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Emergent Behavior of a Second-Order Lohe Matrix Model on the Unitary Group

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Abstract

We study a second-order extension to the first-order Lohe matrix model on the unitary group which can be reduced to the second-order Kuramoto model with inertia as a special case. For the proposed second-order model, we present several sufficient frameworks leading to the emergence of the complete and practical synchronizations in terms of the initial data and the system parameters. For the identical hamiltonians, we show that the complete synchronization emerges asymptotically. In contrast, for the non-identical hamiltonians, the practical synchronization occurs for some class of initial data when the product of the coupling strength and inertia is sufficiently small.

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Authors

Corresponding author

Correspondence to Dohyun Kim.

Additional information

Communicated by Irene Giardina.

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The work of S.Y. Ha is supported by the National Research Foundation of Korea (NRF-2017R1A5A1015626).

Appendices

Proof of Lemma 6

In this appendix, we provide a proof of Lemma 6. First, we write

$$\begin{aligned} u := \frac{1}{\sqrt{y}} \quad \text {or}\quad y=\frac{1}{u^2}. \end{aligned}$$

Then, u satisfies

$$\begin{aligned} {\dot{y}} = -\frac{2u{\dot{u}}}{u^4} = -\frac{2{\dot{u}}}{u^3},\quad -{\ddot{y}} = \left( \frac{2{\dot{u}}}{u^3}\right) ' = \frac{2{\ddot{u}} u^3 - 2{\dot{u}} \cdot 3u^2 {\dot{u}}}{u^6} = \frac{2{\ddot{u}}}{u^3} - \frac{6|{\dot{u}}|^2}{u^4}. \end{aligned}$$
(1)

Then, we substitute (1) into (24) to obtain a differential inequality for u:

$$\begin{aligned} a\left( -\frac{2{\ddot{u}}}{u^3} + \frac{6|{\dot{u}}|^2}{u^4}\right) + b\left( -\frac{2{\dot{u}}}{u^3}\right) + c\left( \frac{1}{u^2}\right) \le d \frac{1}{u^3} + \varepsilon (t). \end{aligned}$$

This can be rewritten as

$$\begin{aligned} 2a{\ddot{u}} + 2b{\dot{u}} - cu \ge -d -\varepsilon u^3 +\frac{6a |{\dot{u}}|^2}{u} \ge -d -\varepsilon u^3 =: -f(t). \end{aligned}$$

Then, we have the following second-order differential inequality:

$$\begin{aligned} 2a {\ddot{u}} + 2b {\dot{u}} - cu + f \ge 0, \quad t>0. \end{aligned}$$

Now, we recall that our goal is to show that u(t) diverges to infinity so that y(t) converges to zero. For \(f=f(t)\), we have two possible cases:

$$\begin{aligned} \text {Case A:}~~f(t)\text { diverges to infinity;}\quad \text {Case B:}~~f(t)\text { is uniformly bounded.} \end{aligned}$$
  •  Case A: Suppose that Case A occurs. Then, we have

    $$\begin{aligned} \lim _{t\rightarrow \infty } G(t):= \lim _{t\rightarrow \infty } \frac{\varepsilon (t)}{y(t)^\frac{3}{2}}=\infty ,\quad \text {or,} \quad \lim _{t\rightarrow \infty } y(t)^\frac{3}{2} = \lim _{t\rightarrow \infty } \frac{\varepsilon (t)}{G(t)} =0. \end{aligned}$$

    This establishes our proof.

