24 T. Detroux et al. with 1 D 2 6 6 6 6 6 6 6 6 4 C C 2! M 2! M C : : : C 2NH ! M 2NH ! M C 3 7 7 7 7 7 7 7 7 5 ; 2 D I .2NHC1/ ˝M (3.19) can be rewritten as a linear eigenvalue problem of double size B1 2 4 3 5 B2 D0 (3.20) with B1 D 1 Jz I 0 ; B2 D 2 0 0 I (3.21) The eigenvalues can thus be found among the eigenvalues of the matrix BDB 1 2 B1 (3.22) As will be explained in Sect. 3.2.4, the matrixBhas a key role in the detection and tracking of bifurcations. After sorting the obtained eigenvalues in order to eliminate the numerical ones, as proposed by Lazarus et al. [23], one can directly relates the remaining eigenvalues to the Floquet exponents. The term “Floquet exponents” will be used throughout this paper; it should however be kept in mind that these exponents are computed with the Hill’s method. 3.2.4 Detection and Tracking of Bifurcations In this work the detection and tracking of LPs and NSs is sought, based on the Floquet exponents. A LP bifurcation is detected when a Floquet exponent cross the imaginary axis along the real axis. It can also be detected when the component of the tangent prediction related to the parameter ! changes sign. As a consequence, the jacobian matrix Jz is singular at a LP bifurcation. In order to continue a LP bifurcation with respect to a parameter of the system, one has to append one equation to (3.3). A condition imposing the singularity of the jacobian matrix is often used: jJzjD0 (3.23) Nevertheless, under this form the extra equation could lead to scaling problems. To overcome this issue, Doedel et al. [24] proposed the use of the so-calledbordering technique. The idea of this technique is to use an extra equation of the form gLP D0 (3.24) with gLP a scalar function which vanishes simultaneously with the determinant of Jz. Such a function can be obtained by solving Jz pLP q LP 0 w gLP D 0 1 (3.25) where denotes a conjugate transpose, and where pLP and qLP are chosen to ensure the nonsingularity of the matrix. The second type of bifurcation studied in this paper, the NS bifurcation, is detected when a pair of Floquet exponents cross the imaginary axis as a pair of complex conjugates. Using the theory of the bialternate matrix product [25] Pˇ of a m mmatrixP Pˇ DPˇI m (3.26)
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