Emergent Behaviors of Lohe Tensor Flocks

Abstract

We present a new aggregation model on the space of rank-m tensors with the same size, and study emergent dynamics of the proposed model. Our proposed new aggregation model is general enough to encompass Lohe type models such as the Kuramoto model, the Lohe sphere model and the Lohe matrix models for the ensemble of real rank-0, rank-1 and rank-2 tensors, respectively. In this regard, we call our proposed model as the Lohe tensor model for rank-m tensors with the same size. For emergent dynamics of the proposed model, we employ the ensemble diameter as a Lyapunov functional to derive a Riccati type differential inequality. As a direct application of these differential inequalities, we provide two sufficient frameworks leading to the emergent dynamics for homogeneous and heterogeneous ensembles in terms of system parameters and initial data.

Appendix A: Proof of Lemma 4.4

In this appendix, we provide a proof of Lemma 4.4.

(i) We use antisymmetric property (4.2) of $A_i$ and $A_j$ and exchange symmetry $\alpha _{*0} \longleftrightarrow \alpha _{*1}$ to find

$$\begin{aligned} \begin{aligned} \mathcal {L}_{ij}&:= -[T_i]_{\alpha _{*0}} [A_j]_{\alpha _{*1}\alpha _{*0}} [{{\bar{T}}}_j]_{\alpha _{*1}} - [T_j]_{\alpha _{*0}} [A_i]_{\alpha _{*1}\alpha _{*0}} [{\bar{T}}_i]_{\alpha _{*1}} + [{{\bar{T}}}_j]_{\alpha _{*0}} [A_i]_{\alpha _{*0}\alpha _{*1}} [T_i]_{\alpha _{*1}} \\&\quad + [{{\bar{T}}}_i]_{\alpha _{*0}} [A_j]_{\alpha _{*0}\alpha _{*1}} [T_j]_{\alpha _{*1}} \\&=-[T_i]_{\alpha _{*1}} [A_j]_{\alpha _{*0}\alpha _{*1}} [{{\bar{T}}}_j]_{\alpha _{*0}} - [T_j]_{\alpha _{*1}} [A_i]_{\alpha _{*0}\alpha _{*1}} [{{\bar{T}}}_i]_{\alpha _{*0}} + [{{\bar{T}}}_j]_{\alpha _{*0}} [A_i]_{\alpha _{*0}\alpha _{*1}} [T_i]_{\alpha _{*1}} \\&\quad + [{{\bar{T}}}_i]_{\alpha _{*0}} [A_j]_{\alpha _{*0}\alpha _{*1}} [T_j]_{\alpha _{*1}} \\&= \Big ( [A_i]_{\alpha _{*0}\alpha _{*1}} - [A_j]_{\alpha _{*0}\alpha _{*1}} \Big ) [{{\bar{T}}}_j]_{\alpha _{*0}} [T_i]_{\alpha _{*1}} - \Big ( [A_i]_{\alpha _{*0}\alpha _{*1}} - [A_j]_{\alpha _{*0}\alpha _{*1}} \Big ) [{{\bar{T}}}_i]_{\alpha _{*0}} [T_j]_{\alpha _{*1}} \\&= [A_i-A_j]_{\alpha _{*0}\alpha _{*1}} \Big ( [{\bar{T}}_j]_{\alpha _{*0}} [T_i]_{\alpha _{*1}} - [{\bar{T}}_i]_{\alpha _{*0}} [T_j]_{\alpha _{*1}} \Big ). \end{aligned} \end{aligned}$$