  •  Case B: Without loss of generality, we assume that f(t) is uniformly bounded, i.e., there exists a uniform constant \(\alpha >d\) such that

    $$\begin{aligned} f(t) < \alpha , \quad t>0. \end{aligned}$$

    For \(h<0\) to be determined later, we observe

    $$\begin{aligned} 2a {\ddot{u}} + 2b{\dot{u}} - h{\dot{u}} \ge - h{\dot{u}} + cu - f(t) \ge -h{\dot{u}} + cu -\alpha , \end{aligned}$$

    and this can be rewritten as

    $$\begin{aligned} {\ddot{u}} + \frac{2b-h}{2a}{\dot{u}} \ge -\frac{h}{2a} \left( {\dot{u}} - \frac{c}{h} u\right) -\alpha . \end{aligned}$$
    (2)

    We choose \(h<0\) to satisfy

    $$\begin{aligned} \frac{2b-h}{2a} = -\frac{c}{h} \quad \text {or}\quad h^2 -2bh - 2ac =0, \quad \text {which yields}\quad h = b\pm \sqrt{b^2+2ac}. \end{aligned}$$

    Hence, we choose \(h= b-\sqrt{b^2+2ac}\) and write

    $$\begin{aligned} A= \frac{2b-h}{2a} = -\frac{c}{h} >0, \quad L = {\dot{u}} + Au. \end{aligned}$$

    Then, (2) yields

    $$\begin{aligned} {\dot{L}} \ge -\frac{h}{2a} L - \alpha , \quad t>0. \end{aligned}$$

    We multiply \(e^{\frac{h}{2a}t}\) with the above relation to obtain

    $$\begin{aligned} {\dot{u}}(t) + Au(t) = L(t) \ge \left( L(0) + \frac{2a}{h}\alpha \right) e^{-\frac{h}{2a}t} - \frac{2a}{h}\alpha . \end{aligned}$$

    Then, we again multiply \(e^{At}\) with the above relation to see

    $$\begin{aligned} e^{At} u(t) - u(0)&\ge \left( L(0) + \frac{2a}{h}\alpha \right) \int _0^t e^{-\frac{h}{2a}s} e^{As}ds - \frac{2a}{h}\alpha \int _0^t e^{As}ds \\&= \left( L(0) + \frac{2a}{h}\alpha \right) \frac{1}{A-\frac{h}{2a}} \Big ( e^{At} e^{-\frac{h}{2a}t} -1\Big ) -\frac{2a}{h}\alpha (e^{At}-1). \end{aligned}$$

    Hence, we have

    $$\begin{aligned} u(t) \ge u(0)e^{-At} + \left( L(0) + \frac{2a}{h}\alpha \right) \frac{1}{A-\frac{h}{2a}} \Big ( e^{-\frac{h}{2a}t} - e^{-At} \Big ) - \frac{2a}{h}\alpha (1-e^{-At}). \end{aligned}$$
    (3)

    Since \(h<0\), the right-hand side of (3) diverges. Therefore, u(t) also diverges. This completes the proof.

Proof of Lemma 8

In this appendix, we provide a proof of Lemma 8 in several steps.

  • Step A (Derivation of an equation for \({\mathcal {D}}_{ij}\)): To derive an equation for \({{\mathcal {D}}}_{ij}\), we recall (25) and find

    $$\begin{aligned} \begin{aligned} m {\ddot{U}}_i U_j^\dagger =&- m {\dot{U}}_i {\dot{U}}_i^\dagger U_i U_j^\dagger - \gamma {\dot{U}}_i U_j^\dagger -{\mathrm {i}}H_i U_i U_j^\dagger \\&+ \frac{m{\mathrm {i}}}{\gamma }(H_i {\dot{U}}_i U_j^\dagger - {\dot{U}}_i U_i^\dagger H_i U_iU_j^\dagger )+ \frac{\kappa }{2N} \sum _{k=1}^N ( U_kU_j^\dagger - U_iU_k^\dagger U_i U_j^\dagger ), \\ m U_i {\ddot{U}}_j^\dagger =&-m U_iU_j^\dagger {\dot{U}}_j {\dot{U}}_j^\dagger - \gamma U_i {\dot{U}}_j^\dagger + {\mathrm {i}}U_iU_j^\dagger H_j \\&+\frac{m{\mathrm {i}}}{\gamma }( U_iU_j^\dagger H_j U_j {\dot{U}}_j^\dagger - U_i{\dot{U}}_j^\dagger H_j) + \frac{\kappa }{2N}\sum _{k=1}^N (U_iU_k^\dagger - U_iU_j^\dagger U_k U_j^\dagger ) . \end{aligned} \end{aligned}$$
    (1)

