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

128 T. Kalmár-Nagy et al.Separating the real and imaginary parts yieldsAs 1 (ϑ) =−ωs 2 (ϑ),As 2 (ϑ) = ωs 1 (ϑ).Using the definition of A results the following boundary value problemddϑ s 1(ϑ) = −ωs 2 (ϑ),ddϑ s 2(ϑ) = ωs 1 (ϑ), (14)Ls 1 (0) + Rs 1 (−τ) = −ωs 2 (0),Ls 2 (0) + Rs 2 (−τ) = ωs 1 (0). (15)The general solution to the differential equation (14) iss 1 (ϑ) = cos(ωϑ)c 1 − sin(ωϑ)c 2 ,s 2 (ϑ) = sin(ωϑ)c 1 + cos(ωϑ)c 2 .The boundary conditions (15) result in a system of linear equations for some of the unknowncoefficients:( ) ( )c 1L + cos(ωτ)R ωI + sin(ωτ)R = 0. (16)c 2The center manifold reduction also requires the calculation of the ‘left-hand side’ criticalreal eigenfunctions n 1,2 of A that satisfy the adjoint problemA ∗ n 1 (σ ) = ωn 2 (σ ),A ∗ n 2 (σ ) = −ωn 1 (σ ).This boundary value problem has the general solutionn 1 (σ ) = cos(ωσ )d 1 − sin(ωσ )d 2 ,n 2 (σ ) = sin(ωσ )d 1 + cos(ωσ )d 2 ,while the boundary conditions simplify to(L T + cos(ωτ)R T − ωI − sin(ωτ)R T ) ( )d 1= 0. (17)d 2With the help of the bilinear form (13), the ‘orthonormality’ conditions(n 1 , s 1 ) = 1, (n 1 , s 2 ) = 0 (18)provide two more equations.