

    MATH  MathSciNet  Google Scholar 

  39. Chandrasesekhar, S.: Hydrodynamics and Hydrodynamic Stability. Dover, New York (1961)

    Google Scholar 

  40. Channon, S., Lebowitz, J.: Numerical experiments in stochasticity and homoclinic oscillations. Ann. New York Acad. Sci. 357, 108–118 (1980)

    Google Scholar 

  41. Chow, S.N., Hale, J.K., Mallet-Paret, J.: An example of bifurcation to homoclinic orbits. J. Differ. Equ. 37, 351–373 (1980)

    MATH  MathSciNet  Google Scholar 

  42. Coppel, W.A.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978)

    Google Scholar 

  43. Dellnitz, M., Junge, O.: Almost invariant sets in Chua’s circuit. Int. J. Bif. Chaos 7, 2475–2485 (1997)

    MATH  MathSciNet  Google Scholar 

  44. d’Ovidio, F., Isern, J., López, C., Hernández-García, C., García-Ladona, E.: Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from the altimetry in the Algerian basin. Deep-Sea Res. 56, 15–31 (2009)

    Google Scholar 

  45. Duc, L., Siegmund, S.: Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals. Int. J. Bifurcation Chaos 18, 641–674 (2008)

    MATH  MathSciNet  Google Scholar 

  46. Farazmand, M., Haller, G.: Computing Lagrangian coherent structures from variational LCS theory. Chaos 22, 013,128 (2012)

    Google Scholar 

  47. Farazmand, M., Haller, G.: Erratum and addendum to “A variational theory for Lagrangian Coherent Structures”. Phys. D 241, 439–441 (2012)

    MATH  MathSciNet  Google Scholar 

  48. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971/1972)

    Google Scholar 

  49. Forgoston, E., Billings, L., Yecko, P., Schwartz, I.: Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant. Chaos 21, 013,116 (2011)

    MathSciNet  Google Scholar 

  50. Franco, E., Pekerek, D., Peng, J., Dabiri, J.: Geometry of unsteady fluid transport during fluid-structure interactions. J. Fluid Mech. 589, 125–145 (2007)

    MATH  MathSciNet  Google Scholar 

  51. Froyland, G., Padberg, K.: Almost-invariant sets and invariant manifolds: connecting probabilistic and geometric descriptions of coherent structures in flows. Phys. D 238, 1507–1523 (2009)

    MATH  MathSciNet  Google Scholar 

  52. Froyland, G., Padberg, K.: Finite-time entropy: a probabilistic method for measuring nonlinear stretching. Physica D 241(19), 1612–1628 (2012)

    MATH  Google Scholar 

  53. Froyland, G., Lloyd, S., Santitissadeekorn, N.: Coherent sets for nonautonomous dynamical systems. Phys. D 239, 1527–1541 (2010)

    MATH  MathSciNet  Google Scholar 

  54. Froyland, G., Santitissadeekorn, N., Monahan, A.: Transport in time-dependent dynamical systems: finite-time coherent sets. Chaos 20, 043116 (2010)

    MathSciNet  Google Scholar 

  55. Gouillart, E., Dauchot, O., Thiffeault, J.L.: Measures of mixing quality in open flows with chaotic advection. Phys. Fluids 23, 013604 (2011)

    Google Scholar 

  56. Green, M., Rowley, C., Smits, A.: The unsteady three-dimensional wake produced by a trapezoidal pitching panel. J. Fluid Mech. 685, 117–145 (2011)

    MATH  Google Scholar 

  57. Gruendler, J.: The existence of homoclinic orbits and the method of Melnikov for systems in \({\mathbb{R}}^{n}\). SIAM J. Math. Anal. 16, 907–931 (1985)

    MATH  MathSciNet  Google Scholar 

  58. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)

