Abstract
In this paper we first prove the equivalence between the system of coupled Volterra gyrostats and a special class of energy-conserving low-order models. We then extend the definition of the classical Volterra gyrostat to include nonlinear feedback, resulting in a class of generalized Volterra gyrostats. Using this new class of gyrostats as a basic building block, we present an algorithm for converting a general class of energy-conserving low-order models that routinely arise in fluid dynamics, turbulence, and atmospheric sciences into a system of coupled generalized Volterra gyrostats with nonlinear feedback.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Achatz, V., Schmitz, G., Greisinger, K.M.: Principal interaction patterns in baroclinic wave life cycles. J. Atmos. Sci. 52, 3201–3213 (1995)
Bohr, T., Jensen, M.H., Paladin, G., Vulpiani, A.: Dynamical Systems Approach to Turbulence. Cambridge University Press, Cambridge (1998), 350 pp.
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domain. Springer, New York (2006), 563 pp.
Charney, J.G., Devore, J.G.: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci. 36, 1205–1216 (1979)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2004), 1180 pp.
Crommelin, D.T., Majda, A.J.: Strategies for model reduction: Comparing different optimal bases. J. Atmos. Sci. 61, 2206–2217 (2004)
De Swart, H.E.: Low-order spectral models of the atmospheric circulation—a survey. Acta Appl. Math. 11, 49–96 (1988)
DelSole: Optimally persistent patterns in time varying fields. J. Atmos. Sci. 58, 1341–1356 (2001)
Farell, B.F., Ioannou, P.J.: Accurate low-dimensional approximation of the linear dynamics of fluid flow. J. Atmos. Sci. 58, 2771–2789 (2001)
Friedrich, R., Haken, H.: Nonequilibrium phase transition in a system with chaotic dynamics. The ABCDE model. Phys. Lett. A 164, 299–304 (1992)
Gluhovsky, A.: Nonlinear systems that are superpositions of gyrostats. Sov. Phys. Dokl. 27, 823–825 (1982)
Gluhovsky, A.: Energy conserving and Hamiltonian low-order models in geophysical fluid dynamics. Nonlinear Process. Geophys. 13, 125–133 (2006)
Gluhovsky, A., Tong, C.: The structure of energy conserving low-order models. Phys. Fluids 11, 334–343 (1999)
Gluhovsky, A., Tong, C., Agee, E.: Selection of modes in convective low-order models. J. Atmos. Sci. 59, 1383–1393 (2002)
Haken, H.: Analogy between higher instabilities in fluids and lasers. Phys. Lett. A 53, 77–79 (1975)
Hemati, N.: Strange attractors in brushless DC motors. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 41, 40–45 (1994)
Hermiz, K.B., Guzdar, P.N., Finn, J.M.: Improved low-order model for shear flow driven by Rayleigh-Bernard convection. Phys. Rev. E 51, 325–331 (1995)
Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996), 420 pp.
Howard, L.N., Krishnamurti, R.: Large-scale flow in turbulent convection: a mathematical model. J. Fluid Mech. 170, 385–410 (1986)
Knobloch, E.: Chaos in segmented disc dynamo. Phys. Lett. A 82, 439–440 (1981)
Krishnamurti, T.N., Bedi, H.S., Hardiker, V.M., Ramaswarny, L.: An Introduction to Global Spectral Modeling, 2nd edn. Springer, New York (2006), 317 pp.
Kundu, P.K.: Fluid Mechanics. Academic Press, New York (1990), 638 pp.
Kwasniok, F.: The reduction of complex dynamical systems using principal interaction patterns. Physica D 92, 28–60 (1996)
Kwasniok, F.: Low-dimensional models of the Ginsburg-Landau equation. SIAM J. Appl. Math. 61, 2063–2079 (2001)
Kwasniok, F.: Empirical low-order models of barotropic flow. J. Atmos. Sci. 61, 235–245 (2004)
Lakshmivarahan, S., Wang, Y.: On the structure of energy conserving low-order models and their relation to Volterra gyrostat. Nonlinear Anal. Ser. B: Real World Appl. (2007, to appear)
Lakshmivarahan, S., Baldwin, M.E., Zheng, T.: Further analysis of Lorenz’s maximum simplification equation. J. Atmos. Sci. 63, 2673–2699 (2006)
Landau, L.D., Lifshitz, E.M.: Mechanics. Addison-Wesley, Reading (1960)
Legras, B., Ghil, M.: Persistent anomalies, blocking and variations in atmospheric predictability. J. Atmos. Sci. 42, 433–471 (1985)
Leipnik, R.B., Newton, T.A.: Double strange attractors in rigid body motion with linear feedback control. Phys. Lett. A 86, 63–67 (1981)
Li, Q., Wang, H.: Has chaos implied by macrovariable equations been justified. Phys. Rev. E 58, R1191–1194 (1998)
Lorenz, E.N.: Maximum simplification of the dynamic equations. Tellus 12, 243–254 (1960)
Lorenz, E.N.: Simplified dynamic equations applied to the rotating-basin experiments. J. Atmos. Sci. 19, 39–51 (1962)
Lorenz, E.N.: Deterministic nonperiodic flows. J. Atmos. Sci. 20, 130–141 (1963)
Lorenz, E.N.: Low-order models of atmospheric circulation. J. Meteorol. Soc. Jpn. 60, 255–267 (1982)
Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer, New York (1999), 582 pp.
Obukhov, A.M.: On the problem of nonlinear interactions in fluid dynamics. Gerlands Beitr. Geophysics 82, 282–290 (1973)
Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987), 710 pp.
Peña, M., Kalnay, E.: Separating fast and slow modes in coupled chaotic systems. Nonlinear Process. Geophys. 11, 319–327 (2004)
Platzman, G.W.: An exact integral of complete spectral equation for unsteady one dimensional flow. Tellus 16, 422–431 (1964)
Rega, G., Troger, H.: Dimension reduction of dynamical systems: Methods, models, application, special issue on dimension reduction of dynamical systems. Nonlinear Dyn. 41, 1–15 (2005)
Rinne, J., Karhilla, V.: A spectral barotropic model in horizontal empirical orthogonal functions. Q. J. Roy. Meteorol. Soc. 101, 365–382 (1975)
Roebber, P.J.: Climate variability in a low-order coupled atmosphere-ocean model. Tellus A 47, 473–494 (1995)
Saltzman, B.: Finite amplitude free convection as an initial value problem, I. J. Atmos. Sci. 19, 329–341 (1962)
Silberman, I.: Planetary waves in the atmosphere. J. Meteorol. 11, 27–34 (1954)
Smith, T.R., Moehlis, J., Holmes, P.: Low-dimensional modeling of turbulence using proper orthogonal decomposition: A tutorial. Special issue on dimension reduction of dynamical systems. Nonlinear Dyn. 41, 275–307 (2005)
Starostin, E.L.: Three-dimensional shapes of looped DNA. Meccanica 31, 235–271 (1996)
Thiffeault, J.L., Horton, W.: Energy-conserving truncations for convection with shear flow. Phys. Fluids 8, 1715–1719 (1996)
Tong, C., Gluhovsky, A.: Energy-conserving low-order models for three-dimensional Rayleigh–Bénard convection. Phys. Rev. E 65, 04306-1–04396-11 (2002)
Wang, Y., Lakshmivarahan, S.: Analysis of bifurcation and stability of equilibria in energy conserving low-order models: Volterra gyrostat. In: SIAM Conference on the Applications of the Dynamical Systems, Snowbird, Utah, USA, 28 May–1 June 2007
Wittenberg, J.: Dynamics of Systems of Rigid Bodies. B.G. Teubner, Stuttgar (1977), 224 pp.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Holmes.
Rights and permissions
About this article
Cite this article
Lakshmivarahan, S., Wang, Y. On the Relation between Energy-Conserving Low-Order Models and a System of Coupled Generalized Volterra Gyrostats with Nonlinear Feedback. J Nonlinear Sci 18, 75–97 (2008). https://doi.org/10.1007/s00332-007-9006-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-007-9006-6