Skip to main content

Quadratic Volume-Preserving Maps: (Un)stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations

Abstract

We implement a semi-analytic scheme for numerically computing high order polynomial approximations of the stable and unstable manifolds associated with the fixed points of the normal form for the family of quadratic volume-preserving diffeomorphisms with quadratic inverse. We use this numerical scheme to study some hyperbolic dynamics associated with an invariant structure called a vortex bubble. The vortex bubble, when present in the system, is the dominant feature in the phase space of the quadratic family, as it encloses all invariant dynamics. Our study focuses on visualizing qualitative features of the vortex bubble such as bifurcations in its geometry, the geometry of some three-dimensional homoclinic tangles associated with the bubble, and the “quasi-capture” of homoclinic orbits by neighboring fixed points. Throughout, we couple our results with previous qualitative numerical studies of the elliptic dynamics within the vortex bubble of the quadratic family.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  • Archer, P.J., Thomas, T.G., Coleman, G.N.: Direct numerical simulation of vortex ring evolution from the laminar to the early turbulent regime. Ann. Phys. 598, 201–226 (2008)

    MathSciNet  Google Scholar 

  • Arioli, G., Zgliczyński, P.: Symbolic dynamics for the Hénon-Heiles Hamiltonian on the critical level. J. Differ. Equ. 171(1), 173–202 (2001)

    Article  Google Scholar 

  • Baldomá, I., Fontich, E., de la Llave, R., Martín, P.: The parameterization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points. Discrete Contin. Dyn. Syst. 17(4), 835–865 (2007)

    MathSciNet  Article  Google Scholar 

  • Barge, M.: Homoclinic intersections and indecomposability. Proc. Am. Math. Soc. 101 (1987)

  • Belbruno, E.: Sun-perturbed earth-to-moon transfers with ballistic capture. J. Guid. Control Dyn. 16, 770–775 (1993)

    Article  Google Scholar 

  • Belbruno, E.: Ballistic lunar capture transfers using the fuzzy boundary and solar perturbations: a survey. J. Br. Interplanet. Soc. 47, 73–80 (1994)

    Google Scholar 

  • Berz, M., Hoffstätter, G.: Computation and application of Taylor polynomials with interval remainder bounds. Reliab. Comput. 4(1), 83–97 (1998)

    MathSciNet  Article  Google Scholar 

  • Berz, M., Makino, K.: Cosy infinity (2012). http://www.cosyinfinity.org

  • Beyn, W., Kleinkauf, J.: The numerical computation of homoclinic orbits for maps. SIAM J. Numer. Anal 34(3), 1207–1236 (1997a)

    MathSciNet  Article  Google Scholar 

  • Beyn, W., Kleinkauf, J.: Numerical approximation of homoclinic chaos. In: Dynamical Numerical Analysis (Atlanta, GA, 1995). Numer. Algorithms 14(1–3), 25–53 (1997b)

  • Bollt, E., Meiss, J.D.: Targeting chaotic orbits to the moon. Phys. Lett. A 204, 373–378 (1995)

    Article  Google Scholar 

  • Bücker, H.M., Corliss, G.F.: A bibliography of automatic differentiation. In: Automatic Differentiation: Applications, Theory, and Implementations. Lect. Notes Comput. Sci. Eng., vol. 50, pp. 321–322. Springer, Berlin (2006)

    Chapter  Google Scholar 

  • Burns, K., Weiss, H.: A geometric criterion for positive topological entropy. Commun. Math. Phys. 172(1), 95–118 (1995)

    MathSciNet  Article  Google Scholar 

  • Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J. 52(2), 283–328 (2003a)

    MathSciNet  Article  Google Scholar 

  • Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. II. Regularity with respect to parameters. Indiana Univ. Math. J. 52(2), 329–360 (2003b)

    MathSciNet  Article  Google Scholar 

  • Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds. III. Overview and applications. J. Differ. Equ. 218(2), 444–515 (2005)

