Archive for Rational Mechanics and Analysis

, Volume 197, Issue 3, pp 1033–1051 | Cite as

Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation

  • Gianni ArioliEmail author
  • Hans Koch


We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.


Bifurcation Diagram Open Neighborhood Bifurcation Point Unstable Manifold Positive Real Part 
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  1. 1.
    Kuramoto Y., Tsuzuki T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progr. Theor. Phys. 55, 365–369 (1976)ADSGoogle Scholar
  2. 2.
    Sivashinsky G.I.: Nonlinear analysis of hydrodynamic instability in laminal flames— I. Derivation of basic equations. Acta Astr. 4, 1177–1206 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kevrekidis J.G., Nicolaenko B., Scovel J.C.: Back in the saddle again: a computer assisted study of the Kuramoto–Sivashinsky equation. SIAM J. Appl. Math. 50, 760–790 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Jolly M.S., Kevrekidis J.G., Titi E.S.: Approximate inertial manifolds for the Kuramoto–Sivashinsky equation: analysis and computations. Physica D 44, 38–60 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Ilyashenko Y.S.: Global analysis of the phase portrait for the Kuramoto–Sivashinski equation. J. Dyn. Differ. Equ. 4, 585–615 (1992)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Collet P., Eckmann J.-P., Epstein H., Stubbe J.: A global attracting set for the Kuramoto-Sivashinsky equation. Commun. Math. Phys. 152, 203–214 (1993)zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Grujić Z.: Spatial analyticity on the global attractor for the Kuramoto–Sivashinsky equation. J. Dyn. Differ. Equ. 12, 217–228 (2000)zbMATHCrossRefGoogle Scholar
  8. 8.
    Zgliczyński P., Mischaikow K.: Rigorous numerics for partial differential equations: the Kuramoto–Sivashinsky equation. Found. Comp. Math. 1, 255–288 (2001)zbMATHGoogle Scholar
  9. 9.
    Zgliczyński P., Mischaikow K.: Towards a rigorous steady states bifurcation diagram for the Kuramoto–Sivashinsky equation—a computer assisted rigorous approach. Preprint, available at, 2003
  10. 10.
    Arioli G., Koch H., Terracini S.: Two novel methods and multi-mode periodic solutions for the Fermi-Pasta-Ulam model. Comm. Math. Phys. 255, 1–19 (2005)zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)zbMATHGoogle Scholar
  12. 12.
    Bates, P.W., Lu, K., Zeng, C.: Existence and persistence of invariant manifolds for semiflows in Banach space. Mem. Amer. Math. Soc. 135(645) (1998)Google Scholar
  13. 13.
    de la Llave R.: A smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearities. J. Dyn. Diff. Eq. 21, 371–415 (2009)zbMATHCrossRefGoogle Scholar
  14. 14.
    Dunford N., Schwartz J.T.: Linear Operators. Part I: General Theory. Wiley- Interscience, New Edition (1988)zbMATHGoogle Scholar
  15. 15.
    The GNU NYU Ada 9X Translator. Available at and many other places

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica and MOXPolitecnico di MilanoMilanItaly
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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