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On the Two Methods for Finding 4-Dimensional Duck Solutions
Institute of Administration Engineering, Ltd., Tokyo, Japan
- 1 Institute of Administration Engineering, Ltd., Tokyo, Japan
Applied Mathematics·Volume 05 (2013)·Pages 16–24·Published 25 December 2013·DOI10.4236/am.2014.51003
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Abstract
This paper gives the existence of a duck solution in a slow-fast system in R 2+2 using two ways. One is an indirect way and the other is a direct way. In the indirect way, the original system is once reduced to the slow-fast system in R 2+1 . In the direct one, it has a 4-dimensional duck solution when having an efficient local model. This is already published in [1,2]. Some sufficient conditions are given to get such a good model.
KeywordsSlow-Fast SystemSingular PerturbationDuck SolutionsBlowing-UpNonstandard Analysis
- K. Tchizawa, “On a Local Model for Finding 4-Dim Duck Solutions,” Selected Topics in Mathematical Methods and Computational Techniques in Electrical Engineering, WSEAS Press, 2010, pp. 177-183.
- K. Tchizawa, “On Relative Stability in 4-Dimensional Duck Solutions,” Journal of Mathematics and System Science, Vol. 2, No. 9, 2012, pp. 558-563.
- K. Tchizawa and S. A. Campbell, “On Winding Duck Solutions in R4,” Proceedings of Neural, Parallel, and Scientific Computations, Vol. 2, 2002, pp. 315-318.
- S. A. Campbell and M. Waite, “Multistability in Coupled Fitzhugh-Nagumo Oscillators,” Nonlinear Analysis, Vol. 47, No. 2, 2000, pp. 1093-1104. http://dx.doi.org/10.1016/S0362-546X(01)00249-8
- H. Miki, H. Nishino and K. Tchizawa, “On the Possible Occurrence of Duck Solutions in Domestic and Two-Region Business Cycle Models,” Nonlinear Studies, Vol. 19, No. 1, 2012, pp. 39-55.
- F. Diener and M. Diener, “Nonstandard Analysis in Practice,” Springer-Verlag, Berlin, 1995.
- A. K. Zvonkin and M. A. Shubin, “Non-Standard Analysis and Singular Perturbations of Ordinary Differential Equations,” Russian Mathematical Surveys, Vol. 39, No. 2, 1984, pp. 69-131. http://dx.doi.org/10.1070/RM1984v039n02ABEH003091
- E. Nelson, “Internal Set Theory,” Bulletin of the American Mathematical Society, Vol. 83, No. 6, 1977, pp. 1165-1198. http://dx.doi.org/10.1090/S0002-9904-1977-14398-X
- E. Benoit, “Canards et Enlacements,” Publications Mathématiques de l’IHéS, Vol. 72, No. 1, 1990, pp. 63-91. http://dx.doi.org/10.1007/BF02699131
- E. Benoit, “Canards en un Point Pseudo-Singulier Noeud,” Bulletin de la Société Mathématique de France, 1999, pp. 2-12.