scholarly journals A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Jingjing Feng ◽  
Qichang Zhang ◽  
Wei Wang ◽  
Shuying Hao
Keyword(s):  
Dynamic Systems ◽  
Approximation Method ◽  
Padé Approximation ◽  
Global Bifurcation ◽  
Heteroclinic Orbits ◽  
Symmetric System ◽  
Pade Approximation ◽  
New Approach ◽  
Homoclinic And Heteroclinic Orbits ◽  
Single Orbit

In dynamic systems, some nonlinearities generate special connection problems of non-Z2symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2symmetric nonlinear quintic systems (orbit with one cusp); and Z2symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.

Improved bounds on the conductivity of composites by interpolation

1994 ◽  
Vol 444 (1921) ◽  
pp. 363-374 ◽  
Keyword(s):  
Composite Material ◽  
Lower Bounds ◽  
Approximation Method ◽  
Interpolation Method ◽  
Padé Approximation ◽  
Upper Bounds ◽  
Pade Approximation ◽  
Two Component ◽  
Null Lagrangian

Different bounds on the conductivity of a composite material may improve on each other in different conductivity régimes. If so, the question arises of how to efficiently interpolate between the bounds. In this paper I show how to do an interpolation with a two-point Padé approximation method. For bounds on two-component composites the interpolation method is shown to be, in a sense, the best possible. The method is discussed in the context of equiaxed polycrystals where the classic Hashin-Shtrikman bounds and the more recent null-lagrangian bounds, partly improve on each other. Denoting the principal conductivities of the crystallite σ 1 ≼ σ 2 ≼ σ 3 , the method gives improved lower bounds for equiaxed polycrystals which have σ 2 (0.77σ 1 + 0.23σ 3 ) ≽ σ 1 σ 3 . The method also gives improved upper bounds.

scholarly journals New Refinements for the Error Function with Applications in Diffusion Theory

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2017
Author(s):  
Gabriel Bercu
Keyword(s):  
Fourier Series ◽  
Approximation Method ◽  
Diffusion Theory ◽  
Error Function ◽  
Padé Approximation ◽  
Absolute Error ◽  
Series Method ◽  
Pade Approximation ◽  
Fourier Series Method ◽  
The Absolute

In this paper we provide approximations for the error function using the Padé approximation method and the Fourier series method. These approximations have simple forms and acceptable bounds for the absolute error. Then we use them in diffusion theory.

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