    MATH  Google Scholar 

  59. Haller, G.: Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Phys. D 149, 248–277 (2001)

    MATH  MathSciNet  Google Scholar 

  60. Haller, G.: Lagrangian coherent structures from approximate velocity data. Phys. Fluids A 14, 1851–1861 (2002)

    MathSciNet  Google Scholar 

  61. Haller, G.: A variational theory for Lagrangian coherent structures. Phys. D 240, 574–598 (2011)

    MATH  MathSciNet  Google Scholar 

  62. Haller, G., Beron-Vera, F.: Geodesic theory of transport barriers in two-dimensional flows. Phys. D 241, 1680–1702 (2012)

    MATH  Google Scholar 

  63. Haller, G., Mezić, I.: Reduction of three-dimensional, volume-preserving flows by symmetry. Nonlinearity 11, 319–339 (1998)

    MATH  MathSciNet  Google Scholar 

  64. Haller, G., Poje, A.: Finite time transport in aperiodic flows. Phys. D 119, 352–380 (1998)

    MATH  MathSciNet  Google Scholar 

  65. Haller, G., Sapsis, T.: Lagrangian coherent structures and the smallest finite-time Lyapunov exponent. Chaos 21, 023,115 (2011)

    MathSciNet  Google Scholar 

  66. Haller, G., Yuan, G.C.: Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys. D 147, 352–370 (2000)

    MATH  MathSciNet  Google Scholar 

  67. Hirsch, M., Pugh, C., Shub, M.: Invariant manifolds. Bull. Am. Math. Soc. 76, 1015–1019 (1970)

    MATH  MathSciNet  Google Scholar 

  68. Holmes, P.: Averaging and chaotic motions in forced oscillations. SIAM J. Appl. Math. 38, 65–80 (1980)

    MATH  MathSciNet  Google Scholar 

  69. Hughes, C., Thompson, A., Wilson, C.: Identification of jets and mixing barriers from sea level and vorticity measurements using simple statistics. Ocean Model. 32, 44–57 (2010)

    Google Scholar 

  70. Jimenez-Madrid, J., Mancho, A.: Distinguished trajectories in time dependent vector fields. Chaos 19, 013,111 (2009)

    MathSciNet  Google Scholar 

  71. Joseph, B., Legras, B.: Relation between kinematic boundaries, stirring and barriers for the Antarctic polar vortex. J. Atmos. Sci. 59, 1198–1212 (2002)

    Google Scholar 

  72. Kasten, J., Petz, C., Hotz, I., Hege, H.C., Noack, B., Tadmor, G.: Lagrangian feature extraction of the cylinder wake. Phys. Fluids 22, 091,108 (2010)

    Google Scholar 

  73. Katija, K., Beaulieu, W., Regula, C., Colin, S., Costello, J., Dabiri, J.: Quantification of flows generated by the hydromedusa Aequorea victoria: a Lagrangian coherent structure analysis. Mar. Ecol. Prog. Ser. 435, 111–123 (2011)

    Google Scholar 

  74. Kelley, D., Ouellette, N.: Separating stretching from folding in fluid mixing. Nat. Phys. 7, 477–480 (2011)

    Google Scholar 

  75. Kelley, D., Ouellette, N.: Spatiotemporal persistence of spectral fluxes in two-dimensional weak turbulence. Phys. Fluids 23, 115,101 (2011)

    Google Scholar 

  76. Kourentis, L., Konstantinidis, E.: Uncovering large-scale coherent structures in natural and forced turbulent wakes by combining PIV, POD, and FTLE. Exp. Fluids 52, 749–763 (2012)

    Google Scholar 

  77. Kovacic, G.: Lobe area via an action formalism in a class of Hamiltonian systems. Phys. D 51, 226–233 (1991)

    MATH  MathSciNet  Google Scholar 

  78. Krauskopf, B., Osinga, H., Doedel, E., Henderson, M., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., Junge, O.: A survey of method’s for computing (un)stable manifold of vector fields. Int. J. Bif. Chaos 15, 763–791 (2005)

