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Infinite dimensional dynamical systems

Part of the Lecture Notes in Mathematics book series (LNM,volume 1007)

Keywords

  • Periodic Orbit
  • Equilibrium Point
  • Parabolic Equation
  • Unstable Manifold
  • Functional Differential Equation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Billotti, J. and J.P. LaSalle, Periodic dissipative processes. Bull. Am. Math. Soc. 6(1971), 1082–1089.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Brunovsky, P. and S.N. Chow, Generic properties of stationary states of reaction diffusion equations. J. Differential Equations. To appear.

    Google Scholar 

  3. Cartwright, M.L., Almost periodic flows and solutions of differential equations. Proc. London Math. Soc. (3) 17(1967), 355–380, Corrigenda (3) 17(1967), 769.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. _____, Almost periodic differential equations and almost periodic flows. J. Differential Eqns. 5(1969), 167–181.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Chafee, N. and E.F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type. J. Math. Anal. Appl. 4(1974), 17–37.

    MathSciNet  MATH  Google Scholar 

  6. Glass, L. and M.C. Mackey, Pathological conditions arising from instabilities in physiological control systems. Annals N.Y. Acad. Sci. 316(1979), 214–235.

    CrossRef  MATH  Google Scholar 

  7. Hale, J.K., Theory of Functional Differential Equations, Applied Math. Sci., Vol. 3, 2nd Edition, Springer-Verlag, 1977.

    Google Scholar 

  8. _____, Topics in Dynamic Bifurcation Theory, CBMS Regional Conference Series in Math., No. 47(1981), Am. Math. Soc., Providence, R. I.

    Google Scholar 

  9. _____, Some recent results on dissipative systems. Functional Differential Equations and Bifurcation (Ed. A.F. Ize), Lecture Notes in Math. Vol 799(1980), Springer-Verlag, pp. 152–172.

    Google Scholar 

  10. _____, Generic properties of an integro-differential equation. Am. J. Math. To appear.

    Google Scholar 

  11. Hale, J.K. and O. Lopes, Fixed point theorems and dissipative processes. J. Differential Eqns. 13(1973), 391–402.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Hale, J.K, and P. Massatt, Asymptotic behavior of gradient-like systems. Proc. 2nd Fla. Conf. on Dyn. Sys. Feb. 1981. To appear.

    Google Scholar 

  13. Hale, J.K. and K. Rybakowski, On a gradient-like integro-differential equation. Proc. Royal Soc. Edinburgh, Ser. A. To appear.

    Google Scholar 

  14. Henry, D., Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math., Vol.840(1981), Springer-Verlag.

    Google Scholar 

  15. _____, Gradient flows defined by parabolic equations, pp. 122–128 in Nonlinear Diffusion (Eds. Fitzgibbon and Walker), Pitman, 1977.

    Google Scholar 

  16. Kurzweil, J., Global solutions of functional differential equations, in Lecture Notes in Math., Vol. 144(1970), Springer-Verlag.

    Google Scholar 

  17. Ladyzenskaya, O.A., A dynamical system generated by the Navier-Stokes equation, J. Sov. Math., 3(1975), 458–479.

    CrossRef  Google Scholar 

  18. Lasota, A., Ergodic theorems in biology, Asterique 50(1977), 239–250.

    MathSciNet  Google Scholar 

  19. Lasota, A. and M. Wazewska-Czyzewska, Matematyczne problemy dynamiki ukladu krwinek czerwonych, Mat. Stosowana 6(1976), 23–40.

    MathSciNet  Google Scholar 

  20. Levin, J.J. and J. Nohel, On a nonlinear delay equation. J. Math. Ana. Appl. 8(1964), 31–44.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Mallet-Paret, J., Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Differential Equations 22(1976), 331–348.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. _____, Generic properties of retarded functional differential equations. Bull. Am. Math. Soc. 81(1975), 750–752.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. _____, Generic periodic solutions of functional differential equations, J. Differential Equations 25 (1977), 163–183.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Mañé, R., On the dimension of the compact invariant set of certain nonlinear maps. Preprint.

    Google Scholar 

  25. Manselli, P. and K. Miller, Dimensionality reduction methods for efficient numerical solution, backward in time, of parabolic equations with variable coefficients. SIAM J. Math. Anal. 11(1980), 147–159.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Marsden, J.E. and M. McCracken, The Hopf Bifurcation and Its Applications, Appl. Math. Sci. Vol.19, Springer-Verlag, 1976.

    Google Scholar 

  27. Massatt, P., Stability and fixed points of dissipative systems. J. Differential Equations 40(1981), 217–231.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. _____, Attractivity properties of α-contractions. J. Differential Equations. To appear.

    Google Scholar 

  29. _____, Asymptotic behavior of a strongly damped nonlinear wave equation. J. Differential Equations. To appear.

    Google Scholar 

  30. Matano, H., Covergence of solutions of one-dimensional semi-linear parabolic equations, J. Math. Kyoto Univ. 18(1978), 221–227.

    MathSciNet  MATH  Google Scholar 

  31. _____, Nonincrease of the lap number of a solution for a one-dimensional semilinear parabolic equation. Pub. Fac. Sci. Univ. Tokyo. To appear.

    Google Scholar 

  32. Miller, K., Nonunique continuation for certain ODE's in Hilbert space and for uniformly parabolic and elliptic equations in self-adjoint divergence form, p.85–101, Lecture Notes in Math., Vol.316(1973), Springer-Verlag.

    Google Scholar 

  33. Nussbaum, R., Periodic solutions of analytic functional differential equations are analytic. Mich. Math. J. 20(1973), 249–255.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. _____, Periodic solutions of nonlinear autonomous functional differential equations, p.283–325 in Functional Differential Equations and Approximation of Fixed Points, (Eds. Peitgen and Walther), Lecture Notes in Math., Vol.730, 1979.

    CrossRef  MathSciNet  Google Scholar 

  35. Oliva, W.M., The behavior at infinity and the set of global solutions of retarded functional differential equations. Symposium on Functional Differential Equations, São Carlos, Brisil (1975), Coleção ATAS, Vol.8 Soc. Brasileira de Mat.

    Google Scholar 

  36. Peters, H., Comportement chaotique d'une équation différéntielle retardée. C.R. Acad. Sci. Paris 290(1980), Ser. A., 1119–1122.

    MATH  Google Scholar 

  37. Smoller, J. and A. Wasserman, Global bifurcation of steady state solutions. J. Differential Equations.

    Google Scholar 

  38. Walther, H.O., Homoclinic solution and chaos in x(t)=f(x(t−1)). Nonlin. Anal. 5(1981), 775–788.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1983 Springer-Verlag

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Hale, J.K. (1983). Infinite dimensional dynamical systems. In: Palis, J. (eds) Geometric Dynamics. Lecture Notes in Mathematics, vol 1007. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0061425

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  • DOI: https://doi.org/10.1007/BFb0061425

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  • Print ISBN: 978-3-540-12336-1

  • Online ISBN: 978-3-540-40969-4

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