    We calculate (1)\(_1 + \)(1)\(_2\) and add \(2m {\dot{U}}_i {\dot{U}}_j^\dagger \) in both sides of the resulting relation to find

    $$\begin{aligned} \begin{aligned}&m {\ddot{U}}_i U_j^\dagger + m {\ddot{U}}_j U_i^\dagger +2m {\dot{U}}_i {\dot{U}}_j^\dagger \\&\quad = 2m {\dot{U}}_i {\dot{U}}_j^\dagger - m ({\dot{U}}_i {\dot{U}}_i^\dagger U_i U_j^\dagger + U_iU_j^\dagger {\dot{U}}_j {\dot{U}}_j^\dagger ) - \gamma ({\dot{U}}_i U_j^\dagger +U_i {\dot{U}}_j^\dagger ) \\&\qquad -{\mathrm {i}}(H_i U_i U_j^\dagger -U_iU_j^\dagger H_j) + \frac{m{\mathrm {i}}}{\gamma }(H_i {\dot{U}}_i U_j^\dagger - U_i {\dot{U}}_j^\dagger H_j \\&\qquad + U_iU_j^\dagger H_j U_j {\dot{U}}_j^\dagger - {\dot{U}}_i U_i^\dagger H_ i U_iU_j^\dagger ) \\&\qquad + \frac{\kappa }{2N} \sum _{k=1}^N \Big ( U_kU_j^\dagger - U_iU_k^\dagger U_i U_j^\dagger + U_iU_k^\dagger - U_iU_j^\dagger U_kU_j^\dagger \Big ). \end{aligned} \end{aligned}$$
    (2)

    For the last term in the summand of right-hand side, we can estimate the term as follows:

    $$\begin{aligned}&U_kU_j^\dagger - U_iU_k^\dagger U_i U_j^\dagger + U_iU_k^\dagger - U_iU_j^\dagger U_kU_j^\dagger \\&\quad = U_kU_j^\dagger - I_d + I_d -(U_iU_k^\dagger - I_d + I_d)(U_iU_j^\dagger - I_d+I_d) \\&\qquad + U_iU_k^\dagger - I_d + I_d - (U_iU_j^\dagger -I_d+I_d)(U_kU_j^\dagger -I_d+I_d) \\&\quad = {\mathcal {D}}_{kj} + I_d -({\mathcal {D}}_{ik} + I_d)({{\mathcal {D}}}_{ij} + I_d) + {\mathcal {D}}_{ik} + I_d -({\mathcal {D}}_{ij}+I_d)({\mathcal {D}}_{kj}+I_d) \\&\quad = -2{\mathcal {D}}_{ij} - {\mathcal {D}}_{ik}{\mathcal {D}}_{ij} - {\mathcal {D}}_{ij} {\mathcal {D}}_{kj}. \end{aligned}$$

    Recall that

    $$\begin{aligned} {\mathcal {D}}_{ij} = U_iU_j^\dagger - I_d, \quad \dot{{\mathcal {D}}}_{ij} = {\dot{U}}_i U_j^\dagger + U_i {\dot{U}}_j^\dagger , \quad \ddot{{\mathcal {D}}}_{ij} = {\ddot{U}}_i U_j^\dagger + U_i {\ddot{U}}_j^\dagger + 2{\dot{U}}_i {\dot{U}}_j^\dagger . \end{aligned}$$