    MATH  MathSciNet  Google Scholar 

  79. Lekien, F., Coulliette, C., Mariano, A., Ryan, E., Shay, L., Haller, G., Marsden, J.: Pollution release tied to invariant manifolds: a case study for the coast of Florida. Phys. D 210, 1–20 (2005)

    MATH  MathSciNet  Google Scholar 

  80. Levnajić, Z., Mezić, I.: Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets. Chaos 20, 033,114 (2010)

    Google Scholar 

  81. Lipinski, D., Mohseni, K.: A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures. Chaos 20, 017,504 (2010)

    MathSciNet  Google Scholar 

  82. MacKay, R., Meiss, J.: Relation between quantum and classical thresholds for multiphoton ionization of excited atoms. Phys. Rev. A 37, 4702–4706 (1988)

    Google Scholar 

  83. MacKay, R., Meiss, J., Percival, I.: Transport in Hamiltonian systems. Phys. D 13, 55–81 (1984)

    MATH  MathSciNet  Google Scholar 

  84. Malhotra, N., Wiggins, S.: Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time dependence, with applications to Rossby wave flow. J. Nonlin. Sci. 8, 401–456 (1998)

    MATH  MathSciNet  Google Scholar 

  85. Mancho, A., Hernández-García, E., Small, D., Wiggins, S., Fernández, V.: Lagrangian transport through an ocean front in the northwestern Mediterranean sea. J. Phys. Oceanography 38, 1222–1237 (2008)

    Google Scholar 

  86. Mathew, G., Mezić, I., Petzold, L.: A multiscale measure for mixing. Physica D 211, 23–46 (2005)

    MATH  MathSciNet  Google Scholar 

  87. Melnikov, V.K.: On the stability of the centre for time-periodic perturbations. Trans. Moscow Math. Soc. 12, 1–56 (1963)

    Google Scholar 

  88. Mendoza, C., Mancho, A.: Hidden geometry of ocean flows. Phys. Rev. Lett. 105, 038,501 (2010)

    Google Scholar 

  89. Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Phys. D 197, 101–133 (2004)

    MATH  MathSciNet  Google Scholar 

  90. Mezić, I., Wiggins, S.: On the integrability and perturbation of three-dimensional flows with symmetry. J. Nonlin. Sci. 4, 157–194 (1994)

    MATH  Google Scholar 

  91. Mezić, I., Wiggins, S.: A method for visualization of invariant sets of dynamical systems based on the ergodic partition. Chaos 9, 213–218 (1999)

    MATH  MathSciNet  Google Scholar 

  92. Mezić, I., Loire, S., Fonoberov, V., Hogan, P.: A new mixing diagnostic and Gulf oil spill movement. Science 330, 486–489 (2010)

    Google Scholar 

  93. Mosovky, B., Meiss, J.: Transport in transitory dynamical systems. SIAM J. Appl. Dyn. Syst. 10, 35–65 (2011)

    MathSciNet  Google Scholar 

  94. Mosovky, B., Meiss, J.: Transport in transitory three-dimensional Liouville flows. SIAM J. Appl. Dyn. Syst. 11, 1785–1816 (2012)

    MathSciNet  Google Scholar 

  95. Moura, A., Feudel, U., Gouillart, E.: Mixing and chaos in open flows. Adv. Appl. Mech. 45, 1–50 (2012)

    Google Scholar 

  96. Olascoaga, M., Haller, G.: Forecasting sudden changes in environmental pollution patterns. Proc. Nat. Acad. Sci. 109, 4738–4743 (2012)

    Google Scholar 

  97. Palmer, K.: Exponential dichotomies and transversal homoclinic points. J. Diff. Equ. 55, 225–256 (1984)

    MATH  MathSciNet  Google Scholar 

  98. Parks, H., Ermentrout, B., Rubin, J.: The dynamics of forced coupled network of active elements. Phys. D 240, 554–567 (2011)

