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

132 T. Kalmár-Nagy et al.with⎛P 6×6 = ⎝⎞⎛L 0 00 L 0 ⎠ − C 6×6 , Q 6×6 = ⎝0 0 L⎞R 0 00 R 0 ⎠ ,0 0 Rr =− ( 0 f 11 0 f 12 0 f 22) T.K is found by substituting the general solution (28) into Equation (32)(P + Q e −τC )K = r + p − PM − Q(cos(ωτ)M − sin(ωτ)N). (33)Despite its hideous look Equation (33) simplifies, becausep − PM − Q(cos(ωτ)M − sin(ωτ)N) = 0.For our system⎛ ⎞⎛K 19 + 32ζ + 32ζ 2 ⎞⎝ K 3⎠6ζ=⎝ ω(3 + 4ζ) ⎠ . (34)5(9 + 33ζ + 32ζK 2 )5 9 + 34ζ + 32ζ 2Finally, Equations (30, 31, 34) are substituted into Equation (28) resulting in the secondorderapproximation of the center manifold (24). It is not necessary to express the centermanifold approximation in its full form, since we only need the values of its components atϑ = 0and−τ in the transformed operator equation (20, 21, 22). For example,w 1 (0) = 1 2 ((M 1 + K 1 )y 2 1 + 2(M 3 + K 3 )y 1 y 2 + (M 5 + K 5 )y 2 2 ),while the expression for w 1 (−τ) is somewhat more lengthy.6. The Hopf BifurcationIn order to restrict a third-order approximation of system (20–22) to the two-dimensionalcenter manifold calculated in the previous section, the second-order approximation w(y 1 ,y 2 )of the center manifold has to be substituted into the two scalar equations (20) and (21). Thenthese equations will assume the formẏ 1 = ωy 2 + a 20 y 2 1 + a 11y 1 y 2 + a 02 y 2 2 + a 30y 3 1 + a 21y 2 1 y 2 + a 12 y 1 y 2 2 + a 03y 3 2 ,ẏ 2 = −ωy 1 + b 20 y 2 1 + b 11y 1 y 2 + b 02 y 2 2 + b 30y 3 1 + b 21y 2 1 y 2 + b 12 y 1 y 2 2 + b 03y 3 2 . (35)Using 10 out of these 14 coefficients a jk ,b jk , the so-called Poincaré–Lyapunov constant can be calculated as shown in [5] or [7] = 18ω [(a 20 + a 02 )(−a 11 + b 20 − b 02 ) + (b 20 + b 02 )(a 20 − a 02 + b 11 )]+ 1 8 (3a 30 + a 12 + b 21 + 3b 03 ).