Seminumerical Algorithms for Computing Invariant Manifolds of Vector Fields at Fixed Points

Haro, Àlex; Mondelo, Josep–Maria

doi:10.1007/978-3-319-29662-3_2

Cited by 2 publications

(3 citation statements)

References 32 publications

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“…Returning back to the problem of numerical approximation of the stable and unstable manifolds, as now there are no analytical expressions, due to the nonlinear dependence of g s and g u on z s and z u , the coefficients a of the polynomial approximation of the stable z u = h(z s ) (and correspondingly of the unstable) manifold can be determined numerically by solving in an iterative way the system of nonlinear homological equations given by (19).…”

Section: Theorem 3 Let Us Assume That the Functionsmentioning

confidence: 99%

“…This involves the expansion of the invariant manifold as series and the construction of a system of homological equations for the coefficients of the series. Based on this approach, Haro et al [19] addressed a numerical approach for the computation of the coefficients of high-order power series expansions of parametrizations of two-dimensional invariant manifolds. Breden et al [4] employed the parametrization method to compute stable and unstable manifolds of vectors fields.…”

Section: Introductionmentioning

confidence: 99%

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A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

Siettos

Russo

2021

Numer Algor

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We address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.

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Section: Theorem 3 Let Us Assume That the Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

Siettos

Russo

2021

Numer Algor

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show abstract

“…This involves the expansion of the invariant manifold as series and the construction of a system of homological equations for the coefficients of the series. Based on this approach, Haro et al (2016) [15] addressed a numerical approach for the computation of the coefficients of high order power series expansions of parametrizations of two-dimensional invariant manifolds. Breden et al (2016) [16] employed the parametrization method to compute stable and unstable manifolds of vectors fields.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Method for the Parametrization of Stable and Unstable Manifolds of Microscopic Simulators

Siettos,

Russo

2019

Preprint

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We address a numerical methodology for the computation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a "good" macroscopic description in the form of Ordinary (ODEs) and/or Partial differential equations (PDEs) does not explicitly/ analytically exists in a closed form. Thus, the assumption is that we have a detailed microscopic simulator of a complex system in the form of Monte-Carlo, Brownian dynamics, Agent-based models e.t.c. (or a black-box large-scale discrete time simulator) but due to the inherent complexity of the problem, we don't have explicitly an accurate model in the form of ODEs or PDEs. Our numerical scheme is a three-tier one including: (a) the "on demand" detection of the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the parametrization of the semi-local invariant stable and unstable manifolds by the numerical solution of the homological/functional equations for the coefficients of the truncated series approximation of the manifolds.

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Seminumerical Algorithms for Computing Invariant Manifolds of Vector Fields at Fixed Points

Cited by 2 publications

References 32 publications

A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

A Numerical Method for the Parametrization of Stable and Unstable Manifolds of Microscopic Simulators

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