Validated continuation over large parameter ranges for equilibria of PDEs

Gameiro, Marcio; Lessard, Jean‐Philippe; Mischaikow, Konstantin

doi:10.1016/j.matcom.2008.03.014

Cited by 34 publications

(70 citation statements)

References 3 publications

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“…This method has been successfully used in [14] and [17], but here we need to extend it considerably in three crucial aspects. First, the requirement E = 0 means that, besides satisfying the differential equation, the solution must obey an additional requirement.…”

Section: +1 Block 1 Block 2 Blockmentioning

confidence: 99%

“…Second, rigorous continuation is required in order to not just obtain result for isolated values of ν (cf. [14,17,35]), but for the entire parameter interval ν ∈ [0, 2]. Note that in [12], a result about an entire parameter interval was also obtained, but at a much more computationally expensive price.…”

Section: +1 Block 1 Block 2 Blockmentioning

confidence: 99%

“…This will be done via validated continuation (cf. [14,17]). One important advantage of this validated continuation method is that it becomes natural to prove the existence of chaos for a continuous range of parameter values.…”

Section: Introductionmentioning

confidence: 99%

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Chaotic Braided Solutions via Rigorous Numerics: Chaos in the Swift–Hohenberg Equation

Berg

Lessard

2008

SIAM J. Appl. Dyn. Syst.

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We prove that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semi-conjugacy to a subshift of finite type shows that the dynamics is chaotic.

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Section: +1 Block 1 Block 2 Blockmentioning

confidence: 99%

Section: +1 Block 1 Block 2 Blockmentioning

confidence: 99%

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Chaotic Braided Solutions via Rigorous Numerics: Chaos in the Swift–Hohenberg Equation

Berg

Lessard

2008

SIAM J. Appl. Dyn. Syst.

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“…In this case, all terms in (20) and (21) will be finite sums that can be evaluated using interval arithmetic.…”

Section: Construction Of the Bounds Required For The Radii Polynomialsmentioning

confidence: 99%

“…Note that all the discrete convolutions in Table 3 are finite sums that can be rigorously estimated combining interval arithmetic with the fast Fourier transform (FFT) algorithm (e.g. see [20]). We then obtain rigorous upper bounds Y (i,j) (21).…”

Section: Two-dimensional Manifold Of Equilibria Of Cahn-hilliardmentioning

confidence: 99%

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

2015

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In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a twodimensional manifold of equilibria of the Cahn-Hilliard equation.

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