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

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136 T. Kalmár-Nagy et al.where we used the linearity of the scalar product and the identity(u, Av) = (A ∗ u, v).This result is equivalent with Equation (A.3).The coordinates y 1 (t), y 2 (t) of the linear system ẏ(t) = Jy(t) describe stable (but notasymptotically stable) solutions, while the other coordinates represent exponentially decayingones. In other words the linear equationẏ(t) = Jy(t)has a two-dimensional attractive invariant subspace (a plane). To obtain the two real vectorsthat span this plane we first find the two complex conjugate eigenvectors that satisfyJs = iωs,J¯s ∗ =−iω¯s ∗ .(A.6)(A.7)These are⎛s = ⎝1i0⎞⎛⎠ , s ∗ = ⎝⎞1−i ⎠ .0The two real vectors⎛ ⎞⎛ ⎞10s 1 = Re s = ⎝ 0 ⎠ , s 2 = Im s = ⎝ 1 ⎠00span the plane in question. n, s satisfy the orthonormality conditions(n 1 , s 1 ) = (n 2 , s 2 ) = 1, (A.8)(n 1 , s 2 ) = (n 2 , s 1 ) = 0. (A.9)Note that since (n, s) = (n 1 , s 1 ) + (n 2 , s 2 ) + i((n 2 , s 1 ) − (n 1 , s 2 )) Equations (A.8) and (A.9)are equivalent to(n, s) = 2.Equations (A.6) and (A.7) can also be written asJs 1 = −ωs 2 ,Js 2 = ωs 1 ,which can then be solved for the real vectors.
CANONICAL FORM FOR DDESSubcritical Hopf Bifurcation in the Delay Equation Model 137Let us first consider a linear scalar autonomous delay differential equation (and the correspondinginitial function) of the formẋ(t) = Lx(t) + Rx(t − τ)+ f(x(t),x(t − τ)),x(t) = φ(t) t ∈[−τ,0). (A.10)The first goal is to put Equation (A.10) into a form similar to Equation (A.1). Here an attemptednumerical solution could give a clue. Discretizing Equation (A.10) leads todx idϑ ≈ 1dϑ (x i−1 − x i ), i = 1,...,N,dx idϑ ∣ = Lx 0 + Rx N + f(x 0 ,x N ).ϑ=0Taking the limit dϑ → 0, and defining the ‘shift of time’ (a chunk of a function)x t (ϑ) = x(t + ϑ),we have the followingddϑ x t(ϑ) = ddϑ x t(ϑ), ϑ ∈[−τ,0),ddϑ x t(ϑ)∣ = Lx t (0) + Rx t (−τ)+ f(x t (0), x t (−τ)).ϑ=0This motivates our definition of the linear differential operator AAu(ϑ) ={ ddϑ u(ϑ),ϑ ∈[−τ,0),Lu(0) + Ru(−τ), ϑ = 0,(A.11)and the nonlinear operator F{ 0, ϑ ∈[−τ,0),F (u)(ϑ) =f(u(0)), ϑ = 0,(A.12)Since d/dϑ = d/dt, we can rewrite Equation (A.10) asddt x t(ϑ) =ẋ t (ϑ) = Ax t (ϑ) + F (x t )(ϑ).This operator formulation can be extended to multidimensional systems as wellẋ t (ϑ) = Ax t (ϑ) + F (x t )(ϑ).(A.13)This form is very similar to the system of ODEs (A.1). There is one very important difference,though. Equation (A.13) represents an infinite-dimensional system. However, the infinitedimensionalphase space of its linear part can also be split into stable, unstable and centersubspaces corresponding to eigenvalues having positive, negative and zero real part. If we
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