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

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Walther H.-O., Yebdri M.

Instytut Matematyczny Polskiej Akademii Nauk

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

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1 Walther H.-O.

1 Yebdri M.

1 1997

Wyniki wyszukiwania

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