    Article  Google Scholar 

  • Cheney, W.: Analysis for Applied Mathematics. Graduate Texts in Mathematics, vol. 208. Springer, Berlin (2001)

    Book  Google Scholar 

  • Chernikov, A.A., Neĭshtadt, A.I., Rogal’sky, A.V., Yakhnin, V.Z.: Adiabatic chaotic advection in nonstationary 2D flows. In: Maiakovskiĭ, V. (ed.) Nonlinear Dynamics of Structures, 1990, pp. 337–345. World Sci. Publ., River Edge (1991)

    Google Scholar 

  • Circi, C., Teofilatto, P.: On the dynamics of weak stability boundary lunar transfers. Celest. Mech. Dyn. Astron. 79, 41–72 (2001)

    Article  Google Scholar 

  • Crow, S.: Stability theory for a pair of trailing vortices. AAIA J. 8, 2172–2179 (1970)

    Article  Google Scholar 

  • Davies, P.A., Koshel, K.V., Sokolovskiy, M.A.: Chaotic advection and nonlinear resonances in a periodic flow above submerged obstacle. In: IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence. IUTAM Bookser., vol. 6, pp. 415–423. Springer, Dordrecht (2008)

    Chapter  Google Scholar 

  • de la Llave, R., González, A., Jorba, À., Villanueva, J.: KAM theory without action-angle variables. Nonlinearity 18(2), 855–895 (2005)

    MathSciNet  Article  Google Scholar 

  • Dullin, H.R., Meiss, J.D.: Nilpotent normal form for divergence-free vector fields and volume-preserving maps. Physica D 237(2), 156–166 (2008)

    MathSciNet  Article  Google Scholar 

  • Dullin, H.R., Meiss, J.D.: Quadratic volume-preserving maps: invariant circles and bifurcations. SIAM J. Appl. Dyn. Syst. 8(1), 76–128 (2009)

    MathSciNet  Article  Google Scholar 

  • Fontich, E., de la Llave, R., Sire, Y.: Construction of invariant whiskered tori by a parameterization method. I. Maps and flows in finite dimensions. J. Differ. Equ. 246(8), 3136–3213 (2009)

    Article  Google Scholar 

  • Gidea, M., Masdemont, J.: Geometry of homoclinic connections in a planar circular restricted three-body problem. Int. J. Bifurc. Chaos 17 (2007)

  • Gomez, G., Koon, W.S., Lo, M.W., Marsden, J.E., Masdemont, J., Ross, S.D.: Connecting orbits and invariant manifolds in the spatial restricted three-body problem. Nonlinearity 17, 1571–1606 (2004)

    MathSciNet  Article  Google Scholar 

  • Gonchenko, S.V., Meiss, J.D., Ovsyannikov, I.I.: Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation. Regul. Chaotic Dyn. 11(2), 191–212 (2006)

    MathSciNet  Article  Google Scholar 

  • Haro, À., de la Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms. Discrete Contin. Dyn. Syst., Ser. B 6(6), 1261–1300 (2006a)

    MathSciNet  Article  Google Scholar 

  • Haro, A., de la Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results. J. Differ. Equ. 228(2), 530–579 (2006b)

    Article  Google Scholar 

  • Hayashi, T., Mizuguchi, N., Sato, T.: Magnetic reconnection and relaxation phenomena in a spherical tokamak. Earth Planets Space 53, 561–564 (2001)

    Google Scholar 

  • Hénon, M.: Numerical study of quadratic area-preserving mappings. Q. Appl. Math. 27, 291–312 (1969)

    Google Scholar 

  • Johnson, T., Tucker, W.: A note on the convergence of parametrised non-resonant invariant manifolds. Qual. Theory Dyn. Syst. 10, 107–121 (2011)

