Optimization by Continuation

Kernevez, J. P.; Liu, Y.; Seoane, M. L.; Doedel, Eusebius J.

doi:10.1007/978-94-009-0659-4_23

Cited by 4 publications

(7 citation statements)

References 8 publications

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“…Recursivity-based optimization was addressed by Schütze et al [52] who proposed a recursive subdivision technique to perform multi-objective and multi-parametric optimization. Since then, multi-parametric optimization was coupled with continuation, Kernevez et al [53] used a descent optimization algorithm coupled with a continuation method to perform the optimization of nonlinear systems. Later, Balaram et al [54] combined the method of Kernevez et al with a genetic algorithm in order to provide a global algorithm of optimization by continuation.…”

Section: Introductionmentioning

confidence: 99%

“…However, it turned out that ISs can be generated when increasing the order of the nonlinear additional spring. Kernevez et al [53] proposed a continuationbased algorithm to control the position of ISs in a reaction-diffusion chemical system. Their strategy consisted in following curves of isola centers with respect to one system parameter.…”

Section: Introductionmentioning

confidence: 99%

“…The method presented here is intended as an extension of [41] to the multi-parametric case. The method of Kernevez et al [53] can be viewed as a particular case of this method and corresponds to the second level of recursivity of limit points. Unlike the method in [53], the proposed algorithm is not restricted to limit points and can deal with any of the bifurcations presented in [41].…”

Section: Introductionmentioning

confidence: 99%

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A multi-parametric recursive continuation method for nonlinear dynamical systems

Grenat

Baguet

Lamarque

et al. 2019

Mechanical Systems and Signal Processing

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The aim of this paper is to provide an efficient multi-parametric recursive continuation method of specific solution points of a nonlinear dynamical system such as bifurcation points. The proposed method explores the topology of specific points found on the frequency response curves by tracking extremum points in the successive codimensions of the problem with respect to multiple system parameters. To do so, the characterization of extremum points by a constraint equation and its associated extended system are presented. As a result, a recursive algorithm is generated by successively appending new constraint equations to the extended system at each new level of continuation. Then, the methodology is applied to a nonlinear tuned vibration absorber (NLTVA). The limit of existence of isolated solutions and extremum points optimizing the region without isolated solution are found and used to improve the NLTVA.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A multi-parametric recursive continuation method for nonlinear dynamical systems

Grenat

Baguet

Lamarque

et al. 2019

Mechanical Systems and Signal Processing

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“…Indeed, Dellwo et al [33] explore this idea and use a perturbation method to solve for isola centres, with an application to chemical reactors. Later on, Kernevez et al [34] applied continuation of isola formation points with a design perspective on a similar chemical system with the help of the software AUTO. Both of these papers formulate extended systems equivalent to (32), but differ in their approach to tackle the problem.…”

Section: Sub-harmonic Isolasmentioning

confidence: 99%

Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances

et al. 2019

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In this paper, a frequency-domain characterisation of the period-doubling bifurcation is proposed. This allows an efficient detection and localisation of such points along frequency-response curves computed through continuation and the harmonic balance method. A simple strategy for branch switching to sub-harmonic regimes is presented as well. Furthermore, these bifurcations are tracked in a twodimensional parameter space, and extremum points with respect to the tracking parameter are characterized and linked to sub-harmonic isola formation. As a test case for these methods, a forced Duffing oscillator with asymmetric clearances is studied numerically. The results, which include the prediction of period-doubling cascades and sub-harmonic isolas, are then compared to experimental results, yielding an excellent agreement.

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“…Even though an initial guess might not be available, one can always search for a formation center and find a nearby isola with a small perimeter. The detection of isola formation centers can be done via solving an appropriate extended system during a two-parameter continuation (see [20,27] and [21] for a detailed discussion on the implementation details, including continuation of isola centers in a third parameter).…”

Section: Computing Initial Guesses In the Proximity Of Isola Centersmentioning

confidence: 99%

On the Numerical Continuation of Isolas of Equilibria

Avitabile

Desroches

Rodrigues

2012

Int. J. Bifurcation Chaos

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Abstract. We present a numerical strategy to compute one-parameter families of isolas of equilibrium solutions in ODEs. Isolas are solution branches closed in parameter space. Numerical continuation is required to compute one single isola since it contains at least one unstable segment. We show how to use pseudo-arclength predictor-corrector schemes in order to follow an entire isola in parameter space, as an individual object, by posing a suitable algebraic problem. We continue isolas of equilibria in a two-dimensional dynamical system, the so-called continuous stirred tank reactor model, and also in a three-dimensional model related to plasma physics. We then construct a toy model and follow a family of isolas past a fold and illustrate how to initiate the computation close to a formation center, using approximate ellipses in a model inspired by the Van der Pol equation. We also show how to introduce node adaptivity in the discretization of the isola, so as to concentrate nodes in region with higher curvature. We conclude by commenting on the extension of the proposed numerical strategy to the case of isolas of periodic orbits.

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Optimization by Continuation

Cited by 4 publications

References 8 publications

A multi-parametric recursive continuation method for nonlinear dynamical systems

A multi-parametric recursive continuation method for nonlinear dynamical systems

Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances

On the Numerical Continuation of Isolas of Equilibria

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