Floquet Multipliers of a Periodic Solution Under State-Dependent Delay.

We consider a periodic function p : R → R of minimal period 4 which satisfies a family of delay differential equations 0.1 x ′ (t) = g (x (t - d Δ (x t))) , Δ ∈ R , with a continuously differentiable function g : R → R and delay functionals d Δ : C ([ - 2 , 0 ] , R) → (0 , 2). The solution segment x t in Eq. (0.1) is given by x t (s) = x (t + s) . For every Δ ∈ R the solutions of Eq. (0.1) defines a semiflow of continuously differentiable solution operators S Δ , t : x 0 ↦ x t , t ≥ 0 , on a continuously differentiable submanifold X Δ of the space C 1 ([ - 2 , 0 ] , R) , with codim X Δ = 1 . At Δ = 0 the delay is constant, d 0 (ϕ) = 1 everywhere, and the orbit O = { p t : 0 ≤ t < 4 } ⊂ X 0 of the periodic solution is extremely stable in the sense that the spectrum of the monodromy operator M 0 = D S 0 , 4 (p 0) is σ 0 = { 0 , 1 } , with the eigenvalue 1 being simple. For | Δ | ↗ ∞ there is an increasing contribution of variable, state-dependent delay to the time lag d Δ (x t) = 1 + ⋯ in Eq. (0.1). We study how the spectrum σ Δ of M Δ = D S Δ , 4 (p 0) changes if | Δ | grows from 0 to ∞ . A main result is that at Δ = 0 an eigenvalue Λ (Δ) < 0 of M Δ bifurcates from 0 ∈ σ 0 and decreases to - ∞ as | Δ | ↗ ∞ . Moreover we verify the spectral hypotheses for a period doubling bifurcation from the periodic orbit O at the critical parameter Δ ∗ where Λ (Δ ∗) = - 1 . [ABSTRACT FROM AUTHOR]