A hybrid symbolic-numerical approach to the center-focus problem

Mahdi, Adam; Pessoa, Claudio; Hauenstein, Jonathan D.

doi:10.1016/j.jsc.2016.11.019

Cited by 16 publications

(10 citation statements)

References 59 publications

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“…It is well known since Poincaré [34] and Liapunov [24] that system (3) has a center at the origin if and only if there exists a local analytic first integral of the form H (x, ) = x 2 + 2 + F (x, ) defined in a neighborhood of the origin, where F starts with terms of order higher than 2. For the well known center problem, see [31,36].…”

Section: Basic Theory Of the Averaging Methodsmentioning

confidence: 99%

An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems

Huang

Yap

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows to transform the considered differential systems to the normal formal of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms that determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic manipulation; • Symbolic and algebraic algorithms → Symbolic calculus algorithms.

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Section: Basic Theory Of the Averaging Methodsmentioning

confidence: 99%

An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems

Huang

Yap

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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“…The Hilbert basis theorem implies that a finite subset of equations exists that gives necessary and sufficient conditions to determine the unique solution to the entire infinite set of equations (see e.g. [61]). The simplest, smallest candidate for this subset consists of (2.7), two equations from jump conditions for conservation of waves (3.4), and modified jump conditions for conservation of momentum in x (3.10 a ) and y (3.10 b ).…”

Section: Shock Solutions Of the Soliton Modulation Equationsmentioning

confidence: 99%

Modulation theory for soliton resonance and Mach reflection

2022

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Resonant Y-shaped soliton solutions to the Kadomtsev–Petviashvili II (KPII) equation are modelled as shock solutions to an infinite family of modulation conservation laws. The fully two-dimensional soliton modulation equations, valid in the zero dispersion limit of the KPII equation, are demonstrated to reduce to a one-dimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs limits to a travelling discontinuity. This discontinuity is a multivalued, weak solution satisfying modified Rankine–Hugoniot jump conditions for the one-dimensional modulation equations. These results are applied to analytically describe the dynamics of the Mach reflection problem, V-shaped initial conditions that correspond to a soliton incident upon an inward oblique corner. Modulation theory results show excellent agreement with direct KPII numerical simulation.

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“…The Hilbert basis theorem implies that a finite subset of equations exists that gives necessary and sufficient conditions to determine the unique solution to the entire infinite set of equations (see e.g. [64]). The simplest, smallest candidate for this subset consists of ( 9), two equations from jump conditions for conservation of waves (15), and modified jump conditions for conservation of momentum in x (21a) and y (21b).…”

Section: Properties Of the Y Soliton Shock Solution 451 Construction ...mentioning

confidence: 99%

Modulation theory for soliton resonance and Mach reflection

Ryskamp,

Hoefer,

Biondini

2021

Preprint

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Resonant Y-shaped soliton solutions to the Kadomtsev-Petviashvili II (KPII) equation are modelled as shock solutions to an infinite family of modulation conservation laws. The fully two-dimensional soliton modulation equations, valid in the zero dispersion limit of the KPII equation, are demonstrated to reduce to a one-dimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs limits to a travelling discontinuity. This discontinuity is a multivalued, weak solution satisfying modified Rankine-Hugoniot jump conditions for the onedimensional modulation equations. These results are applied to analytically describe the dynamics of the Mach reflection problem, V-shaped initial conditions that correspond to a soliton incident upon an inward oblique corner. Modulation theory results show excellent agreement with direct KPII numerical simulation.

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A hybrid symbolic-numerical approach to the center-focus problem

Cited by 16 publications

References 59 publications

An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems

An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems

Modulation theory for soliton resonance and Mach reflection

Modulation theory for soliton resonance and Mach reflection

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