    Then, we rewrite (2) in terms of \({\mathcal {D}}_{ij}\):

    $$\begin{aligned} \begin{aligned}&m\ddot{{\mathcal {D}}}_{ij} + \gamma \dot{{\mathcal {D}}}_{ij} + \kappa {\mathcal {D}}_{ij} \\&=\Big [2m {\dot{U}}_i {\dot{U}}_j^\dagger - m ({\dot{U}}_i {\dot{U}}_i^\dagger U_i U_j^\dagger + U_iU_j^\dagger {\dot{U}}_j {\dot{U}}_j^\dagger )\Big ] -{\mathrm {i}}(H_i U_i U_j^\dagger -U_iU_j^\dagger H_j) \\&\quad + \frac{m{\mathrm {i}}}{\gamma }(H_i {\dot{U}}_i U_j^\dagger - U_i {\dot{U}}_j^\dagger H_j + U_iU_j^\dagger H_j U_j {\dot{U}}_j^\dagger - {\dot{U}}_i U_i^\dagger H_ i U_iU_j^\dagger )\\&\quad -\frac{\kappa }{2N}\sum _{k=1}^N \Big ({\mathcal {D}}_{ik} {\mathcal {D}}_{ij} + {\mathcal {D}}_{ij} {\mathcal {D}}_{kj}\Big ) \\&=:{\mathcal {I}}_{21} + {\mathcal {I}}_{22} + {\mathcal {I}}_{23} + {\mathcal {I}}_{24}. \end{aligned} \end{aligned}$$
    (3)
  • Step B (Derivation of an equation for \(\Vert {\mathcal {D}}_{ij}\Vert _\text {F}^2\)): In (3), we apply Lemma 5 to obtain

    $$\begin{aligned} \begin{aligned}&m\frac{d^2}{dt^2} \Vert {{\mathcal {D}}}_{ij}\Vert _\text {F}^2 + \gamma \frac{d}{dt} \Vert {{\mathcal {D}}}_{ij}\Vert _\text {F}^2 + 2\kappa \Vert {{\mathcal {D}}}_{ij}\Vert _\text {F}^2 \\&\quad = 2\text {Re}\,\text {Tr}\Big (({\mathcal {I}}_{21} + {\mathcal {I}}_{22}+ {\mathcal {I}}_{23}+ {\mathcal {I}}_{24}) {\mathcal {D}}_{ij}^\dagger \Big ) + 2m \Vert \dot{{\mathcal {D}}}_{ij}\Vert _\text {F}^2. \end{aligned} \end{aligned}$$
    (4)

    Below, we present estimates of \(\text {Re}\,\text {Tr}\,({\mathcal {I}}_{2k}{\mathcal {D}}_{ij}^\dagger ),~k=1,\ldots ,4\), separately.

  • (Estimate of \(\text {Re}\,\text {Tr}\,({\mathcal {I}}_{21}{\mathcal {D}}_{ij}^\dagger ))\): We observe

    $$\begin{aligned}&\text {Re}\,\text {Tr}\,({\mathcal {I}}_{21}{\mathcal {D}}_{ij}^\dagger ) \\&\quad =m \text {Re}\,\text {Tr} \Big [( 2{\dot{U}}_i {\dot{U}}_j^\dagger - {\dot{U}}_i {\dot{U}}_i^\dagger U_iU_j^\dagger - U_iU_j^\dagger {\dot{U}}_j{\dot{U}}_j^\dagger )(U_jU_i^\dagger -I_d) \Big ] \\&\quad = m\text {Re}\,\text {Tr}\,\Big [ 2{\dot{U}}_i {\dot{U}}_j^\dagger U_j U_i^\dagger - 2 {\dot{U}}_i {\dot{U}}_j^\dagger - {\dot{U}}_i {\dot{U}}_i^\dagger + {\dot{U}}_i {\dot{U}}_i^\dagger U_i U_j^\dagger - {\dot{U}}_j {\dot{U}}_j^\dagger + {\dot{U}}_j {\dot{U}}_j^\dagger U_i U_j^\dagger \Big ] \\&\quad = -m\Vert {\dot{U}}_i\Vert _\text {F}^2 - m\Vert {\dot{U}}_j\Vert _\text {F}^2 + m\text {Re}\,\text {Tr}\,\Big [ 2{\dot{U}}_i {\dot{U}}_j^\dagger U_j U_i^\dagger - 2{\dot{U}}_i {\dot{U}}_j^\dagger \\&\qquad + {\dot{U}}_i {\dot{U}}_i^\dagger U_i U_j^\dagger + {\dot{U}}_j {\dot{U}}_j^\dagger U_i U_j^\dagger \Big ] \\&\quad \le -m\Vert {\dot{U}}_i\Vert _\text {F}^2 - m\Vert {\dot{U}}_j\Vert _\text {F}^2 + 2m \Vert {\dot{U}}_i\Vert _\text {F} \Vert {\dot{U}}_j\Vert _\text {F} + 2m \Vert {\dot{U}}_i\Vert _\text {F} \Vert {\dot{U}}_j\Vert _\text {F} + m \Vert {\dot{U}}_i\Vert _\text {F}^2 + m\Vert {\dot{U}}_j\Vert _\text {F}^2 \\&\quad \le 4m \Vert {\dot{U}}_i\Vert _\text {F} \Vert {\dot{U}}_j\Vert _\text {F} \le 4m D(\dot{{\mathcal {U}}})^2. \end{aligned}$$
  • (Estimate of \(\text {Re}\,\text {Tr}\,({\mathcal {I}}_{22}{\mathcal {D}}_{ij}^\dagger ))\): We use the fact that \(H_i\)’s are real-valued Hermitian matrices to see

