Correction to: Asymptotic analysis of passive mitigation of dynamic instability using a nonlinear energy sink network
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1 Correction to: Nonlinear Dyn (2018) 94:1501–1522 https://doi.org/10.1007/s1107101844380
Remark
In the sequel, the equation numbering follows that of the original paper.
In the original version of the manuscript [1], Result 3.3 which states the conditions to ensure that the system is in a harmless situation fails to describe some possible situations, especially when very different NESs are considered. Therefore, Result 3.3 must be corrected, just like Result 3.4 which results from it and gives a quantitative characterization of the mitigation limit. These corrections do not affect the results presented in Sect. 4 and the numerical validation performed in Sect. 5.
To know the steadystate regime from a given set of parameters, the superslow behavior of the system must be checked in every successive subspace \(I_k^a\) (see Eq. 30) along the trajectory of the system and not only in the last one (\(I_{2N}^a\)) as it is done in the original version of the paper. The behavior of the system in a given subspace \(I_k^a\) is determined by the relative position between the arrival point \(\mathbf{c}=\left[ c_{1},\ldots ,c_{N}\right] \) (i.e., the point on the critical manifold S in \(I_k^a\) on which the trajectory arrives after a jump at the slow timescale, it is described more precisely below) and the fixed points (stable and unstable). Note that if \(I_k^a\) does not contain fixed point, the direction of the superslow flow on the critical manifold S is determined by the sign of the function \(\left. f\left( r_1,\ldots ,r_N\right) \right _\mathbf{r=c}\) (see Eq. 39).
An easy calculus shows that M identical NESs with parameters a, \(\mu \) and \(\alpha \) are equivalent to one NES with parameters Ma, \(M \mu \) and \(M \alpha \). This is consistent with the Eq. (53). Therefore, in the NES network, all groups of identical NESs can be replaced by one equivalent NES and the previous reasoning can be used to describe the trajectory of the system.
After these preliminary comments, the corrected versions of Results 3.3 and 3.4 are now given.
Result 3.3
Definition 3.1 states that we consider a set of initial conditions \(\mathbf{r_0}=\left[ r_1(0),\ldots ,r_N(0)\right] \) as a small perturbation of the trivial solution, i.e., \(\mathbf{r}_{\mathbf{0}}\in I_1^a\).
If condition (86) is not respected, the trajectory reaches the jump point of \(I_1^a\). Note that \(I_1^a\) has only one jump point. The coordinates of the corresponding arrival point \(\mathbf{c}\), contained in the next crossed \(I_k^a\), are determined from those of the jump point through Eq. (85).
 Situation 1.

There are no fixed points in \(I_k^a\), then, depending of the sign of the function \(\left. f\left( r_1,\ldots ,r_N\right) \right _\mathbf{r=c}\), \(\mathbf{b}^{(\mathbf{i})}\) or \(\mathbf{b}^{(\mathbf{j})}\) is reached and the trajectory leaves \(I_k^a\).
 Situation 2.

The arrival point is between an unstable fixed point and \(\mathbf{b}^{(\mathbf{i})}\) (resp. \(\mathbf{b}^{(\mathbf{j})}\)), then \(\mathbf{b}^{(\mathbf{i})}\) (resp. \(\mathbf{b}^{(\mathbf{j})}\)) is reached and the trajectory leaves \(I_k^a\).
 Situation 3.

A stable fixed point is the first neighboring point of the arrival point \(\mathbf{c}\), and then the stable fixed point is reached.
 [HlS]: The system is in a harmless situation if

[HlS(a)]: situation 1 or 2 holds, and along the trajectory a \(I_k^a\) is met again. In this case, the system is in a harmless situation corresponding to a mitigation through strongly modulated response (SMR).

[HlS(b)]: situation 3 holds in \(I_k^a\) with \(k\ne 2N\). In this case, the reached fixed point has a small amplitude and the system is in a harmless situation corresponding to a mitigation through periodic response.


[HfS]: The system is in a harmful situation if situation 3 holds in \(I_k^a=I_{2N}^a\). Indeed, in this case, the reached fixed point has a large amplitude close to that of the case without NES.
Result 3.4
From a given set of parameters, Result 3.3 gives a theoretical prediction of the resulting steadystate regime and therefore allows to know if the system is a harmless or harmful situation. Consequently, the mitigation limit, denoted \(\rho _{ml}\), can be predicted theoretically as the first value of \(\rho \) for which situation 3 holds in \(I_k^a=I_{2N}^a\) (i.e., scenario HfS).
As usual in the example shown in Figs. 7 and 8 of the original paper, the mitigation limit corresponds to the value of the parameter \(\rho \) corresponding to the transition from harmless to harmful situation. In this example, three NESs are considered (\(N=3\)) and the last harmless situation before the transition (with respect to \(\rho \)) to harmful situation corresponds to a SMR in which the trajectory reaches \(I_{2N=6}^a\). The subinterval \(I_{6}^a\) contains unstable fixed points and the larger one is denoted \(\mathbf{r_u^*}=\left[ r_{u,1}^*,r_{u,2}^*,r_{u,3}^*\right] \). Therefore, the harmless situation corresponds to the scenario HlS(a) described above. Moreover, the arrival point \(\mathbf{c}\) in \(=I_{6}^a\) is due to a jump in direction \(r_3\) from a jump point \(\mathbf{b}=\left[ b_{1},b_{2},r_3^M\right] \). Consequently \(\mathbf{c}=\left[ b_{1},b_{2},r_3^u\right] \) and the mitigation limit is the solution of \(r_{u,3}^*(\rho )= r_{3}^u\). Indeed, at this special value, the response of the system switches from HlS(a) to HfS in which \(\mathbf{r_u^*}\) is reached. On Fig. 7, the mitigation limit corresponds therefore to the intersection between the branch of the \(r_3\) coordinate of the larger unstable fixed point and \(r_3^u\) as it is defined in the original version of the paper.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest concerning the publication of this manuscript.
Reference
 1.Bergeot, B., Bellizzi, S.: Asymptotic analysis of passive mitigation of dynamic instability using a nonlinear energy sink network. Nonlinear Dyn. 94(2), 1501–1522 (2018). https://doi.org/10.1007/s1107101844380 CrossRefGoogle Scholar