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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Erratic solutions of simple delay equations

Author: Bernhard Lani-Wayda
Journal: Trans. Amer. Math. Soc. 351 (1999), 901-945
MSC (1991): Primary 34K15, 58F13, 70K50
MathSciNet review: 1615995
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Abstract: We give an example of a smooth function $g:\mathbb{R}\longrightarrow \mathbb{R}$ with only one extremum, with $\mathrm{ sign }\ g(x) = - \mathrm{ sign }\ g(-x)$ for $x \neq 0$, and the following properties: The delay equation $\dot x (t) = g(x(t-1))$ has an unstable periodic solution and a solution with phase curve transversally homoclinic to the orbit of the periodic solution. The complicated motion arising from this structure, and its robustness under perturbation of $g$, are described in terms of a Poincaré map. The example is minimal in the sense that the condition $g' < 0$ (under which there would be no extremum) excludes complex solution behavior. Based on numerical observations, we discuss the role of the erratic solutions in the set of all solutions.

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  • [AdHW] U. An der Heiden and H.-O. Walther, Existence of chaos in control systems with delayed feedback, J. Diff. Equs. 47 (1983), 273-295. MR 85a:58061
  • [BC1] M. Benedicks and L. Carleson, On iterations of $1-ax^{2}$ on $(-1,1) $, Ann. of Math. 122 (1985), 1-25. MR 87c:58058
  • [BC2] M. Benedicks and L. Carleson, Dynamics of the Hénon map, Ann. of Math. 133 (1991), 73-169. MR 92d:58116
  • [DL-W] P. Dormayer and B. Lani-Wayda, Floquet multipliers and secondary bifurcations in functional differential equations: Numerical and analytical results, Z. Angew. Math. Phys. (ZAMP) 46 (1995), 823-858. MR 97f:34047
  • [Dr] R.D. Driver, Ordinary and delay differential equations, (Applied Mathematical Sciences 20) Springer Verlag, New York, 1977. MR 57:16897
  • [DvGV-LW] O. Diekmann, S.A. van Gils, S.M. Verduyn-Lunel, and H.-O. Walther, Delay Equations, (Applied Mathematical Sciences 110) Springer-Verlag, New York, 1995. MR 97a:34001
  • [G] T. Gedeon, Cyclic feedback systems, Mem. AMS (to appear). CMP 97:07
  • [GP] P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Physica 9D (1983), 189-208. MR 85i:58071
  • [H] J.K. Hale, Functional Differential Equations, (Applied Mathematical Sciences 3), Springer Verlag, New York, 1971. MR 57:6711
  • [HL1] J.K. Hale and X.B. Lin, Symbolic dynamics and nonlinear semiflows, Ann. Mat. Pura Appl. (4) 144 (1986), 229-259. MR 89g:58130
  • [HL2] J.K. Hale and X.B. Lin, Examples of transverse homoclinic orbits in delay equations, Nonlinear Analysis 10 (1986), 693-709. MR 87i:34087
  • [HS] J.K. Hale and N. Sternberg, Onset of Chaos in Differential Delay Equations, J. Computational Phys. 77 No.1 (1988), 221-239. MR 89h:58115
  • [HV-L] J.K. Hale and S.M. Verduyn-Lunel, Introduction to Functional Differential Equations, (Applied Mathematical Sciences 99), Springer Verlag, New York, 1993. MR 94m:34169
  • [ILW] A. Ivanov, B. Lani-Wayda and H.-O. Walther, Unstable hyperbolic periodic solutions of differential delay equations, Recent Trends in Differential Equations, ed. R.P. Agarwal, World Scientific, Singapore, 1992, pp. 301-316. MR 93h:34125
  • [KS] U. Kirchgraber and D. Stoffer, Chaotic behavior in simple dynamical systems, SIAM Review 32 No. 3 (1990), 424-452. MR 91e:58141
  • [KY] J.L. Kaplan and J.A. Yorke, On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Analysis 6 (1975), 268-282. MR 50i:13812