This and (4.9) yield

$$\begin{aligned} \begin{aligned} |\mathcal {L}_{ij}|^2&=\left| \sum _{\alpha _{*0}, \alpha _{*1}} [A_i-A_j]_{\alpha _{*0}\alpha _{*1}}\left( \bar{[T_j]}_{\alpha _{*0}}[T_i]_{\alpha _{*1}}-\bar{[T_i]}_{\alpha _{*0}}[T_j]_{\alpha _{*1}}\right) \right| ^2\\&\le ||A_i-A_j||_F^2\sum _{\alpha _{*0}, \alpha _{*1}}\left| \bar{[T_j]}_{\alpha _{*0}}[T_i]_{\alpha _{*1}}-\bar{[T_i]}_{\alpha _{*0}}[T_j]_{\alpha _{*1}}\right| ^2\\&=||A_i-A_j||_F^2\sum _{\alpha _{*0}, \alpha _{*1}}\left| \bar{[T_j]}_{\alpha _{*0}}[T_i]_{\alpha _{*1}}-\bar{[T_j]}_{\alpha _{*0}}[T_j]_{\alpha _{*1}}+\bar{[T_j]}_{\alpha _{*0}}[T_j]_{\alpha _{*1}}-\bar{[T_i]}_{\alpha _{*0}}[T_j]_{\alpha _{*1}}\right| ^2\\&=||A_i-A_j||_F^2\sum _{\alpha _{*0}, \alpha _{*1}} \left| \bar{[T_j]}_{\alpha _{*0}}[T_i-T_j]_{\alpha _{*1}}+[\bar{T_j}-\bar{T_i}]_{\alpha _{*0}}[T_j]_{\alpha _{*1}} \right| ^2\\&\le 4||A_i-A_j||_F^2 \cdot ||T_i-T_j||_F^2 \cdot ||T_j||_F^2 = 4||A_i-A_j||_F^2 \cdot ||T_i-T_j||_F^2. \end{aligned} \end{aligned}$$

This implies

$$\begin{aligned} |\mathcal {L}|\le 2||A_i-A_j||_F \cdot ||T_i-T_j||_F \le 2D(A)D(T). \end{aligned}$$

(ii) Now, we estimate the term in the sigma notation.

$$\begin{aligned} \begin{aligned}&\big |\bar{[T_j]}_{\alpha _{*0}}[T_c]_{\alpha _{*i_*}}\bar{[T_i]}_{\alpha _{*1}}[T_i]_{\alpha _{*(1-i_*)}} -\bar{[T_j]}_{\alpha _{*0}}[T_i]_{\alpha _{*i_*}}\bar{[T_c]}_{\alpha _{*1}}[T_i]_{\alpha _{*(1-i_*)}}\\&\qquad +\bar{[T_i]}_{\alpha _{*0}}[T_c]_{\alpha _{*i_*}}\bar{[T_j]}_{\alpha _{*1}}[T_j]_{\alpha _{*(1-i_*)}} -\bar{[T_i]}_{\alpha _{*0}}[T_j]_{\alpha _{*i_*}}\bar{[T_c]}_{\alpha _{*1}}[T_j]_{\alpha _{*(1-i_*)}}\\&\qquad +{[T_j]}_{\alpha _{*0}}\bar{[T_c]}_{\alpha _{*i_*}}{[T_i]}_{\alpha _{*1}}\bar{[T_i]}_{\alpha _{*(1-i_*)}} -{[T_j]}_{\alpha _{*0}}\bar{[T_i]}_{\alpha _{*i_*}}{[T_c]}_{\alpha _{*1}}\bar{[T_i]}_{\alpha _{*(1-i_*)}}\\&\qquad +{[T_i]}_{\alpha _{*0}}\bar{[T_c]}_{\alpha _{*i_*}}{[T_j]}_{\alpha _{*1}}\bar{[T_j]}_{\alpha _{*(1-i_*)}} -{[T_i]}_{\alpha _{*0}}\bar{[T_j]}_{\alpha _{*i_*}}{[T_c]}_{\alpha _{*1}}\bar{[T_j]}_{\alpha _{*(1-i_*)}}\big |\\&\quad \le \big |-{1\over 2}\bar{[T_c]}_{\alpha _{*0}}[T_i-T_j]_{\alpha _{*i_*}}\overline{[T_i+T_j]}_{\alpha _{*1}}[T_i-T_j]_{\alpha _{*(1-i_*)}} \\&\qquad +{1\over 2}\bar{[T_c]}_{\alpha _{*0}}[T_i+T_j]_{\alpha _{*i_*}}\overline{[T_i-T_j]}_{\alpha _{*1}}[T_i-T_j]_{\alpha _{*(1-i_*)}}\\&\qquad +{1\over 2}\overline{[T_i-T_j]}_{\alpha _{*0}}[T_i-T_j]_{\alpha _{*i_*}}\overline{[T_i+T_j]}_{\alpha _{*1}}{[T_c]}_{\alpha _{*(1-i_*)}} \\&\quad -{1\over 2}\overline{[T_i-T_j]}_{\alpha _{*0}}[T_i+T_j]_{\alpha _{*i_*}}\overline{[T_i-T_j]}_{\alpha _{*1}}{[T_c]}_{\alpha _{*(1-i_*)}} \big |\\&\quad \le 2||T_i-T_j||_F^2 \cdot ||T_i+T_j||_F \cdot ||T_c||_F\\&\quad \le 4D(T)^2||T_i||_F \cdot ||T_c||_F. \end{aligned} \end{aligned}$$

