Subcritical Hopf Bifurcation in the Delay Equation Model for Machine ...

140 T. Kalmár-Nagy et al.gives the definition of the adjoint operator{−du(γ ),A ∗ dγu(γ ) =γ ∈ (0,τ],L ∗ u(0) + R ∗ u(τ), γ = 0,with the two complex conjugate eigenfunctionsA ∗ n(γ ) =−iωIn(γ ),A ∗ ¯n ∗ (γ ) = iωI¯n ∗ (γ ).(A.22)(A.23)Introducing the real functionsn 1 (γ ) = Re n(γ ),n 2 (γ ) = Im n(γ ).Equations (A.22) and (A.23) can be rewritten asA ∗ n 1 (γ ) = ωIn 2 (γ ),A ∗ n 2 (γ ) =−ωIn 1 (γ ).Since Equation (A.21) requires functions from two different spaces (C 1 [−1,0] and C1 [0,1] )itisa bilinear form instead of an inner product. The ‘orthonormality’ conditions (see Equations(A.8) and (A.9)) are(n 1 , s 1 ) = (n 2 , s 2 ) = 1,(n 2 , s 1 ) = (n 1 , s 2 ) = 0.The new coordinates y 1 , y 2 can be found by the projections (instead of Equation (A.16))y 1 (t) = y 1t (0) = (n 1 , x t )| ϑ=0 ,y 2 (t) = y 2t (0) = (n 2 , x t )| ϑ=0 .Now x t (ϑ) can be decomposed asx t (ϑ) = y 1 (t)s 1 (ϑ) + y 2 (t)s 2 (ϑ) + w(t)(ϑ)(A.24)and the operator differential equation (A.13) can be transformed into the ‘canonical form’ẏ 1 = (n 1 , ẋ t )| ϑ=0 = (n 1 , Ax t + F (x t ))| ϑ=0= (n 1 , Ax t )| ϑ=0 + (n 1 , F (x t ))| ϑ=0= (A ∗ n 1 , x t )| ϑ=0 + (n 1 , F (x t ))| ϑ=0= ω(n 2 , x t )| ϑ=0 + (n 1 , F (x t ))| ϑ=0 = ωy 2 + n T 1 (0)F,ẏ 2 = (n 2 , ẋ t )| ϑ=0 =−ωy 1 + n T 2 (0), F