  • [L-W1] B. Lani-Wayda, Persistence of Poincaré mappings in functional differential equations (with application to structural stability of complicated behavior), J. Dyn. Diff. Equs. 7 No. 1 (1995), 1-71. MR 96e:34118
  • [L-W2] B. Lani-Wayda, Hyperbolic Sets, Shadowing and Persistence for Noninvertible Mappings in Banach spaces, Research Notes in Mathematics No. 334, Longman Group Ltd., Harlow, Essex, 1995. CMP 98:01
  • [L-WW1] B. Lani-Wayda and H.-O. Walther, Chaotic motion generated by delayed negative feedback, Part I: A transversality criterion, Diff. Int. Equs. 8 No. 6 (1995), 1407-52. MR 96c:58115
  • [L-WW2] B. Lani-Wayda, and H.-O. Walther, Chaotic motion generated by delayed negative feedback, Part II: Construction of nonlinearities, Math. Nachr. 180 (1996), 141-211. MR 97g:58147
  • [La] A. Lasota, Ergodic problems in biology, Asterisque 50 (1977), 239-250. MR 58:9378
  • [Laz] V.A. Lazutkin, Positive Entropy for the Standard Map I, Preprint 94-47, Université de Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay, France, 1994.
  • [LWCz] A. Lasota and M. Wazewska-Czyzewska, Matematyczne problemy dynamiki ukladu krwinek czerwonych, Mat. Stosowana 6 (1976), 23-40.
  • [LY] T.Y. Li and J.A. Yorke, Period three implies chaos, Am. Math. Monthly 82 (1977), 985-992. MR 52:5898
  • [MG] M.C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197 (1977), 287-295.
  • [M] M. Morse, A one-to-one representation of geodesics on a surface of negative curvature, Am. J. Math. 43 (1921), 33-51.
  • [MH] M. Morse and G. Hedlund, Symbolic Dynamics , Am. J. Math. 60 (1938), 815-866.
  • [M-P] J. Mallet-Paret, Morse decompositions for delay differential equations, J. Diff. Equs. 72 (1988), 270-315. MR 89m:58182
  • [M-PS] J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Diff. Eqns. 125 (1996), 441-489. MR 97a:34193b
  • [M-PSm] J. Mallet-Paret and H.L. Smith, The Poincaré-Bendixson theorem for monotone cyclic feedback systems, J. Dyn. Diff. Equs. 2 (1990), 367-421. MR 91k:58098
  • [M-PW] J. Mallet-Paret and H.-O. Walther, Rapid oscillations are rare in scalar systems governed by monotone negative feedback with a time lag, Preprint, Math. Inst. Univ. Giessen, 1994.
  • [P] K.J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, U. Kirchgraber and H.-O. Walther (eds.), Dynamics Reported, vol. I, Teubner-Wiley, Stuttgart/Chichester, 1988, pp. 265-306. MR 89j:58060
  • [S] S. Smale, Differentiable Dynamical Systems, Bull. AMS 73 (1967), 747-817. MR 37:3598
  • [SW] H. Steinlein and H.-O. Walther, Hyperbolic Sets, Transversal Homoclinic Trajectories, and Symbolic Dynamics for $C^{1}-$maps in Banach Spaces, J. Dyn. Diff. Equs. 2 (1990), 325-365. MR 92b:58170
  • [W1] H.-O. Walther, Homoclinic solution and chaos in $\dot x(t) = f(x(t-1))$, Nonlinear Analysis 5 (1981), 775-788. MR 83a:58060
  • [W2] H.-O. Walther, Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations, Memoirs AMS 402 (1989). MR 90m:58184
  • [W3] H.-O. Walther, The 2-dimensional attractor of $x'(t)=-\mu x(t)+f(x(t-1))$, Memoirs AMS 544 (1995). MR 95f:58070

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Additional Information

Bernhard Lani-Wayda
Affiliation: Mathematisches Institut der Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany

Received by editor(s): September 4, 1996
Additional Notes: Supported by the Deutsche Forschungsgemeinschaft within the Schwerpunkt Analysis, Ergodentheorie und Effiziente Simulation Dynamischer Systeme.
Article copyright: © Copyright 1999 American Mathematical Society

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