We use the above estimate to see

$$\begin{aligned} \begin{aligned} \mathcal {M}&=\sum _{i_*\ne 0}\kappa _{i_*} \big ( \bar{[T_j]}_{\alpha _{*0}}[T_c]_{\alpha _{*i_*}}\bar{[T_i]}_{\alpha _{*1}}[T_i]_{\alpha _{*(1-i_*)}} -\bar{[T_j]}_{\alpha _{*0}}[T_i]_{\alpha _{*i_*}}\bar{[T_c]}_{\alpha _{*1}}[T_i]_{\alpha _{*(1-i_*)}}\\&\quad +\bar{[T_i]}_{\alpha _{*0}}[T_c]_{\alpha _{*i_*}}\bar{[T_j]}_{\alpha _{*1}}[T_j]_{\alpha _{*(1-i_*)}} -\bar{[T_i]}_{\alpha _{*0}}[T_j]_{\alpha _{*i_*}}\bar{[T_c]}_{\alpha _{*1}}[T_j]_{\alpha _{*(1-i_*)}}\\&\quad +{[T_j]}_{\alpha _{*0}}\bar{[T_c]}_{\alpha _{*i_*}}{[T_i]}_{\alpha _{*1}}\bar{[T_i]}_{\alpha _{*(1-i_*)}} -{[T_j]}_{\alpha _{*0}}\bar{[T_i]}_{\alpha _{*i_*}}{[T_c]}_{\alpha _{*1}}\bar{[T_i]}_{\alpha _{*(1-i_*)}}\\&\quad +{[T_i]}_{\alpha _{*0}}\bar{[T_c]}_{\alpha _{*i_*}}{[T_j]}_{\alpha _{*1}}\bar{[T_j]}_{\alpha _{*(1-i_*)}} -{[T_i]}_{\alpha _{*0}}\bar{[T_j]}_{\alpha _{*i_*}}{[T_c]}_{\alpha _{*1}}\bar{[T_j]}_{\alpha _{*(1-i_*)}} \big )\quad (:=\mathcal {J}_1)\\&\quad +\kappa _{0}\big ( \bar{[T_j]}_{\alpha _{*0}}[T_c]_{\alpha _{*0}}\bar{[T_i]}_{\alpha _{*1}}[T_i]_{\alpha _{*1}} -\bar{[T_j]}_{\alpha _{*0}}[T_i]_{\alpha _{*0}}\bar{[T_c]}_{\alpha _{*1}}[T_i]_{\alpha _{*1}}\\&\quad +\bar{[T_i]}_{\alpha _{*0}}[T_c]_{\alpha _{*0}}\bar{[T_j]}_{\alpha _{*1}}[T_j]_{\alpha _{*1}} -\bar{[T_i]}_{\alpha _{*0}}[T_j]_{\alpha _{*0}}\bar{[T_c]}_{\alpha _{*1}}[T_j]_{\alpha _{*1}}\\&\quad +{[T_j]}_{\alpha _{*0}}\bar{[T_c]}_{\alpha _{*0}}{[T_i]}_{\alpha _{*1}}\bar{[T_i]}_{\alpha _{*1}} -{[T_j]}_{\alpha _{*0}}\bar{[T_i]}_{\alpha _{*0}}{[T_c]}_{\alpha _{*1}}\bar{[T_i]}_{\alpha _{*1}}\\&\quad +{[T_i]}_{\alpha _{*0}}\bar{[T_c]}_{\alpha _{*0}}{[T_j]}_{\alpha _{*1}}\bar{[T_j]}_{\alpha _{*1}} -{[T_i]}_{\alpha _{*0}}\bar{[T_j]}_{\alpha _{*0}}{[T_c]}_{\alpha _{*1}}\bar{[T_j]}_{\alpha _{*1}} \big )\quad (:=\mathcal {J}_2) \end{aligned} \end{aligned}$$

