Morse decomposition for delay–differential equations with positive feedback

Morse Decompositions for Delay-Difference Equations

“…The proof of Theorem 4.1 is based on several tools, whose logical structure is borrowed from [19]. However, there are certain differences: For instance, the finite dimensional state space of (1.2) allows to simplify various arguments based on the Arzelà-Ascoli theorem.…”

“…For instance, discrete Lyapunov functionals are used to obtain convergence to equilibria in tridiagonal ODEs [24] and scalar parabolic equations [18], but also a Poincaré-Bendixson theory for a class of ODEs [14] in R n , n > 2, reaction-diffusion equations [4] and delaydifferential equations [16]. Finally, a Morse decomposition of global attractors for delaydifferential equations is constructed in [15,19]. In conclusion, the existence of such a discrete Lyapunov functional imposes a serious constraint on the possible long-term behavior of various systems.…”

The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

“…[5,6,7,9,10,11]. A −2,0 and A 0,2 admit Morse decompositions [18]. Further technical conditions regarding f ensure that A −2,0 and A 0,2 have spindle-like structures [5,9,10,11]: A 0,2 is the closure of the unstable set ofξ 1 containing the equilibrium pointsξ 0 ,ξ 1 ,ξ 2 , periodic orbits in C 0,2 and heteroclinic orbits among them.…”

“…(3.1) so that v 0 = ϕ and V (v t ) = 2 for all t ∈ R. Proposition 5. 18. Let * ∈ {k, p, q} and set r : R → R to be the periodic solution of Eq.…”

Periodic Orbits and Global Dynamics for Delay Differential Equations

“…As we need it later, we also note that if (2.4) holds, and ϕ ∈ A \ ξ belongs to the stable set W s ξ = ϕ : ω (ϕ) exists and ω (ϕ) =ξ ofξ, then ϕ −ξ has at least three sign changes on [−1, 0], see Lemma 3.9 in [40] for a proof.…”