    MathSciNet  Article  Google Scholar 

  • Kaper, T.J., Wiggins, S.: Lobe area in adiabatic Hamiltonian systems. Physica D 51(1–3), 205–212 (1991). Nonlinear science: the next decade (Los Alamos, NM, 1990)

    MathSciNet  Article  Google Scholar 

  • Katok, A.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press (2012). With a supplementary chapter by Katok and Leonardo Mendoza

  • Kennedy, J.A., Yorke, J.A.: The topology of stirred fluids. Topol. Appl. 80 (1997)

  • Krauskopf, B., Osinga, H.M., Doedel, E.J., Henderson, M.E., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., Junge, O.: A survey of methods for computing (un)stable manifolds of vector fields. Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3), 763–791 (2005)

    MathSciNet  Article  Google Scholar 

  • Krutzsch, C.: Uber eine experimentell beobachtete Erscheinung an Wirbelringen bei ihrer translatorischen Bewegung in wirklichen Flussigkeiten. Ann. Phys. 5, 497–523 (1939)

    Article  Google Scholar 

  • Lessard, J.P., Mireles James, J.D., Reinhardt, Ch.: Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields (2013, submitted)

  • Lomelí, H.E., Meiss, J.D.: Quadratic volume-preserving maps. Nonlinearity 11(3), 557–574 (1998)

    MathSciNet  Article  Google Scholar 

  • Lomelí, H.E., Meiss, J.D.: Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps. Chaos 10(1), 109–121 (2000). Chaotic kinetics and transport (New York, 1998)

    MathSciNet  Article  Google Scholar 

  • Lomelí, H.E., Meiss, J.D.: Resonance zones and lobe volumes for exact volume-preserving maps. Nonlinearity 22(8), 1761–1789 (2009)

    MathSciNet  Article  Google Scholar 

  • Lomelí, H.E., Ramírez-Ros, R.: Separatrix splitting in 3D volume-preserving maps. SIAM J. Appl. Dyn. Syst. 7, 1527–1557 (2008)

    MathSciNet  Article  Google Scholar 

  • MacKay, R.S.: Transport in 3D volume-preserving flows. J. Nonlinear Sci. 4, 329–354 (1994)

    MathSciNet  Article  Google Scholar 

  • Makino, K., Berz, M.: Taylor models and other validated functional inclusion methods. Int. J. Pure Appl. Math. 6(3), 239–316 (2003)

    MathSciNet  Google Scholar 

  • Mezić, I.: Chaotic advection in bounded Navier–Stokes flows. J. Fluid Mech. 431, 347–370 (2001)

    MathSciNet  Article  Google Scholar 

  • Mireles James, J.D., Lomelí, H.: Computation of heteroclinic arcs for the volume preserving Hénon map. SIAM J. Appl. Dyn. Syst. 9(3), 919–953 (2010)

    MathSciNet  Article  Google Scholar 

  • Mireles James, J.D., Mischaikow, K.: Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps. In: SIADS (2013, to appear)

  • Mullowney, P., Julien, K., Meiss, J.D.: Blinking rolls: chaotic advection in a three-dimensional flow with an invariant. SIAM J. Appl. Dyn. Syst. 4(1), 159–186 (2005) (electronic)

    MathSciNet  Article  Google Scholar 

  • Mullowney, P., Julien, K., Meiss, J.D.: Chaotic advection and the emergence of tori in the Küppers–Lortz state. Chaos 18(3), 033104 (2008)

    MathSciNet  Article  Google Scholar 

  • Neishtadt, A.I., Vainshtein, D.L., Vasiliev, A.A.: Chaotic advection in a cubic Stokes flow. Physica D 111(1–4), 227–242 (1998)

    MathSciNet  Article  Google Scholar 

  • Neumaier, A., Rage, T.: Rigorous chaos verification in discrete dynamical systems. Physica D 67(4), 327–346 (1994)

