2.1.3 Computation of derivatives during localization and tracking of bi-
furcation . . . 54 2.1.4 Examples of bifurcation tracking . . . 56 2.2 Multi-parametric recursive continuation . . . 66 2.2.1 Extremum point . . . 66 2.2.2 Recursive continuation of extremum points in increasing codimen-
sion. . . 68 2.2.3 Algorithm of multi-parametric recursive continuation . . . 69 2.2.4 Results interpretation . . . 69 2.3 Multi-parametric analysis of IS in a NLTVA . . . 69 2.3.1 Level-1 of LP continuation: ISs of the NLTVA . . . 71 2.3.2 Level-2 of LP continuation: continuation of the coincident birth
and merging of IS . . . 73 2.3.3 Level-3 of continuation: continuation of the coincident birth and
merging points of IS . . . 77 2.3.4 Suitability of the designs of interest . . . 79 2.4 Conclusion . . . 82
Bifurcation analysis
This chapter builds on the theoretical aspects introduced in the previous chapter to address advanced topics such as detection and the multi-parametric tracking of bifurcation points.
First, indicators for bifurcation detection along a followed curve are presented. Then, augmented systems for the accurate calculation of bifurcations are described. By using both continuation techniques and augmented systems characterizing bifurcation points, an approach for direct parametric analysis by tracking bifurcation points of nonlinear dy- namical systems is recalled. Instead of computing multiple responses curves with various values of system parameters, bifurcation tracking permits performing an efficient para- metric analysis with respect to one varying system parameter. The method is based on the HBM for the calculation of periodic solutions. Then, bifurcation tracking is applied to a Jeffcott rotor and a NLTVA to perform parametric analysis and observe the evolution of LP and NS bifurcations.
Secondly, an algorithm based on multi-parametric recursive continuation of extremum points is presented. The notion of extremum point is used to propose an original method of multi-parametric recursive continuation. Then, the characterization of the extremum point with an equality constraint function from continuated curve method is described. Then, this characterization is used to establish a recursive extremum point in successive increasing codimension. To finish, the multi-parametric recursive continuation algorithm is presented.
Finally, the multi-parametric recursive continuation method is applied in Section #2.3 to the NLTVA to optimize Isolated Solutions (IS). By applying the multi-parametric re- cursive continuation method to the NLTVA, the topological skeleton and extremum points of the IS are obtained. The limit of existence of IS and extremum points optimizing the zone without IS are found and used to improve the NLTVA.
Conclusions are drawn in the last section.
2.1
Bifurcation analysis
For nonlinear dynamical systems, an important role on the dynamics is played by nonlin- ear phenomena such as bifurcations. Bifurcations are periodic solutions where the implicit function theorem is ill-posed and where most of the dynamical changes of the systems oc- cur. Therefore, a parametric analysis based on bifurcation tracking seems a good way to apprehend the nonlinear dynamic of the system. In this subsection, the detection, local- ization and tracking of bifurcation points using indicators and standard extended system are presented. First, codimension-1 bifurcations are recalled based on the works from [XIE 16a]. Then two codimension-2 bifurcations, Resonance 1:1 (R1) and Limit Point Neimark Sacker (LPNS) bifurcations, are presented. The characterization of bifurcation points with respect to Hill’s method combined with the equation of motion in HBM is used to compute the bifurcation points.
2.1.1
Detection of bifurcations
A bifurcation point appears when a Floquet exponent λ crosses the imaginary axis, see 1.6. As stated in the state of the art Subsection 1.2.3, codimension-1 bifurcations can be distinguished from each other by the way the Floquet exponents cross the imaginary axis. In Subsection 2.1.2, the localization and the tracking of these bifurcations is recalled. During the continuation, the augmented system used for bifurcation tracking can again be ill-posed at some points. These points are called codimension-2 bifurcations. The codimension-2 bifurcations R1 and LPNS are treated in this Section.
Codimention-1 bifurcations To detect codimention-1 bifurcation points along the con- tinuated curve of periodic solutions, the following properties are used:
• Regular Points are well define, therefore RX is not singular.
• Limit Points (LP) are characterized by a singularity in the jacobian RX. Its deter-
minant det(RX) can be used as an indicator in order to detect such bifurcations.
The bifurcation is detected when the corresponding indicator det(RX) changes its
sign along the continuated curve. To be sure that no additional branch appears from the bifurcation, the jacobian J of the extended system defined in Eq. (1.19), must be invertible, i.e., det(J ) 6= 0.
• Branch Points (BP) are associated with the birth of a new branch of solutions. When a new branch of solutions appears, the jacobian J becomes singular. Consequently, the determinant det(J ) can be used as an indicator of BP along the continuated curve.
• Neimarck-Sacker bifurcations (NS) are associated with the birth of quasi-periodic regime. At a NS bifurcation, a quasi-periodic branch of solutions appears and the stability of the periodic solution on the main branch changes. A NS bifurcation appears when a pair of Floquet exponent crosses the imaginary axis with λ = ±iκ =
2πω2
ω1 with
ω1
ω2 ∈ Q. Indicators used for NS detection have been proposed in the/ following documents [SEY 09][KUZ 13][GOV 00]. These indicators also detect SR with ω1 ω2 ∈ N./ ϕNS1=
∏
1≤ j<i≤n λi+ λj ϕNS2= det(2JB I2n) (2.1)where JBis a diagonal matrix composed by the 2n Floquet exponents and stands
for the bialternate product. However, these indicators can also detected Neutral Saddle points where (λi, λj) form a pair of opposite reals.
Bifurcation analysis
Codimension-2 bifurcations Codimension-2 bifurcations are detected on the curve ob- tained by bifurcation tracking. To detect codimention-2 bifurcation points onto the con- tinuated curve of bifurcations, the following properties are used:
• A 1 : 1 Resonance (R1) appears during the continuation of LPs when a sec- ond eigenvalues becomes zero. Therefore, there exist two singular eigenvalues λ1 = λ2 = 0. When the jacobian RX contains two singularities, the quadratic
eigenvalue problem has a singular eigenvalue with algebraic multiplicity equal to two and geometric multiplicity equal to one. In order to detect a R1 using the LP indicator, a small modification needs to be done to the jacobian. Since the jacobian RX already presents a singularity associated with an eigenvector v, the regularized
jacobian RX+ vvT can be used to detect a R1 along the LP tracking curve. There-
fore, a R1 occurs when the determinant det(RX+ vvT) changes sign. Additional
information on possible new branch of solutions can be obtained with respect to the current bifurcation tracking curve. For LP continuation, the birth of a new branch of solution is indicated by a change of sign of the determinant of the jacobian J defined in Eq. (1.19). For BP continuation, the indicator must be regularized with respect to the eigenvector vJ associated to the singularity of J . Then, the appari-
tion of a new branch can be detected by a change of sign of the determinant of the regularized jacobian J + vJvTJ.
• A Limit Point Neimarck Sacker (LPNS) bifurcation can appear either on branches of NS or LP bifurcations. This bifurcation is characterized by the combination of NS and LP bifurcations with λ1= 0 and λ2= 2πωω12 with ωω12 ∈ Q. Therefore, the/
detection of the LPNS bifurcation differs depending on which bifurcation is con- tinuated. For LP continuation, the LPNS is detected when a pair of eigenvalues crosses the imaginary axis with λ = ±2πω2
ω1 with
ω1
ω2 ∈ Q. Therefore the same in-/ dicator as for NS can be used. On the other hand, for NS continuation, the LPNS is detected when a single eigenvalue crosses the imaginary axis. Therefore, the LP indicator can be used.