Abstract
We study the emergent dynamics of a mixed Kuramoto ensemble in the presence of both attractive and repulsive coupling strengths. To be precise, we consider coupled Kuramoto-type systems consisting of two ensembles in which the oscillators in the same group interact attractively with a positive intra-group coupling strength, whereas the oscillators in the different group communicate repulsively with a negative inter-group coupling strength. For the modeling perspective of the Kuramoto model (KM for brevity), two types of systems are treated in this paper: an attractive–repulsive inertial KM–KM and an attractive–repulsive KM–KM. For these two models, we provide sufficient frameworks leading to complete bi-polar synchronization in which asymptotic configuration tends to a bi-polar state. Our estimates mainly rely on the gradient-like flow formulation and the energy estimate.
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Acknowledgements
The work of S.-Y. Ha was supported by the National Research Foundation of Korea (NRF-2017R1A2B2001864), and the work of S. E. Noh was supported by the National Research Foundation of Korea (NRF-2017R1C1B5018312).
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A Proof of Theorem 4
A Proof of Theorem 4
Recall that our goal is to establish
However, since we have already established the complete synchronization in Proposition 1, we have
Hence, it suffices to show that there exists \(\varXi ^\text {e} \in {\mathcal {S}}\) such that
The proof of (41) will split into two parts. Let \((\varXi _0,\varUpsilon _0) \in {\mathbb {R}}^{2N} \times {\mathbb {R}}^{2N}\) be the initial data such that the solution of (8) is global and bounded, and we define the \(\omega \)-limit set of \((\varXi _0,\varUpsilon _0)\) by
As a consequence of zero convergence for \(\Vert {\dot{\varXi }}\Vert \), we have
Let \(\varXi ^\text {e} \in \omega (\varXi _0,\varOmega _0)\), and we may assume that
Moreover, we set
\(\bullet \) (Step A: zero convergence of E): we define an energy-like function E(t) as follows:
where \(\varepsilon >0\) is a positive constant to be determined later. In what follows, we simply write \(\nabla V\) for \(\nabla _\varXi V\) and define the maximum and minimum for \(\{m_j\}\):
Then, we observe
We choose \(\varepsilon \) sufficiently small so that
Then, we have
It is easy to see that there exists \(\varepsilon _0\) such that for all \(\varepsilon \in [0,\varepsilon _0)\),
Thus, we see
and since \(0 \in \omega (\varXi _0,\varUpsilon _0)\), we can derive
\(\bullet \) (Step B: finite length of the phase vector \(\varXi \)): Let \(\rho \) be the Lojasiewicz exponent of V in Theorem 2. We use the Cauchy-Schwarz inequality to have
We now apply Young’s inequality to find
Thus, (44) becomes
Since \(0= \varXi ^\text {e} \in \omega (\varXi _0,\varUpsilon _0)\), we can choose a sequence \(\{t_n\}\) such that \(\varXi (t_n) \rightarrow 0\) as \(n \rightarrow \infty \). Then, zero convergence of \(\Vert {\dot{\varXi }}(t)\Vert \) and (43) yield that for any \(r>0\), there exists \(n_0 >0\) such that
where C is a positive constant:
Note that (46) follows from the zero convergences of \(\Vert \varXi (t_n)\Vert \) and energy-like function E, and (47) follows from the zero convergence of \(\Vert {\dot{\varXi }}\Vert \). We set
Our claim is to prove that \(T=\infty \). By using Lojasiewicz’s inequality in Theorem 2, we obtain from (45) that
where we used (47) for the last inequality. Together with (42) and (49), we find
We integrate (50) on the time-interval \((t_{n_0},T)\) to get
If \(T<\infty \), we have
Then, it follows from (46) and (51) that \(\Vert \varTheta (T)\Vert <r\). This contradicts (48). Hence, we finally have
The relation (51) with \(T=\infty \) implies that the trajectory \(\varTheta (t)\) has a finite length. Hence, we have that \(\varTheta (t)\) is convergent and indeed, \(\varTheta _e\) is the limit. On the other hand, (50) easily gives
Similarly, we integrate (52) over \((t_{n_0},T=\infty )\) as in (51) to find \({{\overline{\varPhi }}}(t)\) also have a finite length and it converges to \({{\overline{\varPhi }}}_e\). Hence, we conclude that \(\varXi ^\text {e} = (\varTheta _e, {{\overline{\varPhi }}}_e)\) is the limit point of the trajectory \(\varXi (t) = (\varTheta (t),{{\overline{\varPhi }}}(t))\).
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Ha, SY., Kim, D., Lee, J. et al. Synchronization Conditions of a Mixed Kuramoto Ensemble in Attractive and Repulsive Couplings. J Nonlinear Sci 31, 39 (2021). https://doi.org/10.1007/s00332-021-09699-0
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DOI: https://doi.org/10.1007/s00332-021-09699-0