Browsing by Author "Shieh, Shih-Feng"

Now showing 1 - 7 of 7

A hyperplane-constrained continuation method for near singularity in coupled nonlinear Schrodinger equations
(2010-05-01)
Kuo, Yueh-Cheng
;
Lin, Wen-Wei
;
Shieh, Shih-Feng
;
Wang, Weichung
;
應用數學系
The continuation method is a useful numerical tool for solving differential equations to obtain multiform solutions and allow bifurcation analysis. However, when a standard continuation method is used to solve a type of time-independent m-coupled nonlinear Schrodinger (NLS) equations that can be used to model nonlinear optics, nearly singular systems arise in the computations of prediction and correction search directions and detections of bifurcations. To overcome the stability and efficiency problems that exist in standard continuation methods, we propose a new hyperplane-constrained continuation method by adding additional constraints to prevent the singularities while tracking the solution curves. Aimed at the 3-coupled DNLS equations, we conduct theoretical analysis to the solutions and bifurcations on the primal stalk solution curve. The proposed algorithms have been implemented successfully to demonstrate numerical solution profiles, energies, and bifurcation diagrams in various settings. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.
A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrodinger equations
(2009-11-01)
Kuo, Yueh-Cheng
;
Lin, Wen-Wei
;
Shieh, Shih-Feng
;
Wang, Weichung
;
應用數學系
We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrodinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly. (C) 2009 Elsevier Inc. All rights reserved.
A structure preserving flow for computing Hamiltonian matrix exponential
(2019-11-01)
Kuo, Yueh-Cheng
;
Lin, Wen-Wei
;
Shieh, Shih-Feng
;
應用數學系
;
Department of Applied Mathematics
This article focuses on computing Hamiltonian matrix exponential. Given a Hamiltonian matrix H, it is well-known that the matrix exponential e(H) is a symplectic matrix and its eigenvalues form reciprocal (lambda, 1/(lambda) over bar). It is important to take care of the symplectic structure for computing e(H). Based on the structure-preserving flow proposed by Kuo et al. (SIAM J Matrix Anal Appl 37:976-1001, 2016), we develop a numerical method for computing the symplectic matrix pair (M, L) which represents e(H).
On the total variation, topological entropy and sensitivity for interval maps
(2008-05-15)
Juang, Jonq
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Shieh, Shih-Feng
;
應用數學系
;
Department of Applied Mathematics
A concept related to total variation termed H-1 condition was recently proposed to characterize the chaotic behavior of an interval map f by Chen, Huang and Huang [G. Chen, T. Huang, Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos 14 (2004) 2161-2186]. In this paper, we establish connections between H-1 condition, sensitivity and topological entropy for interval maps. First, we introduce a notion of restrictiveness of a piecewise-monotone continuous interval map. We then prove that H-1 condition of a piecewise-monotone continuous map implies the non-restrictiveness of the map. In addition, we also show that either H-1 condition or sensitivity then gives the positivity of the topological entropy of f. (C) 2007 Elsevier Inc. All rights reserved.
Piecewise linear maps, Liapunov exponents and entropy
(2008-02-01)
Juang, Jonq
;
Shieh, Shih-Feng
;
應用數學系
;
Department of Applied Mathematics
Let L-A = {f(A,x):x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if f(A,x) is an element of L-A, then the Liapunov exponent lambda(x) of f(A,x) is equal to a measure theoretic entropy h(mA,x) of f(A,x), where m(A,x) is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that max(x) lambda(x) = max(x)h(mA,x) = log(lambda(1)), where lambda(1)is the maximal eigenvalue of A. (c) 2007 Elsevier Inc. All rights reserved.
STRUCTURE-PRESERVING FLOWS OF SYMPLECTIC MATRIX PAIRS
(2016-01-01)
Kuo, Yueh-Cheng
;
Lin, Wen-Wei
;
Shieh, Shih-Feng
;
應用數學系
;
Department of Applied Mathematics
We construct a nonlinear differential equation of matrix pairs (M(t); L(t)) that are invariant (structure-preserving property) in the class of symplectic matrix pairs {(M, L) = ((sic)S-2, (sic)S-1)| X = [X-i j](1
The asymptotic analysis of the structure-preserving doubling algorithms
(2017-10-15)
Kuo, Yueh-Cheng
;
Lin, Wen-Wei
;
Shieh, Shih-Feng
;
應用數學系
;
Department of Applied Mathematics
This paper is the second part of [15]. Taking advantage of the special structure and properties of the Hamiltonian matrix, we apply a symplectically similar transformation introduced by [18] to reduce H to a Hamiltonian Jordan canonical form J. The asymptotic analysis of the structure-preserving flows and RDEs is studied by using e(Jt). The convergence of the SDA as well as its rate can thus result from the study of the structure preserving flows. A complete asymptotic dynamics of the SDA is investigated, including the linear and quadratic convergence studied in the literature [3,12,13]. (C) 2017 Elsevier Inc. All rights reserved.