Floquet Multipliers of a Periodic Solution Under State-Dependent Delay

Abstract

AbstractWe consider a periodic function $$p:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ p : R → R of minimal period 4 which satisfies a family of delay differential equations $$\begin{aligned} x'(t)=g(x(t-d_{\Delta }(x_t))),\quad \Delta \in {\mathbb {R}}, \end{aligned}$$ x ′ ( t ) = g ( x ( t - d Δ ( x t ) ) ) , Δ ∈ R , with a continuously differentiable function $$g:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ g : R → R and delay functionals $$\begin{aligned} d_{\Delta }:C([-2,0],{\mathbb {R}})\rightarrow (0,2). \end{aligned}$$ d Δ : C ( [ - 2 , 0 ] , R ) → ( 0 , 2 ) . The solution segment $$x_t$$ x t in Eq. (0.1) is given by $$x_t(s)=x(t+s)$$ x t ( s ) = x ( t + s ) . For every $$\Delta \in {\mathbb {R}}$$ Δ ∈ R the solutions of Eq. (0.1) defines a semiflow of continuously differentiable solution operators $$S_{\Delta ,t}:x_0\mapsto x_t$$ S Δ , t : x 0 ↦ x t , $$t\ge 0$$ t ≥ 0 , on a continuously differentiable submanifold $$X_{\Delta }$$ X Δ of the space $$C^1([-2,0],{\mathbb {R}})$$ C 1 ( [ - 2 , 0 ] , R ) , with codim $$X_{\Delta }=1$$ X Δ = 1 . At $$\Delta =0$$ Δ = 0 the delay is constant, $$d_0(\phi )=1$$ d 0 ( ϕ ) = 1 everywhere, and the orbit $${{\mathcal {O}}}=\{p_t:0\le t<4\}\subset X_0$$ 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 $$M_0=DS_{0,4}(p_0)$$ M 0 = D S 0 , 4 ( p 0 ) is $$\sigma _0=\{0,1\}$$ σ 0 = { 0 , 1 } , with the eigenvalue 1 being simple. For $$|\Delta |\nearrow \infty $$ | Δ | ↗ ∞ there is an increasing contribution of variable, state-dependent delay to the time lag $$d_{\Delta }(x_t)=1+\cdots $$ d Δ ( x t ) = 1 + ⋯ in Eq. (0.1). We study how the spectrum $$\sigma _{\Delta }$$ σ Δ of $$M_{\Delta }=DS_{\Delta ,4}(p_0)$$ M Δ = D S Δ , 4 ( p 0 ) changes if $$|\Delta |$$ | Δ | grows from 0 to $$\infty $$ ∞ . A main result is that at $$\Delta =0$$ Δ = 0 an eigenvalue $$\Lambda (\Delta )<0$$ Λ ( Δ ) < 0 of $$M_{\Delta }$$ M Δ bifurcates from $$0\in \sigma _0$$ 0 ∈ σ 0 and decreases to $$-\infty $$ - ∞ as $$|\Delta |\nearrow \infty $$ | Δ | ↗ ∞ . Moreover we verify the spectral hypotheses for a period doubling bifurcation from the periodic orbit $${{\mathcal {O}}}$$ O at the critical parameter $$\Delta _{*}$$ Δ ∗ where $$\Lambda (\Delta _{*})=-1$$ Λ ( Δ ∗ ) = - 1 .