    MathSciNet  Article  Google Scholar 

  • Newhouse, S., Berz, M., Grote, J., Makino, K.: On the estimation of topological entropy on surfaces. In: Geometric and Probabilistic Structures in Dynamics. Contemp. Math., vol. 469, pp. 243–270. Amer. Math. Soc., Providence (2008)

    Chapter  Google Scholar 

  • Palis, J. Jr., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, New York (1982). An introduction, Translated from the Portuguese by A.K. Manning

    Book  Google Scholar 

  • Peikert, Sadlo: Topology-Guided Visualization of Constrained Vector Fields. Springer, Berlin (2007)

    Google Scholar 

  • Raynal, F., Wiggins, S.: Lobe dynamics in a kinematic model of a meandering jet. I. Geometry and statistics of transport and lobe dynamics with accelerated convergence. Physica D 223(1), 7–25 (2006)

    MathSciNet  Article  Google Scholar 

  • Robinson, C.: Dynamical Systems, 2nd edn. Studies in Advanced Mathematics. CRC Press, Boca Raton (1999). Stability, symbolic dynamics, and chaos

    Google Scholar 

  • Senet, J., Ocampo, C.: Low-thrust variable specific impulse transfers and guidance to unstable periodic orbits. J. Guid. Control Dyn. 28, 280–290 (2005)

    Article  Google Scholar 

  • Shadden, S.C., Katija, D., Rosenfeld, M., Marsden, J.E., Dabiri, J.O.: Transport and stirring induced by vortex formation. Ann. Phys. 5, 497–523 (2007)

    Google Scholar 

  • Smale, S.: Diffeomorphisms with many periodic points. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 63–80. Princeton Univ. Press, Princeton (1965)

    Google Scholar 

  • Sotiropoulos, F., Ventikos, Y., Lackey, T.C.: Chaotic advection in three-dimensional stationary vortex-breakdown bubbles: Sil’nikov’s chaos and the devil’s staircase. J. Fluid Mech. 444, 257–297 (2001)

    MathSciNet  Article  Google Scholar 

  • van den Berg, J.B., Mireles James, J.D., Lessard, J.P., Mischaikow, K.: Rigorous numerics for symmetric connecting orbits: even homoclinics for the Gray-Scott equation. SIAM J. Math. Anal. 43, 1557–1594 (2011)

    MathSciNet  Article  Google Scholar 

  • Zbigniew, G., Zgliczyński, P.: Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map. Nonlinearity 14(5), 909–932 (2001)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author was supported by NSF grant DMS 0354567, by a DARPA FunBio grant, and also by the University of Texas Department of Mathematics Program in Applied and Computational Analysis RTG Fellowship, during the preparation of this work. The author would like to thank Professor Rafael de la Llave for his continued encouragement and support as this manuscript evolved. The images in the manuscript are generated using the PovRay ray-tracing software. Thanks go to Dr. Jason Chambless for many helpful suggestions and insights on the use of the PovRay package. The author spent a week in February 2010 visiting the University of Colorado at Boulder Department of Applied Mathematics, and conversations with Professor James Meiss greatly improved and influenced the content of this work. Thanks again to Professors Meiss and Dullin for granting their permission to reproduce Fig. 3 in the present manuscript. Thanks go to Professors Meiss and Curry and also to Mr. Brock Alan Mosovsky and Ms. Kristine Snyder for their hospitality during that visit. Finally, the author thanks Professors de la Llave and Meiss for carefully reading the manuscript and for their many helpful comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. D. Mireles James.

Additional information

Communicated by G. Haller.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mireles James, J.D. Quadratic Volume-Preserving Maps: (Un)stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations. J Nonlinear Sci 23, 585–615 (2013). https://doi.org/10.1007/s00332-012-9162-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-012-9162-1

Keywords

  • Volume preserving dynamics
  • Stable and unstable manifolds
  • Parameterization method
  • Homoclinic chaos

Mathematics Subject Classification

  • 37C05
  • 37D10
  • 37D45