Discrete Ljapunov functionals and ω-limit sets
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 23 (1989) no. 3, p. 415-431
@article{M2AN_1989__23_3_415_0,
     author = {Fiedler, Bernold},
     title = {Discrete Ljapunov functionals and $\omega $-limit sets},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {23},
     number = {3},
     year = {1989},
     pages = {415-431},
     zbl = {0688.58041},
     mrnumber = {1014483},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1989__23_3_415_0}
}
Fiedler, Bernold. Discrete Ljapunov functionals and $\omega $-limit sets. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 23 (1989) no. 3, pp. 415-431. http://www.numdam.org/item/M2AN_1989__23_3_415_0/

S. Angenent 1, The Morse-Smale property for a semi linear parabolic equation, J. Diff. Eq. 62 (1986), 427-442. | MR 837763 | Zbl 0581.58026

S. Angenent 2, The zeroset of a solution of a parabolic equation, J. reine angew. Math. 390 (1988), 79-96. | MR 953678 | Zbl 0644.35050

S. Angenent & B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. AMS 307 (1988), 545-568. | MR 940217 | Zbl 0696.35086

I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 24 (1901), 1-88. | JFM 31.0328.03 | MR 1554923

J. E. Billoti & J. P. Lasalle, Periodic dissipative processes, Bull. AMS 6 (1971), 1082-1089. | Zbl 0274.34061

P. Brunovský & B. Fiedler 1, Zero numbers on invariant manifolds in scalar reaction diffusion equations, Nonlin. Analysis TMA 10 (1986), 179-194. | MR 825216 | Zbl 0594.35056

P. Brunovský & B. Fiedler 2, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89. | MR 945964 | Zbl 0679.35047

P. Brunovský & B. Fiedler 3, Connecting orbits in scalar reaction diffusion equations II, to appear in J. Diff. Eq. | MR 945964 | Zbl 0699.35144

E. A. Coddington & N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York 1955. | MR 69338 | Zbl 0064.33002

C. C. Conley & J. Smoller, Topological techniques in reaction diffusion equations, in « Biological Growth and Spread », Jäger & Rost & Tautu (eds.), Lect. Notes Biomath. 38, Springer-Verlag, Heidelberg 1980, 473-483. | MR 609381 | Zbl 0444.35005

C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, in « Nonlinear Evolution Equations », M. G. Crandall (ed.), Academic Press, New York 1978, 103-123. | MR 513814 | Zbl 0499.35015

B. Fiedler & J. Mallet-Paret 1, Connections between Morse sets for delay-differential equations, to appear in J. reine angew. Math. | MR 993217 | Zbl 0659.34077

B. Fiedler & J. Mallet-Paret 2, A Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rational Mech. Analysis, in press. | MR 1004714 | Zbl 0704.35070

J. K. Hale, Ordinary Differential Equations, John Wiley & Sons, New York 1969. | MR 419901 | Zbl 0186.40901

J. K. Hale & L. T. Magalhāes & W. M. Oliva, An Introduction to Infinite imensional Dynamical Systems - Geometrie Theory, Appl. Math. Sc. 47, Springer-Verlag, New York 1984. | MR 725501 | Zbl 0533.58001

P. Hartman, Ordinary Differential Equations, Birkäuser, Boston 1982. | MR 658490 | Zbl 0476.34002

D. Henry 1, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. 840, Springer-Verlag, New York 1981. | MR 610244 | Zbl 0456.35001

D. Henry 2, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. Diff. Eq. 59 (1985), 165-205. | MR 804887 | Zbl 0572.58012

M. W. Hirsch 1, Differential equations and convergence almost everywhere in strongly monotone semiflows, in « Nonlinear Partial Differential Equations », J.Smoller (ed.), AMS, Providence 1983, 267-285. | MR 706104 | Zbl 0523.58034

M. W. Hirsch 2, Systems of differential equations that are competitive or cooperative II. Convergence almost everywhere. SIAM J. Math. Anal. 16 (1985), 423-439. | MR 783970 | Zbl 0658.34023

M. W. Hirsch 3, Stability and convergence in strongly monotone dynamical Systems, J. reine angew. Math. 383 (1988), 1-53. | MR 921986 | Zbl 0624.58017

