Multivalued perturbations of a saddle dynamics

  • Giovanni ColomboEmail author
  • Michal Fečkan
  • Barnabas M. Garay
Original Research


We consider multivalued perturbations both of the discrete and of the continuous time hyperbolic dynamical system of the type
$$ x_{k + 1} \in Xx_k + f\left( {x_k } \right) + G\left( {x_k } \right), \dot x \in Ax + f\left( x \right) + G\left( x \right) $$
, where G is a parameterized multivalued map, i.e., \( G\left( x \right) = g\left( {x,\varepsilon \mathcal{B}_\mathcal{X} } \right) \) with \( \mathcal{B}_\mathcal{X} \) denoting the closed unit ball of a Banach space X and ε > 0. Under the assumptions that f and g are Lipschitz, with small Lipschitz constant, we prove that the saddle-type dynamics persists under the multivalued perturbation. More precisely, we construct analogues of the stable and unstable manifolds, which are typical of a single-valued hyperbolic dynamics and remain graphs of Lipschitz maps in the multivalued setting. Under more stringent assumptions on g, we prove some further topological properties. Also the maximal bounded invariant set is investigated.


Stable and unstable manifolds Multivalued/inflated dynamics 

Mathematics Subject Classification (2000)

37D10 34A60 37B55 39A11 


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  1. 1.
    Arnold L. and Kloeden P. E., Discretization of a random dynamical system near a hyperbolic point, Math. Nachr., 181, 43–72, (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aubin J. P. and Cellina A., Differential Inclusions. Set-Valued Maps and Viability Theory, Springer, Berlin, (1984)zbMATHGoogle Scholar
  3. 3.
    Aubin J. P. and Frankowska H., Set-Valued Analysis, Birkhäuser, Basel, (1990)zbMATHGoogle Scholar
  4. 4.
    Aulbach B., The fundamental existence theorem on invariant fiber bundles, J. Differ. Equations Appl., 3, 501–537, (1998)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Aulbach B. and Garay B. M., Discretization of semilinear differential equations with an exponential dichotomy, Comput. Math. Appl., 28, 23–35, (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Aulbach B., Pötzsche Ch. and Siegmund S., A smoothness theorem for invariant fiber bundles, J. Dynam. Differential Equations, 14, 519–547, (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Aulbach B. and Wanner T., The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spacs, Nonlin. Anal. 40, 91–104, (2000)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Aulbach B. and Wanner T., Integral manifolds for Carathéodory-type differential equations in Banach spacs, in: Aulbach B., Colonius F. (Eds.), Six Lectures on Dynamical Systems, World Scientific, Singapore, 45–119, (1996)Google Scholar
  9. 9.
    Aulbach B. and Wanner T., Topological simplification of nonautonomous difference equations, J. Difference Equations Applications, 12, 283–296, (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Aulbach B. and Wanner T., Invariant foliations and decoupling of non-autonomous difference equations, J. Difference Equations Applications, 9, 459–472, (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Aulbach B. and Wanner T., Invariant foliations for Carathéodory-type differential equations in Banach spacs, in: Martynyuk A. A. (Ed.), Advances in Stability Theory at the End of the 20th Century - Stability and Control: Theory, Methods and Applications, Taylor and Francis, London, 13, 1–14, (2003)Google Scholar
  12. 12.
    Battelli F. and Fečkan M., Global center manifolds in singular systems, Nonlinear Differential Equations Applications, 3, 19–34, (1996)CrossRefGoogle Scholar
  13. 13.
    Battelli F. and Lazzari C., Exponential dichotomies, heteroclinic orbits, and Melnikov functions, J. Differential Equations, 86, 342–366, (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Borsuk K., Theory of Retracts, PWN, Warszawa, (1967)zbMATHGoogle Scholar
  15. 15.
    Chaperon M., Invariant manifold theory via generating maps, C.R. Acad. Sci. Paris, Ser. I, 346, 1175–1180, (2008)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Clarke F. H., Ledyaev Yu. S., Stern R. J. and Wolenski P. R., Nonsmooth Analysis and Control Theory, Springer, Berlin, (1998)zbMATHGoogle Scholar
  17. 17.
    Colombo G., Fečkan M. and Garay B. M., Inflated deterministic chaos and Smale’s horseshoe, (submitted)Google Scholar
  18. 18.
    Colombo G. and Nguyen T. Khai, Quantitative isoperimetric inequalities for a class of nonconvex sets, Calc. Var. PDE’s, 37, 141–166, (2010)zbMATHCrossRefGoogle Scholar
  19. 19.
