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Smoothness of the attractor of almost all solutions of a delay differential equation

Seria
Rozprawy Matematyczne tom/nr w serii: 368 wydano: 1997
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Abstract
Let a C¹-function f:ℝ → ℝ be given which satisfies f(0) = 0, f'(ξ) < 0 for all ξ ∈ ℝ, and sup f < ∞ or -∞ < inf f. Let C = C([-1,0],ℝ). For an open-dense set of initial data the phase curves [0,∞) → C given by the solutions [-1,∞) → ℝ to the negative feedback equation
x'(t) = -μx(t) + f(x(t-1)), with μ > 0,
are absorbed into the positively invariant set S ⊂ C of data ϕ ≠ 0 with at most one sign change. The global attractor A of the semiflow restricted to S̅ is either the singleton {0} or it is given by a Lipschitz continuous map a with domain pA in a 2-dimensional subspace L ⊂ C and range in a complementary subspace Q; pA is homeomorphic to the closed unit disk in ℝ². We show that a is in fact C¹-smooth.
EN
CONTENTS
1. Introduction......................................................................................................................................................................5
2. The delay differential equation and its attractor of almost all solutions............................................................................9
 2.1. The delay differential equation....................................................................................................................................9
 2.2. Slowly oscillating solutions.........................................................................................................................................13
 2.3. The attractor of eventually slowly oscillating solutions...............................................................................................16
 2.4. Floquet multipliers of slowly oscillating periodic solutions and adapted Poincaré maps.............................................21
 2.5. Local invariant manifolds...........................................................................................................................................27
3. A-priori estimates...........................................................................................................................................................33
 3.1. Nonautonomous equations........................................................................................................................................33
 3.2. Vectors tangent to the attractor and to domains of adapted Poincaré maps.............................................................39
4. Transversals on the attractor and smoothness..............................................................................................................41
 4.1. A sufficient condition for smoothness.........................................................................................................................41
 4.2. Smoothness at wandering points...............................................................................................................................42
5. Curves on the attractor emanating from periodic orbits and connecting the stationary point to a periodic orbit............44
 5.1. From lines in the plane L to curves on the graph A which are transversal to the flow................................................44
 5.2. Arcs emanating from periodic orbits..........................................................................................................................45
 5.3. Smooth ends at periodic orbits..................................................................................................................................47
 5.4. A curve on A connecting 0 in K̅ to a periodic orbit.....................................................................................................53
6. Smoothness at periodic orbits.......................................................................................................................................59
 6.1. Interior periodic orbits................................................................................................................................................59
 6.2. Smoothness at the boundary.....................................................................................................................................62
7. Smoothness at the stationary point................................................................................................................................63
 7.1. Cases of no attraction................................................................................................................................................63
 7.2. On the inclination of tangent spaces of the attractor close to the stationary point.....................................................65
 7.3. The cases of attraction..............................................................................................................................................68
References........................................................................................................................................................................71
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 368
Liczba stron
72
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXVIII
Daty
wydano
1997
otrzymano
1996-06-17
poprawiono
1996-11-20
Twórcy
Bibliografia
  • [1] R. Abraham and J. Robbin, Transversal Mappings and Flows, Benjamin, New York, 1967.
  • [2] S. N. Chow and K. Lu, $C^k$ centre unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 303-320.
  • [3] S. N. Chow and H. O. Walther, Characteristic multipliers and stability of symmetric periodic solutions of ẋ(t)=g(x(t-1)), Trans. Amer. Math. Soc. 307 (1988), 127-142.
  • [4] O. Diekmann, S. van Gils, S. Verduyn Lunel and H. O. Walther, Delay Equations: Complex-, Functional- and Nonlinear Analysis, Springer, New York, 1995.
  • [5] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, 1988.
  • [6] J. K. Hale and X. B. Lin, Symbolic dynamics and nonlinear semiflows, Ann. Mat. Pura Appl. 144 (1986), 229-259.
  • [7] G. Iooss, Bifurcation of Maps and Applications, North-Holland Math. Stud. 36, North-Holland, Amsterdam, 1979.
  • [8] B. Lani-Wayda and H. O. Walther, Chaotic motion generated by delayed negative feedback. Part I: A transversality criterion, Differential Integral Equations 8 (1995), 1407-1452.
  • [9] B. Lani-Wayda and M. Yebdri, On local center manifolds, technical report, in preparation.
  • [10] J. Mallet-Paret and G. Sell, Systems of differential delay equations I: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations 125 (1996), 385-440.
  • [11] J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations 125 (1996), 441-489.
  • [12] J. Mallet-Paret and H. O. Walther, Rapid oscillations are rare in scalar systems governed by monotone negative feedback with a time lag, preprint, Universität Gießen, 1994.
  • [13] A. Neugebauer, Invariante Mannigfaltigkeiten und Neigungslemmata für Abbildungen in Banachräumen, diploma thesis, Universität München, 1988.
  • [14] W. Rinow, Lehrbuch der Topologie, Deutscher Verlag der Wiss., Berlin, 1975.
  • [15] I. Tereščák, Dynamical systems with discrete Lyapunov functionals, Ph.D. thesis, Comenius University, Bratislava, 1994.
  • [16] H. O. Walther, On instability, ω-limit sets and periodic solutions of nonlinear autonomous differential delay equations, in: Functional Differential Equations and Approximation of Fixed Points, H. O. Peitgen and H. O. Walther (eds.), Lecture Notes in Math. 730, Springer, Heidelberg, 1979, 489-503.
  • [17] H. O. Walther, An invariant manifold of slowly oscillating solutions for ẋ(t)=-μ x(t)+f(x(t-1)), J. Reine Angew. Math. 414 (1991), 67-112.
  • [18] H. O. Walther, On Floquet multipliers of periodic solutions of delay equations with monotone nonlinearities, in: Proc. Internat. Sympos. on Functional Differential Equations (Kyoto, 1990), T. Yoshizawa and J. Kato (eds.), World Sci., Singapore, 1991, 349-356.
  • [19] H. O. Walther, Unstable manifolds of periodic orbits of a differential delay equation, Oscillations and Dynamics in Delay Equations, J. R. Graef and J. K. Hale (eds.), Amer. Math. Soc., Providence, 1992, 177-240.
  • [20] H. O. Walther, The 2-dimensional attractor of x'(t)=-μ x(t)+f(x(t-1)), Mem. Amer. Math. Soc. 544 (1995).
Języki publikacji
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Uwagi
1991 Mathematics Subject Classification: 34K15, 58F12.
Identyfikator YADDA
bwmeta1.element.zamlynska-c107eab5-123d-4084-9d5c-4965945dcaf0
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ISSN
0012-3862
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DML-PL
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