Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1247.34120
Krisztin, Tibor; Vas, Gabriella
Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback. (English)
[J] J. Dyn. Differ. Equations 23, No. 4, 727-790 (2011). ISSN 1040-7294; ISSN 1572-9222/e

For scalar delay equations of the type $$\dot x(t) = -\mu x(t) + f(x(t-1)),$$ the global attractor $\mathcal{A}$ (within the class of solutions with not more than two zeros per time unit, in other words, within the corresponding level set of a zero-counting Liapunov functional) is studied. The situation is more complex than in previous work of Krisztin, Walther and Wu. Two spindle-like three dimensional subsets of $\mathcal{A}$ (as described in the previous results) exist, joining equilibria $\xi_{-2} < 0 < \xi_2 $ of the equation. There are also unstable equilibria $\xi_{-1}, \xi_{1} $ satisfying $\xi_{-2} < \xi_{-1}< 0 < \xi_{1}< \xi_2 $; these correspond to the zero equilibrium in the quoted earlier work. It is shown that, for suitable $ \mu>0$ and $C^1$-nonlinearity $f$, the global attractor $\mathcal{A}$ contains, in addition, two periodic orbits with large-amplitude in the sense that their range includes the interval $ (\xi_{-1}, \xi_{1})$. Further, the dynamics on $\mathcal{A}$ can be described completely. All the heteroclinic connections between different periodic orbits, or between periodic orbits and equilibria, which are not excluded by the discrete Liapunov functional do actually exist. The nonlinearities $f$ are smoothed step functions, a feature that is probably important for the proofs (in that it allows explicit calculations), but not essential for the results. The given examples contribute much to the deeper understanding of the attractors for this type of infinite-dimensional dynamical systems.
[Bernhard Lani-Wayda (Giessen)]

MSC 2000:: *34K13 Periodic solutions of functional differential equations
37C70 Attractors and repellers, topological structure
37D05 Hyperbolic orbits and sets
37L25 Inertial manifolds and other invariant attracting sets
37L45 Hyperbolicity; Lyapunov functions
34K25 Asymptotic theory of functional-differential equations
34D45 Attractors
34C45 Method of integral manifolds
34C37 Homoclinic and heteroclinic solutions of ODE
34K21

Keywords: delay differential equation; positive feedback; global attractor; large amplitude periodic orbits; heteroclinic connections

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]