    MATH  Google Scholar 

  99. Peacock, T., Dabiri, J.: Introduction to focus issue: Lagrangian coherent structures. Chaos 20, 017501 (2010)

    Google Scholar 

  100. Pedlosky, J.: Geophysical Fluids Dynamics. Springer, New York (1979)

    Google Scholar 

  101. Peng, J., Peterson, R.: Attracting structures in volcanic ash transport. Atmos. Environ. 48, 230–239 (2012)

    Google Scholar 

  102. Pierrehumbert, R.: Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves. Geophys. Astrophys. Fluid Dyn. 58, 285–319 (1991)

    Google Scholar 

  103. Pierrehumbert, R., Yang, H.: Global chaotic mixing in isentropic surfaces. J. Atmos. Sci. 50, 2462–2480 (1993)

    Google Scholar 

  104. Pouransari, Z., Speetjens, M., Clercx, H.: Formation of coherent structures by fluid inertia in three-dimensional laminar flows. J. Fluid Mech. 654, 5–34 (2010)

    MATH  MathSciNet  Google Scholar 

  105. Provenzale, A.: Transport by coherent barotropic vortices. Ann. Rev. Fluid Mech. 31, 55–93 (1999)

    MathSciNet  Google Scholar 

  106. Radko, T.: On the generation of large-scale structures in a homogeneous eddy field. J. Fluid Mech. 668, 76–99 (2011)

    MATH  MathSciNet  Google Scholar 

  107. Rodrigue, S., Eschenazi, E.: Lobe transport analysis of the Kelvin-Stuart cat’s eyes driven flow. Chaos 20, 013,101 (2010)

    MathSciNet  Google Scholar 

  108. Rom-Kedar, V., Poje, A.: Universal properties of chaotic transport in the presence of diffusion. Phys. Fluids 11, 2044–2057 (1999)

    MATH  MathSciNet  Google Scholar 

  109. Rom-Kedar, V., Wiggins, S.: Transport in two-dimensional maps. Arch. Ration. Mech. Appl. 109, 239–298 (1990)

    MATH  MathSciNet  Google Scholar 

  110. Rom-Kedar, V., Leonard, A., Wiggins, S.: An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394 (1990)

    MATH  MathSciNet  Google Scholar 

  111. Sandstede, B., Balasuriya, S., Jones, C.K.R.T., Miller, P.D.: Melnikov theory for finite-time vector fields. Nonlinearity 13(4), 1357–1377 (2000)

    MATH  MathSciNet  Google Scholar 

  112. Sapsis, T., Peng, G., Haller, G.: Instabilities on prey dynamics in jellyfish feeding. Bull. Math. Biol. 73, 1841–1856 (2011)

    MATH  MathSciNet  Google Scholar 

  113. Senatore, C., Ross, S.: Detection and characterization of transport barriers in complex flows via ridge extraction of the finite time Lyapunov exponent field. Int. J. Num. Meth. Engin. 86, 1163–1174 (2011)

    MATH  Google Scholar 

  114. Shadden, S., Lekien, F., Marsden, J.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D 212, 271–304 (2005)

    MATH  MathSciNet  Google Scholar 

  115. Shadden, S., Lekien, F., Paduan, J., Chavez, F., Marsden, J.: The correlation between surface drifters and coherent structures based on high-frequency radar data in Monterey bay. Deep Sea Res. II Topic. Stud. Oceanography 56, 161–172 (2009)

    Google Scholar 

  116. Shraiman, B.: Diffusive transport in a Rayleigh-Bénard convection cell. Phys. Rev. A 36, 261–267 (1987)

    Google Scholar 

  117. Shuckburgh, E., Hayes, P.: Diagnosing transport and mixing using a tracer-based coordinate system. Phys. Fluids 15, 3342–3357 (2003)

