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Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback. (English) Zbl 1247.34120

For scalar delay equations of the type

$\stackrel{˙}{x}\left(t\right)=-\mu x\left(t\right)+f\left(x\left(t-1\right)\right),$

the global attractor $𝒜$ (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 $𝒜$ (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 $𝒜$ contains, in addition, two periodic orbits with large-amplitude in the sense that their range includes the interval $\left({\xi }_{-1},{\xi }_{1}\right)$. Further, the dynamics on $𝒜$ 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.

MSC:
 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 (infinite-dimensional dissipative systems) 34K25 Asymptotic theory of functional-differential equations 34D45 Attractors 34C45 Invariant manifolds (ODE) 34C37 Homoclinic and heteroclinic solutions of ODE 34K21 Stationary solutions of functional-differential equations