    $$\begin{aligned} \text {Re}\,\text {Tr}\,({\mathcal {I}}_{22}{\mathcal {D}}_{ij}^\dagger )&= \text {Re}\,\text {Tr} \Big ( -{\mathrm {i}}H_i + {\mathrm {i}}H_j + {\mathrm {i}}H_i U_iU_j^\dagger - {\mathrm {i}}U_i U_j^\dagger H_j\Big ) \\&=\text {Re}\,\text {Tr}\, \Big ( {\mathrm {i}}U_iU_j^\dagger ( H_i-H_j) \Big ) \le \Vert H_i-H_j\Vert _\infty \le 2\Vert {\mathcal {H}}\Vert _\infty . \end{aligned}$$
  • (Estimate of \(\text {Re}\,\text {Tr}\,({\mathcal {I}}_{23}{\mathcal {D}}_{ij}^\dagger ))\): We observe

    $$\begin{aligned}&\text {Re}\,\text {Tr}\,({\mathcal {I}}_{23}{\mathcal {D}}_{ij}^\dagger ) \\&\quad = \text {Re}\,\text {Tr} \biggl [\frac{m{\mathrm {i}}}{\gamma }\Big (H_i {\dot{U}}_i U_j^\dagger - U_i {\dot{U}}_j^\dagger H_j + U_iU_j^\dagger H_j U_j{\dot{U}}_j^\dagger - {\dot{U}}_i U_i^\dagger H_i U_iU_j^\dagger \Big ) \\&\qquad \times (U_jU_i^\dagger -I_d)\biggr ] \\&\quad = \frac{m}{\gamma } \text {Re}\,\text {Tr}\, \Big [ {\mathrm {i}}\Big ( H_i {\dot{U}}_i U_i^\dagger - H_j U_j {\dot{U}}_j^\dagger + H_j U_j {\dot{U}}_j^\dagger - H_i {\dot{U}}_i U_i^\dagger \\&\qquad -H_i {\dot{U}}_i U_j^\dagger - U_i {\dot{U}}_j^\dagger H_j + U_iU_j^\dagger H_j U_j{\dot{U}}_j^\dagger - {\dot{U}}_i U_i^\dagger H_i U_iU_j^\dagger \Big )\Big ] \\&\quad = -\frac{m}{\gamma } \text {Re}\,\text {Tr}\, \Big [{\mathrm {i}}\Big ( H_i {\dot{U}}_i U_j^\dagger + U_i {\dot{U}}_j^\dagger H_j - U_iU_j^\dagger H_j U_j{\dot{U}}_j^\dagger - {\dot{U}}_i U_i^\dagger H_i U_iU_j^\dagger \Big )\Big ]\\&\quad \le \frac{m}{\gamma }\Big ( \Vert H_i\Vert _\infty \Vert {\dot{U}}_i\Vert _\text {F} + \Vert H_j\Vert _\infty \Vert {\dot{U}}_j\Vert _\text {F} + \Vert H_i\Vert _\infty \Vert {\dot{U}}_i\Vert _\text {F} + \Vert H_j\Vert _\infty \Vert {\dot{U}}_j\Vert _\text {F}\Big ) \\&\quad \le \frac{2m}{\gamma }\Vert {\mathcal {H}}\Vert _\infty \Big ( \Vert {\dot{U}}_i\Vert _\text {F} + \Vert {\dot{U}}_j\Vert _\text {F}\Big ) \le \frac{4m\Vert {\mathcal {H}}\Vert _\infty }{\gamma } D(\dot{{\mathcal {U}}}). \end{aligned}$$
  • (Estimate of \(\text {Re}\,\text {Tr}\,({\mathcal {I}}_{24}{\mathcal {D}}_{ij}^\dagger ))\): We directly observe