    Google Scholar 

  118. Solomon, T., Gollub, J.: Chaotic particle transport in time-dependent Rayleigh-Bénard convection. Phys. Rev. A 38, 6280–6286 (1988)

    Google Scholar 

  119. Speetjens, M., de Wispelaere, H., van Steenhoven, A.: Multi-functional Lagrangian flow structures in three-dimensional ac electro-osmotic micro-flows. Fluid Dyn. Res. 43, 035503 (2011)

    MathSciNet  Google Scholar 

  120. Sposito, G.: On steady flows with Lamb surfaces. Int. J. Eng. Sci. 35, 197–209 (1997)

    MATH  MathSciNet  Google Scholar 

  121. Tallapragada, P., Ross, S., Schmale, D.: Lagrangian coherent structures are associated with fluctuations in airborne microbial populations. Chaos 21, 033,122 (2011)

    Google Scholar 

  122. Tang, W., Walker, P.: Finite-time statistics of scalar diffusion in Lagrangian coherent structures. Phys. Rev. E 86, 045201(R) (2012)

    Google Scholar 

  123. Tang, W., Chan, P., Haller, G.: Lagrangian coherent structure analysis of terminal winds detected by Lidar. part I: Turbulence structures. J. Appl. Meteor. Climat. 50, 325–338 (2011)

    Google Scholar 

  124. Tang, W., Chan, P., Haller, G.: Lagrangian coherent structure analysis of terminal winds detected by Lidar. Part II: Structure evolution and comparison with flight data. J. Appl. Meteor. Climat. 50, 2167–2183 (2011)

    Google Scholar 

  125. Thiffeault, J.L.: Using multiscale norms to quantify mixing and transport. Nonlinearity 25, R1–R44 (2012)

    MATH  MathSciNet  Google Scholar 

  126. Titaud, O., Brankart, J., Verron, J.: On the use of finite-time Lyapunov exponents and vectors for direct assimilation of tracer images into ocean models. Tellus Ser. A Dyn. Meteorol. Oceanography 63, 1038–1051 (2011)

    Google Scholar 

  127. Weiss, J., Knobloch, E.: Mass transport by modulated traveling waves. Phys. Rev. A 40, 2579–2589 (1989)

    Google Scholar 

  128. Wiggins, S.: Chaotic Transport in Dynamical Systems. Springer, New York (1992)

    MATH  Google Scholar 

  129. Yagasaki, K.: Invariant manifolds and control of hyperbolic trajectories on infinite- or finite-time intervals. Dyn. Syst. 23, 309–331 (2008)

    MATH  MathSciNet  Google Scholar 

  130. Yi, Y.: A generalized integral manifold theorem. J. Differ. Equ. 102, 153–187 (1993)

    MATH  Google Scholar 

Download references

Acknowledgements

Funding from Connecticut College’s Research Matters fund for travel to the Banff International Research Station (BIRS) for the workshop associated with this volume and funding from BIRS while at Banff are gratefully acknowledged. I also express thanks to the Simons Foundation for a Collaboration Grant for Mathematicians. I also express thanks to the Simons Foundation for a Collaboration grant in Mathematics, and the Australian Research Council for grant FT130100484.

Author information

Authors and Affiliations

  1. Department of Mathematics, Connecticut College, New London, CT, 06320, USA

    Sanjeeva Balasuriya

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

    Sanjeeva Balasuriya

Authors
  1. Sanjeeva Balasuriya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sanjeeva Balasuriya .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Leicestershire, Leicestershire, United Kingdom

    Wael Bahsoun

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    Christopher Bose

  3. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Gary Froyland

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this paper

Cite this paper

Balasuriya, S. (2014). Nonautonomous Flows as Open Dynamical Systems: Characterising Escape Rates and Time-Varying Boundaries. In: Bahsoun, W., Bose, C., Froyland, G. (eds) Ergodic Theory, Open Dynamics, and Coherent Structures. Springer Proceedings in Mathematics & Statistics, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0419-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation