Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-18T18:15:20.680Z Has data issue: false hasContentIssue false
  • English
  • Français

LYAPUNOV EXPONENTS OF THE KURAMOTO–SIVASHINSKY PDE

Part of: Infinite-dimensional dissipative dynamical systems

Published online by Cambridge University Press:  15 July 2019

RUSSELL A. EDSON Open the ORCID record for RUSSELL A. EDSON [Opens in a new window] ,
J. E. BUNDER Open the ORCID record for J. E. BUNDER [Opens in a new window] ,
TRENT W. MATTNER Open the ORCID record for TRENT W. MATTNER [Opens in a new window]  and
A. J. ROBERTS Open the ORCID record for A. J. ROBERTS [Opens in a new window]
RUSSELL A. EDSON
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia email russell.edson@adelaide.edu.au, judith.bunder@adelaide.edu.au, trent.mattner@adelaide.edu.au, anthony.roberts@adelaide.edu.au
J. E. BUNDER*
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia email russell.edson@adelaide.edu.au, judith.bunder@adelaide.edu.au, trent.mattner@adelaide.edu.au, anthony.roberts@adelaide.edu.au
TRENT W. MATTNER
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia email russell.edson@adelaide.edu.au, judith.bunder@adelaide.edu.au, trent.mattner@adelaide.edu.au, anthony.roberts@adelaide.edu.au
A. J. ROBERTS
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia email russell.edson@adelaide.edu.au, judith.bunder@adelaide.edu.au, trent.mattner@adelaide.edu.au, anthony.roberts@adelaide.edu.au
*
✉judith.bunder@adelaide.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence.

Keywords

Lyapunov exponentsdynamical systems

MSC classification

Primary: 37L30: Attractors and their dimensions, Lyapunov exponents
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 3 , July 2019 , pp. 270 - 285
Copyright
© 2019 Australian Mathematical Society 

References

Benettin, G., Galgani, L., Giorgilli, A. and Strelcyn, J.-M., “Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 2. Numerical application”, Meccanica 15 (1980) 21–30; doi:10.1007/BF02128237.Google Scholar
Chertovskih, R., Chimanski, E. V. and Rempel, E. L., “Route to hyperchaos in Rayleigh–Bénard convection”, Europhys. Lett. 112 (14001); doi:10.1209/0295-5075/112/14001.Google Scholar
Chicone, C., Ordinary differential equations with applications, Volume 34 Texts in Applied Mathematics (Springer, New York, 2006); doi:10.1007/0-387-35794-7.Google Scholar
Cross, M. C. and Hohenberg, P. C., “Hohenberg. Pattern formation outside of equilibrium”, Rev. Mod. Phys. 65 (1993) 851–1112; doi:10.1103/RevModPhys.65.851.Google Scholar
Cvitanović, P., Davidchack, R. and Siminos, E., “On the state space geometry of the Kuramoto–Sivashinsky flow in a periodic domain”, SIAM J. Appl. Dyn. Syst. 9 (2010) 1–33; doi:10.1137/070705623.Google Scholar
Dankowicz, H., Holmes, P., Berkooz, G. and Elezgaray, J., “Local models of spatio-temporally complex fields”, Physica D 90 (1996) 387–407; doi:10.1016/0167-2789(95)00245-6.Google Scholar
Dieci, L., Russell, R. and Van Vleck, E., “On the computation of Lyapunov exponents for continuous dynamical systems”, SIAM J. Numer. Anal. 34 (1997) 402–423; doi:10.1137/S0036142993247311.Google Scholar
Eckmann, J.-P. and Ruelle, D., “J. Ergodic theory of chaos and strange attractors”, Rev. Mod. Phys. 57 (1985) 617–656; doi:10.1103/RevModPhys.57.617.Google Scholar
Eguíluz, V. M., Alstrøm, P., Hernández-García, E. and Piro, O., “Average patterns of spatiotemporal chaos: a boundary effect”, Phys. Rev. E 59 (1999) 2822–2825; doi:10.1103/PhysRevE.59.2822.Google Scholar
Foias, C., Nicolaenko, B., Sell, G. R. and Temam, R., Inertial manifolds for the Kuramoto–Sivashinsky equation and an estimate of their lowest dimension”, Technical Report, University of Minnesota Digital Conservancy, 1986, http://hdl.handle.net/11299/4494.Google Scholar
Geist, K., Parlitz, U. and Lauterborn, W., “Comparison of different methods for computing Lyapunov exponents”, Prog. Theor. Phys. 83 (1990) 875–893; doi:10.1143/PTP.83.875.Google Scholar
Grassberger, P. and Procaccia, I., “Measuring the strangeness of strange attractors”, Physica D 9 (1983) 189–208; doi:10.1016/0167-2789(83)90298-1.Google Scholar
Greenside, H. S., “Spatiotemporal chaos in large systems: the scaling of complexity with size”, Technical Report, 1996; https://arxiv.org/abs/chao-dyn/9612004.Google Scholar
Hassanaly, M. and Raman, V., “Ensemble-LES analysis of perturbation response of turbulent partially-premixed flames”, P. Combust Inst. 37 (2018) 2249–2257; doi:10.1016/j.proci.2018.06.209.Google Scholar
Hohenberg, P. C. and Shraiman, B. I., “Chaotic behavior of an extended system”, Physica D 37 (1989) 109–115; doi:10.1016/0167-2789(89)90121-8.Google Scholar
Hyman, J. M. and Nicolaenko, B., “The Kuramoto–Sivashinsky equation: a bridge between PDEs and dynamical systems”, Physica D 18 (1986) 113–126; doi:10.1016/0167-2789(86)90166-1.Google Scholar
Hyman, J. M., Nicolaenko, B. and Zaleski, S., “Order and complexity in the Kuramoto–Sivashinsky model of weakly turbulent interfaces”, Physica D 23 (1986) 265–292; doi:10.1016/0167-2789(86)90136-3.Google Scholar
Kaplan, J. L. and Yorke, J. A., “Chaotic behavior of multidimensional difference equations”, in: Functional differential equations and approximation of fixed points (eds Peitgen, H.-O. and Walther, H.-O.), (Springer, Berlin, 1979) 204–227; doi:10.1007/BFb0064319.Google Scholar
Keefe, L., Moin, P. and Kim, J., “The dimension of attractors underlying periodic turbulent Poiseuille flow”, J. Fluid Mech. 242 (1992) 1–29; doi:10.1017/S0022112092002258.Google Scholar
Kevrekidis, I. G. and Samaey, G., “Equation-free multiscale computation: algorithms and applications”, Annu. Rev. Phys. Chem. 60 (2009) 321–344; doi:10.1146/annurev.physchem.59.032607.093610.Google Scholar
Kuramoto, Y. and Tsuzuki, T., “Persistent propagation of concentration waves in dissipative media far from thermal equilibrium”, Prog. Theor. Phys. 55 (1976) 356–369; doi:10.1143/PTP.55.356.Google Scholar
Lan, Y. and Cvitanović, P., “Unstable recurrent patterns in Kuramoto–Sivashinsky dynamics”, Phys. Rev. E 78 (2008) 026208; doi:10.1103/PhysRevE.78.026208.Google Scholar
Manneville, P., “Liapounov exponents for the Kuramoto–Sivashinsky model”, in: Macroscopic modelling of turbulent flows (eds Frisch, U., Keller, J. B., Papanicolaou, G. C. and Pironneau, O.), (Springer, Berlin, 1985) 319–326; doi:10.1007/3-540-15644-5 26.Google Scholar
Pomeau, Y. and Zaleski, S., “The Kuramoto–Sivashinsky equation: a caricature of hydrodynamic turbulence?” in: Macroscopic modelling of turbulent flows (eds Frisch, U., Keller, J. B., Papanicolaou, G. C. and Pironneau, O.), (Springer, Berlin, 1985) 296–303; doi:10.1007/3-540-15644-5 23.Google Scholar
Rempel, E. L., Chian, A. C.-L., Macau, E. E. N. and Rosa, R. R., “Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto–Sivashinsky equation”, Chaos 14 (2004) 545–556; doi:10.1063/1.1759297.Google Scholar
Ruelle, D., “Ergodic theory of differentiable dynamical systems”, Publ. Math. Inst. Hautes Études Sci. 50 (1979) 27–58; doi:10.1007/bf02684768.Google Scholar
Ruelle, D. and Takens, F., “On the nature of turbulence”, Commun. Math. Phys. 20 (1971) 167–192; doi:10.1007/BF01646553.Google Scholar
Shimada, I. and Nagashima, T., “A numerical approach to ergodic problem of dissipative dynamical systems”, Prog. Theor. Phys. 61 (1979) 1605–1616; doi:10.1143/PTP.61.1605.Google Scholar
Sivashinsky, G. I., “Nonlinear analysis of hydrodynamic instability in laminar flames–I. Derivation of basic equations”, Acta Astron. 4 (1977) 1177–1206; doi:10.1016/0094-5765(77)90096-0.Google Scholar
Sivashinsky, G. I. and Michelson, D. M., “On irregular wavy flow of a liquid film down a vertical plane”, Prog. Theor. Phys. 63 (1980) 2112–2114; doi:10.1143/PTP.63.2112.Google Scholar
Skokos, C., “The Lyapunov characteristic exponents and their computation”, in: Dynamics of small solar system bodies and exoplanets (eds Souchay, J. J. and Dvorak, R.), (Springer, Berlin, 2010) 63–135; doi:10.1007/978-3-642-04458-8_2.Google Scholar
Sprott, J. C., Elegant chaos: algebraically simple chaotic flows (World Scientific, Singapore, 2010); doi:10.1142/7183.Google Scholar
Tajima, S. and Greenside, H. S., “Microextensive chaos of a spatially extended system”, Phys. Rev. E 66 (2002) 017205; doi:10.1103/PhysRevE.66.017205.Google Scholar
Takens, F., Detecting strange attractors in turbulence, Lect. Notes in Math. (Springer, Berlin, 1981) 366–381; doi:10.1007/bfb0091924.Google Scholar
Wittenberg, R. W. and Holmes, P., “Scale and space localization in the Kuramoto–Sivashinsky equation”, Chaos 9 (1999) 452–465; doi:10.1063/1.166419.Google Scholar
Xu, M., “Spatiotemporal chaos in large systems driven far-from-equilibrium: connecting theory with experiment”, Ph. D. Thesis, Virginia Polytechnic Institute and State University, 2017. https://vtechworks.lib.vt.edu/handle/10919/79499.Google Scholar
Yang, H.-L., Takeuchi, K. A., Ginelli, F., Chaté, H. and Radons, G., “Hyperbolicity and the effective dimension of spatially extended dissipative systems”, Phys. Rev. Lett. 102 (2009) 074102; doi:10.1103/PhysRevLett.102.074102.21.Google Scholar