On the stable equilibrium points of gradient systems

Absil, Pierre-Antoine; Kurdyka, Krzysztof

doi:10.1016/j.sysconle.2006.01.002

Cited by 123 publications

(121 citation statements)

References 11 publications

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“…We rst establish an abstract convergence result for arbitrary functions which are not necessary solutions of the ordinary dierential equation (1). Then, we prove the convergence result under a L 1 -condition of the forcing term.…”

Section: Convergence Resultsmentioning

confidence: 97%

“…Let u be a bounded solution of (1). Assume that ∇F is bounded from below and G, ∇F satises the angle and comparability condition, this means there exists a constant ν > 0 such that…”

Section: Proofmentioning

confidence: 99%

“…This system has been studied by many authors such as Absil & Kurdyka [1], Chill [5], Haraux & Jendoubi [11], [12] or Simon [17]. They have proved that if F satises a Lojasiewicz inequality then the bounded solution converges to an equilibrium as t goes to innity.…”

Section: Introductionmentioning

confidence: 99%

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On the Convergence of Bounded Solutions of Non Homogeneous Gradient-like Systems

Tran¹,

Nguyen²

2017

J. ADV. ENG. COMPUT.

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Abstract. We study the long time behavior of the bounded solutions of non homogeneous gradient-like system which admits a strict Lyapunov function. More precisely, we show that any bounded solution of the gradient-like system converges to an accumulation point as time goes to innity under some mild hypotheses. As in homogeneous case, the key assumptions for this system are also the angle condition and the Kurdyka-Lojasiewicz inequality. The convergence result will be proved under a L 1 -condition of the perturbation term. Moreover, if the Lyapunov function satises a Lojasiewicz inequality then the rate of convergence will be even obtained.

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Section: Convergence Resultsmentioning

confidence: 97%

“…Let u be a bounded solution of (1). Assume that ∇F is bounded from below and G, ∇F satises the angle and comparability condition, this means there exists a constant ν > 0 such that…”

Section: Proofmentioning

confidence: 99%

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On the Convergence of Bounded Solutions of Non Homogeneous Gradient-like Systems

Tran¹,

Nguyen²

2017

J. ADV. ENG. COMPUT.

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“…Below, we show that the invariant subspace formed by such solutions is, in fact, asymptotically stable. To this end, we use the properties of the system at hand rather than relying on the general stability result for gradient systems in [2]. Specifically, the analysis below uses on the Lyapunov function U α .…”

Section: Synchronizationmentioning

confidence: 99%

Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

Medvedev

Tang

2015

J Nonlinear Sci

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The Kuramoto model of coupled phase oscillators on complete, Paley, and Erdős-Rényi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the Kuramoto model on these graphs can be qualitatively different. Specifically, we identify twisted states, steady state solutions of the Kuramoto model on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the initial value problems for the Kuramoto model on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the Kuramoto model on Cayley and random graphs.

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“…L'inéga-lité de Łojasiewicz a aussi des applications aux équations différentielles et aux équations aux dérivées partielles [7], [17], [31]. Elle est un outil efficace pour étudier la stabilité des champs de gradient [1].…”

Section: Introductionunclassified