A fast sampling method for estimating the domain of attraction

In dynamical systems, the full stability of fixed point solutions is determined by their basins of attraction. Characterizing the structure of these basins is, in general, a complicated task, especially in high dimensionality. Recent works have advocated to quantify the non-linear stability of fixed points of dynamical systems through the relative volumes of the associated basins of attraction [Wiley et al., Chaos 16, 015103 (2006) and Menck et al. Nat. Phys. 9, 89 (2013)]. Here, we revisit this issue and propose an efficient numerical method to estimate these volumes. The algorithm first identifies stable fixed points. Second, a set of initial conditions is considered that are randomly distributed at the surface of hypercubes centered on each fixed point. These initial conditions are dynamically evolved. The linear size of each basin of attraction is finally determined by the proportion of initial conditions which converge back to the fixed point. Armed with this algorithm, we revisit the problem considered by Wiley et al. in a seminal paper [Chaos 16, 015103 (2006)] that inspired the title of the present manuscript and consider the equal-frequency Kuramoto model on a cycle. Fixed points of this model are characterized by an integer winding number q and the number n of oscillators. We find that the basin volumes scale as (1-4q/n), contrasting with the Gaussian behavior postulated in the study by Wiley et al.. Finally, we show the applicability of our method to complex models of coupled oscillators with different natural frequencies and on meshed networks.

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“…• It does not require a Lyapunov function of the dynamical system considered, 37,38 which is complicated to find in general;…”

Section: Discussionmentioning

confidence: 99%

The size of the sync basin revisited

Delabays

Tyloo

Jacquod

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…This section is dedicated to an evaluative comparative analysis between the synthesised CPSO estimation strategy in this paper and another peer reviewed technique [26]. Table 1 provides the estimated DA features for three dynamical nonlinear systems with quadratic LF(s) taken from the literature [39,40].…”

Section: Results Analysis and Discussionmentioning

confidence: 99%

“…Theorem 1 [26]. A closed set ϑ ∈ R n , which includes the original as equilibrium, may estimate the DA for the system (2) origin provided:…”

Section: Leda and Basic Notationsmentioning

confidence: 99%

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Chaotic Particle Swarm Optimisation for Enlarging the Domain of Attraction of Polynomial Nonlinear Systems

et al. 2020

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A novel technique for estimating the asymptotic stability region of nonlinear autonomous polynomial systems is established. The key idea consists of examining the optimal Lyapunov function (LF) level set that is fully included in a region satisfying the negative definiteness of its time derivative. The minor bound of the biggest achievable region, denoted as Largest Estimation Domain of Attraction (LEDA), can be calculated through a Generalised Eigenvalue Problem (GEVP) as a quasi-convex Linear Inequality Matrix (LMI) optimising approach. An iterative procedure is developed to attain the optimal volume or attraction region. Furthermore, a Chaotic Particular Swarm Optimisation (CPSO) efficient technique is suggested to compute the LF coefficients. The implementation of the established scheme was performed using the Matlab software environment. The synthesised methodology is evaluated throughout several benchmark examples and assessed with other results of peer technique in the literature.

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“…There are two useful methods for studying the qualitative behavior of solutions of the nonlinear systems: (i) the Lyapunov's second method and its various extensions (such as the La Salle's invariance principle), see, e. g. [1], [3], [4], [5], [9], [12], [15], [16], [17] and (ii) the use of variation of constants formula where the behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system ( [7]). This idea is essential in the proof of Lyapunov's indirect method in Corollary 2.43 in [6, p. 160].…”

Section: Introductionmentioning

confidence: 99%

On local asymptotic stabilization of the nonlinear systems with time-varying perturbations by state-feedback control

Vrabel

2018

International Journal of General Systems

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In this paper, we are interested in the relation between the solutions of the control systemẋ = f (x, u) and the solutions of its (potentially unknown) perturbationẋ = f (x, u) + w(x, t). Under the assumption that the linear part of the unperturbed system at the point (0, 0) is controllable and that disturbance w(x, t) is sufficiently small, there exists a state-feedback controller of the form u = −Kx such that the perturbed system preserves the local asymptotic stability of the zero solution of unperturbed system. The main result of this paper gives the sufficient conditions, more specifically, the relations between the important parameters of the system, to ensure this property and at the same time provides the method for calculating the lower bound of region of attraction. Moreover, we obtain a nontrivial extension of the classical result of H. K. Khalil regarding asymptotic behavior of the (uncontrolled) perturbed systems whose nominal part is exponentially asymptotically stable at the origin x = 0.

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A fast sampling method for estimating the domain of attraction

Cited by 48 publications

References 34 publications

The size of the sync basin revisited

The size of the sync basin revisited

Chaotic Particle Swarm Optimisation for Enlarging the Domain of Attraction of Polynomial Nonlinear Systems

On local asymptotic stabilization of the nonlinear systems with time-varying perturbations by state-feedback control

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