    $$\begin{aligned} \text {Re}\,\text {Tr}\,({\mathcal {I}}_{24}{\mathcal {D}}_{ij}^\dagger )= -\frac{\kappa }{2N}\sum _{k=1}^N \text {Re}\,\text {Tr}\Big ( {\mathcal {D}}_{ik} {\mathcal {D}}_{ij} {\mathcal {D}}_{ij}^\dagger + {\mathcal {D}}_{ij} {\mathcal {D}}_{kj} {\mathcal {D}}_{ij}^\dagger \Big ) \le \kappa D({\mathcal {U}})^\frac{3}{2}. \end{aligned}$$

    In (4), we combine the estimates of \({\mathcal {I}}_{2k},~k=1,\ldots ,4\) to obtain the desired estimate.

    $$\begin{aligned}&m\frac{d^2}{dt^2} \Vert {{\mathcal {D}}}_{ij}\Vert _\text {F}^2 + \gamma \frac{d}{dt} \Vert {{\mathcal {D}}}_{ij}\Vert _\text {F}^2 + 2\kappa \Vert {{\mathcal {D}}}_{ij}\Vert _\text {F}^2 \\&\quad \le 2m \Vert \dot{{\mathcal {D}}}_{ij}\Vert _\text {F}^2 + 8m D(\dot{{\mathcal {U}}})^2 + 4\Vert {\mathcal {H}}\Vert _\infty + \frac{8m\Vert {\mathcal {H}}\Vert _\infty }{\gamma }D(\dot{{\mathcal {U}}}) + 2\kappa D({\mathcal {U}})^\frac{3}{2} \\&\quad \le 12m D(\dot{{\mathcal {U}}})^2 + 4\Vert {\mathcal {H}}\Vert _\infty + \frac{8m\Vert {\mathcal {H}}\Vert _\infty }{\gamma }D(\dot{{\mathcal {U}}}) + 2\kappa D({\mathcal {U}})^\frac{3}{2}, \end{aligned}$$

    where we estimated the term \(\Vert \dot{{\mathcal {D}}}_{ij}\Vert _\text {F}^2\) as follows:

    $$\begin{aligned} \Vert \dot{{\mathcal {D}}}_{ij}\Vert _\text {F}^2= \Vert {\dot{U}}_i U_j^\dagger + U_i {\dot{U}}_j^\dagger \Vert _\text {F}^2 \le 2( \Vert {\dot{U}}_i \Vert _\text {F}^2 + \Vert {\dot{U}}_j\Vert _\text {F}^2) \le 4 D(\dot{{\mathcal {U}}})^2. \end{aligned}$$

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Ha, SY., Kim, D. Emergent Behavior of a Second-Order Lohe Matrix Model on the Unitary Group. J Stat Phys 175, 904–931 (2019). https://doi.org/10.1007/s10955-019-02270-y

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