J. L. Kaplan & J. A. Yorke, On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal. 6 (1975), 268-282. | MR 361367 | Zbl 0241.34080

S. Lefschetz, Differential Equations : Geometric Theory, Wiley & Sons, New York 1963. | MR 153903 | Zbl 0107.07101

J. Mallet-Paret, 1 Morse decompositions and global continuation of periodic solutions for singularly perturbed delay equations, in « Systems of Nonlinear Partial Differential Equations », J. M. Ball (ed.), D. Reidel, Dordrecht 1983, 351-366. | MR 725532 | Zbl 0562.34060

J. Mallet-Paret 2, Morse decompositions for delay-differential equations, J. Diff. Eq. 72 (1988), 270-315. | MR 932368 | Zbl 0648.34082

J. Mallet-Paret & H. Smith, A Poincaré-Bendixson theorem for monotone cyclic feedback Systems, preprint 1987. | MR 1073471 | Zbl 0712.34060

P. Massatt, The convergence of scalar parabolic equations with convection to periodic solutions, preprint 1986.

H. Matano 1, Convergence of solutions of one-dimensional parabolic equations, J. Math. Kyoto Univ. 18 (1978), 221-227. | MR 501842 | Zbl 0387.35008

H. Matano 2, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sec. IA 29 (1982), 401-441. | MR 672070 | Zbl 0496.35011

H. Matano 3, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving Systems, J. Fac. Sc. Univ. Tokyo Sec. IA 30 (1984), 645-673. | MR 731522 | Zbl 0545.35042

H. Matano 4, Asymptotic behavior of solutions of semilinear heat equations on S1, in « Nonlinear Diffusion equations and Their Equilibrium States », W.-M. Ni & B. Peletier & J. Serrin (eds.), Springer-Verlag, New York 1988, 139-162. | MR 956085 | Zbl 0671.35039

H. Matano 5, Asymptotic behavior of nonlinear equations, Res. Notes Math., Pitman, to appear.

H. Matano 6, Strongly order-preserving local semi-dynamical Systems - theory and applications, in « Semigroups, Theory and Applications », H. Brezis & M. G. Crandall & F. Kappel (eds.), John Wiley & Sons, New York 1986. | Zbl 0634.34031

K. Mischaikow, Conley's connection matrix, in « Dynamics of Infinite Dimensional Systems », S.-N. Chow & J. K. Hale (eds.), Springer-Verlag, Berlin 1987, 179-186. | MR 921911 | Zbl 0655.34036

K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. reine angew. Math. 211 (1962), 78-94. | MR 146534 | Zbl 0127.31801

H. Poincaré, Œuvres I, Gauthier-Villars, Paris 1928.

G. Pólya, Qualitatives über Wärmeausgleich, Z. Angew. Math. Mech. 13 (1933), 125-128. | JFM 59.0494.01

Sansone & R. Conti, Non-Linear Differential Equations, Pergamon Press, Oxford 1964. | Zbl 0128.08403

H. Smith, Monotone semiflows generated by functional differential equations, J. Diff. Eq. 66 (1987), 420-442. | MR 876806 | Zbl 0612.34067

R. A. Smith 1, The Poincaré-Bendixson theorem for certain differential equations of higher order, Proc. Roy. Soc. Edmburgh A 83 (1979), 63-79. | MR 538586 | Zbl 0408.34042

R. A. Smith 2, Existence of periodic orbits of autonomous ordinary differential equations, Proc. Roy. Soc. Edmburgh A 85 (1980), 153-172. | MR 566073 | Zbl 0429.34040

R. A. Smith 3, Existence of periodic orbits of autonomous retarded functional differential equations, Math. Proc. Camb. Phil. Soc. 88 (1980), 89-109. | MR 569635 | Zbl 0435.34062

J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York 1983. | MR 688146 | Zbl 0508.35002

C. Sturm, Sur une classe d'équations à différences partielles, J. Math. Pure Appl. 1 (1836), 373-444.

T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second order prabolic equation with one space variable, Differential Equations 4, 1 (1968), 17-22. | MR 223758 | Zbl 0232.35053