High-Order Analysis of Canard Explosion in the Brusselator Equations

2020 ◽  
Vol 30 (05) ◽  
pp. 2050078 ◽  
Author(s):  
Bo-Wei Qin ◽  
Kwok-Wai Chung ◽  
Antonio Algaba ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Order Approximation ◽  
Transformation Method ◽  
High Order ◽  
Numerical Continuation ◽  
Chemical System ◽  
High Order Approximation ◽  
Order Analysis ◽  
First Order Approximation ◽  
Strongly Agree ◽  
Nonlinear Time Transformation

The aim of this paper is to obtain a high-order approximation of the canard explosion in the Brusselator equations. This classical chemical system has been extensively studied but, until now, only first-order approximation to the canard explosion has been provided. Here, with the help of the nonlinear time transformation method, we are able to obtain an approximation to any desired order. Our results strongly agree with those obtained by numerical continuation.

High-Order Analysis of Global Bifurcations in a Codimension-Three Takens–Bogdanov Singularity in Reversible Systems

2020 ◽  
Vol 30 (01) ◽  
pp. 2050017 ◽  
Author(s):  
Bo-Wei Qin ◽  
Kwok-Wai Chung ◽  
Antonio Algaba ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Transformation Method ◽  
High Order ◽  
Numerical Continuation ◽  
Reversible Systems ◽  
Efficient Manner ◽  
Codimension Two ◽  
Codimension One ◽  
Melnikov Functions ◽  
Global Connections ◽  
Nonlinear Time Transformation

A codimension-three Takens–Bogdanov bifurcation in reversible systems has been very recently analyzed in the literature. In this paper, we study with the help of the nonlinear time transformation method, the codimension-one and -two homoclinic and heteroclinic connections present in the corresponding unfolding. The algorithm developed allows to obtain high-order approximations for the global connections, in such a way that it supplies in a very efficient manner the coefficients that would be obtained with high-order Melnikov functions. As we show, all our analytical predictions have excellent agreement with the numerical results. In particular we remark that, for the two different codimension-two points, the theoretical approximation coincides in six decimal digits with the numerical continuation, even being quite far from the codimension-three point. The better approximations we provide in this work will help in the study of reversible systems that exhibit this codimension-three Takens–Bogdanov bifurcation.

The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Isotropically Scattering Plane-Parallel Medium: Part 1

2002 ◽  
Vol 124 (4) ◽  
pp. 674-684 ◽  
Author(s):  
Zekeriya Altac¸
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Differential Equations ◽  
Order Approximation ◽  
High Order ◽  
High Order Approximation ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Plane Parallel ◽  
Transfer Problems

A high order approximation, the SKN method—a mnemonic for synthetic kernel—is proposed for solving radiative transfer problems in participating medium. The method relies on approximating the integral transfer kernel by a sum of exponential kernels. The radiative integral equation is then reducible to a set of coupled second-order differential equations. The method is tested for one-dimensional plane-parallel participating medium. Three quadrature sets are proposed for the method, and the convergence of the method with the proposed sets is explored. The SKN solutions are compared with the exact, PN, and SN solutions. The SK1 and SK2 approximations using quadrature Set-2 possess the capability of solving radiative transfer problems in optically thin systems.

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