Below, we estimate $\mathcal {J}_1$ and $\mathcal {J}_2$.

$\bullet$ (Estimate of $\mathcal {J}_1$): By direct estimate, one has

$$\begin{aligned} |\mathcal {J}_1|\le 4D(T)^2||T_i||_F \cdot ||T_c||_F \sum _{i_*\ne 0}\kappa _{i_*} \end{aligned}$$

$\bullet$ (Estimate of $\mathcal {J}_2$): Similarly, one has

$$\begin{aligned} {\mathcal {J}_2\over \kappa _0}&=\langle {T_j, T_c}\rangle _F \langle {T_i, T_i}\rangle _F -\langle {T_j, T_i}\rangle _F \langle {T_c, T_i}\rangle _F + \langle {T_i, T_c}\rangle _F \cdot \langle {T_j, T_j}\rangle _F - \langle {T_i, T_j}\rangle _F \cdot \langle {T_c, T_j}\rangle _F \\&\quad +\langle {T_c, T_j}\rangle _F \cdot \langle {T_i, T_i}\rangle _F - \langle {T_i, T_j}\rangle _F \cdot \langle {T_i, T_c}\rangle _F + \langle {T_c, T_i}\rangle _F \cdot \langle {T_j, T_j}\rangle _F -\langle {T_j, T_i}\rangle _F \cdot \langle T_j, T_c\rangle _F \\&=-\langle T_i, T_j \rangle _F (\langle {T_c, T_j}\rangle _F +\langle {T_i, T_c}\rangle _F)-\langle {T_j, T_i}\rangle _F (\langle {T_c, T_i}\rangle _F +\langle {T_j, T_c}\rangle _F )\\&\quad +\langle {T_i, T_i}\rangle _F (\langle {T_i, T_c}\rangle _F +\langle {T_j, T_c}\rangle _F +\langle {T_c, T_i}\rangle _F +\langle {T_c, T_j}\rangle _F )\\&=(\langle {T_i, T_i}\rangle _F -\langle {T_i, T_j}\rangle _F )(\langle {T_c, T_j}\rangle _F +\langle {T_i, T_c}\rangle _F )+(\langle {T_i, T_i}\rangle _F -\langle {T_j, T_i}\rangle _F )(\langle {T_c, T_i}\rangle _F +\langle {T_j, T_c}_F \rangle )\\&={1\over 2}||T_i-T_j||_F^2 \Big (4\langle {T_i, T_i}\rangle _F +\langle {T_c-T_j, T_j}\rangle _F +\langle {T_c-T_i, T_i}\rangle _F +\langle {T_j, T_c-T_j}\rangle _F +\langle {T_i, T_c-T_i}\rangle _F \Big ). \end{aligned}$$

Therefore, one has

$$\begin{aligned} \big |\mathcal {J}_2-2\kappa _0||T_i-T_j||_F^2 \cdot ||T_i||_F^2\big |\le {}4\kappa _0D(T)^3||T_i||_F. \end{aligned}$$

If we combine the above two estimates of $\mathcal {J}_1$ and $\mathcal {J}_2$, then we can obtain following estimate:

$$\begin{aligned} \big |\mathcal {M}-2\kappa _0||T_i-T_j||_F^2||T_i||_F^2\big |\le 4D(T)^2||T_i||_F \cdot ||T_c||_F \sum _{i_*\ne 0}\kappa _{i_*}+4\kappa _0D(T)^3||T_i||_F. \end{aligned}$$

Now, we use $\Vert T_i \Vert _F = 1$ to get the desired estimate.