The Łojasiewicz inequality and Łojasiewicz exponent reveal a fundamental relation between a potential function and its gradient. In this paper, we explore the Łojasiewicz exponent of Kuramoto model and prove that the exponent is exactly for equilibria located inside a quarter of circle. This implies that the convergence towards such a phase-locked state must be exponentially fast. In contrast, we give an example to see the exponent can be less than for other equilibriums. More precisely, we prove that the exponent for the bi-cluster equilibrium, which is located on the boundary of a quarter of circle, is . This gives an insight for the occurrence of exponential and algebraic convergence of Kuramoto model. We also present a general theorem for exponential convergence of second-order gradient-like system, by which a criterion for the Kuramoto model with inertia is established.
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February 2015
Research Article|
February 18 2015
On the Łojasiewicz exponent of Kuramoto model
Zhuchun Li;
Zhuchun Li
a)
1Department of Mathematics,
Harbin Institute of Technology
, Harbin 150001, China
Search for other works by this author on:
Xiaoping Xue;
Xiaoping Xue
b)
1Department of Mathematics,
Harbin Institute of Technology
, Harbin 150001, China
Search for other works by this author on:
b)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Math. Phys. 56, 022704 (2015)
Article history
Received:
March 05 2014
Accepted:
February 03 2015
Citation
Zhuchun Li, Xiaoping Xue, Daren Yu; On the Łojasiewicz exponent of Kuramoto model. J. Math. Phys. 1 February 2015; 56 (2): 022704. https://doi.org/10.1063/1.4908104
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