    Debussche A. and Temam R., Some new generalizations of inertial manifolds, Discrete Contin. Dynam. Systems, 2, 543–558, (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Duan J., Lu K. and Schmalfuss B., Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16, 949–972, (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Fečkan M., Transversal bounded solutions for difference equations, J. Difference Equations Applications, 8, 33–51, (2002)zbMATHCrossRefGoogle Scholar
  22. 22.
    Federer H., Curvature measures, Trans. Amer. Math. Soc., 93, 418–491, (1959)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Garay B. M. and Kloeden P. E., Discretization near compact invariant sets, Random Comput. Dynam., 5, 93–123, (1997)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Górniewicz L., Topological approach to differential inclusions, in Topological methods in differential equations and inclusions, Kluwer, Dordrecht, 129–190, (1995)Google Scholar
  25. 25.
    Gruendler J., Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations, 122, 1–26, (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Grüne L., Asymptotic Behaviour of Dynamical and Control Systems under Perturbation and Discretization, Springer, Berlin, (2002)Google Scholar
  27. 27.
    Hilger S., Smoothness of invariant manifolds, J. Funct. Anal., 106, 95–129, (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Himmelberg C. J., Measurable relations, Fund. Math., 87, 53–72, (1975)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Irwin M. C., Smooth Dynamical Systems, Academic Press, New York, (1980)zbMATHGoogle Scholar
  30. 30.
    Ivanov G. E. and Balashov M. V., Lipschitz parameterizations of multivalued mappings with weakly convex values, Izv. Ross. Akad. Nauk Ser. Mat., 71, 47–68, (2007) (in Russian; translation in, Izv. Math., 71, 1123–1143, (2007))MathSciNetGoogle Scholar
  31. 31.
    Kirchraber U. and Palmer K., Geometry in the Neighbourhood of Invariant Manifolds of Maps and Flows and Linearization, Pitman, London, (1991)Google Scholar
  32. 32.
    Kloeden P. E. and Kozyakin V. S., The inflation of attractors and their discretization: the autonomous case, Nonlinear Anal., 40, 333–343, (2000)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Łojasiewicz Jr S., Parametrizations of convex sets, in Progress in approximation theory, Academic, Boston, 629–648, (1991)Google Scholar
  34. 34.
    McGehee R. and Sander E., A new proof of the stable manifold theorem, Z. angew Math. Phys. (ZAMP), 47, 497–513, (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Ornelas A., Parametrization of Carathéodory multifunctions, Rend. Sem. Mat. Univ. Padova, 83, 33–44, (1990)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Palmer K. J., Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55, 225–256, (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Palmer K. J., Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported, 1, 265–306, (1988)MathSciNetGoogle Scholar
  38. 38.
    Palmer K. J., Shadowing in Dynamical Systems, Theory and Applications, Kluwer Academic Publishers, Dordrecht, (2000)zbMATHGoogle Scholar
  39. 39.
    Pilyugin S. and Rieger J., Shadowing and inverse shadowing in set-valued dynamical systems. Hyperbolic case, Topol. Methods Nonlin. Anal., 32, 151–164, (2008)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Pilyugin S. and Rieger J., A general approach to hyperbolicity for set-valued maps, J. Math. Sci., (in print)Google Scholar
  41. 41.
    Robinson J. C., Inertial manifolds with and without delay, Discrete Contin. Dynam. Systems, 5, 813–824, (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Sander E., Hyperbolic sets for noninvertible maps and relations, Discr. Cont. Dyn. Sys., 5, 339–357, (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Shub M., Global Stability of Dynamical Systems, Springer, Berlin, (1987)zbMATHGoogle Scholar
  44. 44.
    Tolstonogov A. A., Differential Inclusions in a Banach Space, Kluwer, Dordrecht, (2000)zbMATHGoogle Scholar
  45. 45.
    Yost D., There can be no Lipschitz version of Michael’s selection theorem, Proceedings of the analysis conference, Singapore (1986), North-Holland, Amsterdam, 295–299, (1988)CrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2010

Authors and Affiliations

  • Giovanni Colombo
    • 1
    Email author
  • Michal Fečkan
    • 2
  • Barnabas M. Garay
    • 3
  1. 1.Department of Pure and Applied MathematicsPadova UniversityPadovaItaly
  2. 2.Department of Mathematical Analysis and Numerical MathematicsComenius UniversityBratislavaSlovakia
  3. 3.Faculty of Information TechnologyPPKE and SZTAKIBudapestHungary

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