BRR CASTELLANA POR CLL 29 Nro.20-337 CENTRO COMERCIAL

(-1,0) Period Doubling Limit Point Neimark Sacker Neimark Sacker Branch Point Im(η) Re(η)

Figure 1.5 – Bifurcations with respect to Floquet multipliers displayed on the complex plan.

The methods for bifurcation localization and calculation depend on the domain in which the bifurcation analysis is performed. For time domain, the bifurcation analysis uses the Floquet multipliers obtained from the monodromy matrix, see Subsection. 1.2.1. A bifurcation point occurs when one of the Floquet multiplier crosses the unit circle, see Fig. 1.3. The following types of bifurcations can appears during the continuation of Nonlinear Normal Modes (NNMs) and periodic solution of discrete time dynamical forced system [KUZ 13]:

• A Limit Point (LP) and a Branch Point (BP) are co-dimension 1 bifurcations that appears for η = 1.

• A Neimark Sacker (NS) is a co-dimension 1 bifurcation that appears for η = e2πω2ω1 with ω1

• A Strong resonance (SR) are co-dimension 2 bifurcations characterized by η = 2πω_ω 2

1 with

ω1

ω2 ∈ Q. The period doubling (PD) is a co-dimension 1 bifurcation that is a special case of Strong Resonance (SR) bifurcations with η = −1.

Stable Limit Point Neimark Sacker Neimark Sacker λ= 0 λ= iθ λ= -iθ λ=iω/2 λ Period Doubling Period Doubling =-iω/2 Unstable Im(λ) Re(λ)

Figure 1.6 – Bifurcations with respect to Floquet exponents displayed on the complex plan. θ /∈ 2πQ

In the case of discrete frequency methods such as the HBM, the characterization of these bifurcations is based on Hill’s method. With Hill’s method, a bifurcation appears when a Floquet exponent λ crosses the imaginary axis, see Fig. 1.6. As stated in Sub- section 1.2.2, the Floquet multipliers η and the Floquet exponents λ are linked by Eqs. (1.37) and (1.47). Therefore, performing stability analysis using Hill’s method transforms the unity circle from the Floquet theory into the imaginary axis. Contrarily to the charac- terization of bifurcations obtained in the time domain, Hill’s method provides a frequency domain characterization with new considerations. Bifurcation points in the frequency do- main are characterized as follows:

• A LP appears for λ = 0 and RT ωφ 6= 0

• A BP appears for λ = 0 and RT

ωφ = 0.

• A NS bifurcation appears for λ = 2πω2

ω1 with

ω1

ω2 ∈ Q./ • A SR bifurcation appears for λ =2πω2

ω1 with

ω1

ω2 ∈ Q. One can see from Eq. 1.47, that for ω1

ω2 ∈ N it exists l ∈ [−H, H] such that an eigenvalue Λ = 0. In such case, when the HBM is used to compute the periodic solution, SR bifurcation can be computed as a pitchfork bifurcation which is a particular case of BP. This remark is based on a generalization of the characterization of PD as a pitchfork bifurcation when the HBM is used, see [PIC 94]. The period doubling bifurcation is a special case of Strong Resonance (SR) bifurcations. PD bifurcation appears for λ = −1.

Stability and bifurcation analysis

Limit Point (LP) A co-dimension 1 bifurcation called Limit Point (LP), saddle-node or fold bifurcation, occurs when η = 1, see Fig. 1.5, and no additional branch of solutions exists. The LP is a point where two fixed points respectively stable and unstable collide together and then disappear, see Fig. 1.7. In order to obtain both solutions, resolution methods combined with continuation technique such as the ones presented respectively in presented in Subsections 1.1.1 and 1.1.2 are needed. Such methods permit following the periodic solutions through the LP bifurcation while being able to compute unstable solutions.

Limit Point

X

α

Figure 1.7 – Limit Point bifurcation

Branch point (BP) A co-dimension 1 bifurcation Branch Point (BP) occurs when η = 1, see Fig. 1.5, and an additional branch of solutions exists. Before the BP, a unique solution exits. After the BP, the previous solution changes its stability and two new stable or unstable solutions appear from the critical point, see Fig.1.8. In order to follow the new branch of solutions, branching methods are used. Two types of Branch Point can be highlighted, the transcritical bifurcation and the pitchfork bifurcation. A transcritical bifurcation is a point where the continuated curve loops back on itself. At such point, the stability of the new branch of solutions changes, see Fig.1.8a. A pitchfork bifurcation represents the breaking of an underlying symmetry of the system, see [SEY 09]. Locally, the new branch is symmetric with respect to the main branch, see Fig. 1.8b. The stability of the new branch does not change at the pitchfork bifurcation. Two types of pitchfork bifurcation can occur, the super- and the sub-critical bifurcations which are characterized by the stability of the appearing branch of periodic solutions. For the supercritical pitch- fork bifurcation, the new solution branch is stable, see Fig. 1.8b. Conversely, for the subcritical pitchfork bifurcation, the new solution branch is unstable, see Fig.1.8c.

Neimark Sacker (NS) A co-dimension 1 bifurcation Neimark Sacker (NS), also called secondary Hopf bifurcation [SEY 09], occurs when η = e±iθ and θ =2πω2

ω1 where

ω1

ω2 ∈ Q./ It is not necessary to verify that ω1

ω2 ∈ Q since NS bifurcations are the only co-dimension/ 1 bifurcations with Im(λ ) 6= 0, see Fig. 1.5. At such bifurcation point, the periodic solution changes its stability and a new branch of quasi-periodic solutions appears from the critical point. The quasi-periodic solutions can be represented as a torus composed by

Transcritical X α (a) Transcritical bifurcation Pitchfork supercritical X α (b) Supercritical pitchfork bifurcation Pitchfork subcritical X α (c) Subcritical pitchfork bifurcation

Figure 1.8 – Branch Point bifurcation

the pulsation ω1 of the previous limit cycle and a new pulsation ω2, see [KUZ 13]. The

NS bifurcation can either be supercritical, see Fig. 1.9a, or subcritical, see Fig. 1.9b.

Neimark-Sacker supercritical X Quasi Periodic Quasi Periodic Periodic α (a) Supercritical NS subcritical X Quasi Periodic Quasi Periodic Periodic Neimark-Sacker α (b) Subcritical NS

Figure 1.9 – Neimark-Sacker bifurcation

Strong Resonances (SR) The co-dimension 2 bifurcation Strong Resonance (SR), also called internal resonance, occurs when η = e±iθ with θ = 2πω2

ω1 and

ω1

ω2 ∈ Q. When en- countered on Nonlinear Normal Modes (NNMs) the SRs are also called modal interaction bifurcations. At such a bifurcation point, the solution of period ω1 changes its stability

and a new branch of periodic solutions with a period ω2 appears from the critical point.

At the bifurcation point, the topology of the periodic solutions is the same as the pitchfork bifurcation. Co-dimension 1 bifurcation Flip, also called period doubling, occurs when η = −1, see Fig. 1.5. Its a special case of strong resonances with ω1

ω2 =

1

2. The periodic

solution changes its stability by going through the Flip bifurcation and a new branch of solutions appears. However, the new branch of solutions is now 2T periodic instead of T periodic.

Parametric analysis by continuation of localized point