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Structural Stability in 4-Dimensional Canards

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Advances in Pure Mathematics 2022年第11期12卷 600-613页

作者： Shuya Kanagawa kiyoyuki tchizawa Deparatment of Mathematics Tokyo City University Tokyo Japan Deparatment of Mathematics Tokyo Gakugei University Tokyo Japan Institute of Administration Engineering Ltd. Sotokanda Tokyo Japan

Let us consider higher dimensional canards in a sow-fast system R2+2 with a bifurcation parameter. Then, the slow manifold sometimes shows various aspects due to the bifurcation. Introducing a key notion “symmetry” to the slow-fast system, it becomes clear when the pseudo singular point obtains the structural stability or not. It should be treated with a general case. Then, it will also be given about the sufficient conditions for the existence of the center manifold under being “symmetry”. The higher dimensional canards in the sow-fast system are deeply related to Hilbert’s 16th problem. Furthermore, computer simulations are done for the systems having Brownian motions. As a result, the rigidity for the system is confirmed.

关键词： Canard Solution Slow-Fast System Nonstandard Analysis Hilbert’s 16th Problem Brownian Motion Stochastic Differential Equation

Canards Flying on Bifurcation

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Advances in Pure Mathematics 2023年第6期13卷 412-424页

作者： Shuya Kanagawa kiyoyuki tchizawa Deparatment of Mathematics Tokyo City University Tokyo Japan Institute of Administration Engineering Ltd. Tokyo Japan

There exists a property “structural stability” for “4-dimensional canards” which is a singular-limit solution in a slow-fast system with a bifurcation parameter. It means that the system includes the possibility to have some critical values on the bifurcation parameter. Corresponding to these values, the pseudo-singular point, which is a singular point in the time-scaled-reduced system should be changed to another one. Then, the canards may fly to another pseudo-singular point, if possible. Can the canards fly? The structural stability gives the possibility for the canards flying. The precise reasons why happen are described in this paper.

关键词： Canard Solution Slow-Fast System Nonstandard Analysis Hilbert’s 16th Problem Brownian Motion Stochastic Differential Equation

On the Two Methods for Finding 4-Dimensional Duck Solutions

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Applied Mathematics 2014年第1期5卷 16-24页

作者： kiyoyuki tchizawa Institute of Administration Engineering Ltd. Tokyo Japan

This paper gives the existence of a duck solution in a slow-fast system in R2+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 R2+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.

关键词： Slow-Fast System Singular Perturbation Duck Solutions Blowing-Up Nonstandard Analysis

Extended Wiener Process in Nonstandard Analysis

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Applied Mathematics 2020年第3期11卷 247-254页

作者： Shuya Kanagawa kiyoyuki tchizawa Department of Mathematics Tokyo City University Setagaya Tokyo Japan Institute of Administration Engineering Ltd. Chiyoda-ku Tokyo Japan

Standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at time t obeys the normal distribution N(0,t) by showing the central limit theorem. The essential theory used in the proof is the extended convolution property in nonstandard analysis which is shown by Kanagawa, Nishiyama and tchizawa (2018). When processing the extension by non-standardization, we have already pointed out that it is needed to proceed the second extension for the convolution, not only to do the first extension for the delta function. In Section 2, we shall introduce again the extended convolution as preliminaries described in our previous paper. In Section 3, we shall provide the extended stochastic process using a hyper number N, and it satisfies the conditions being Wiener process. In Section 4, we shall give a new proof for the non-differentiability in the Wiener process.

关键词： Wiener Process Ito’s Process Stochastic Differential Equation S-Continuity Nonstandard Analysis

Proof of Ito’s Formula for Ito’s Process in Nonstandard Analysis

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Applied Mathematics 2019年第7期10卷 561-567页

作者： Shuya Kanagawa kiyoyuki tchizawa Department of Mathematics Tokyo City University Setagaya Tokyo Japan Institute of Administration Engineering Ltd. Tokyo Japan

In our previous paper [1], we proposed a non-standardization of the concept of convolution in order to construct an extended Wiener measure using nonstandard analysis by E. Nelson [2]. In this paper, we consider Ito’s integral with respect to the extended Wiener measure and extend Ito’s formula for Ito’s process. Because of doing the extension of Ito’s formula, we could treat stochastic differential equations in the sense of nonstandard analysis. In this framework, we need the nonstandardization of convolution again. It was not yet proved in the last paper, therefore we shall provide the proof.

关键词： Ito’s Process Stochastic Differential Equation S-Continuity Nonstandard Analysis

Extended Wiener Measure by Nonstandard Analysis for Financial Time Series

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Applied Mathematics 2018年第8期9卷 975-984页

作者： Shuya Kanagawa Ryoukichi Nishiyama kiyoyuki tchizawa Epartment of Mathematics Tokyo City University Tokyo Japan Department of literature Ryukoku University Kyoto Japan Institute of Administration Engineering Ltd. Tokyo Japan

We propose a new approach to construct an extended Wiener measure using nonstandard analysis by E. Nelson. For the new definition we construct non-standardized convolution of probability measure for independent random... 详细信息

关键词： Time Series Black-Sholes Model S-Continuity Nonstandard Analysis Delta-Function

On Relative Stability in 4-Dimensional Duck Solution

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Journal of Mathematics and System Science 2012年第9期2卷 558-563页

作者： kiyoyuki tchizawa Executive Dr. KanriKogaku Kenkyusho Ltd. (Institute of Administration Engineering Ltd.) Tokyo 101-0021 Japan

This paper gives the existence of a relatively stable duck solution in a slow-fast system in R2+2 with an invariant manifold. The slow-fast system in R2+2 has a 2-dimensional slow vector field and a 2-dimensional fast vector field. The fast vector field restricts a feasible region of the slow vector field strictly. In the case of the slow-fast system in R2+1, that is, the fast vector field is 1-dimension, it is classified according to its sign, because it gives only negative(-), positive(+) or zero sign. Then it is attractive, repulsive or stationary. On the other hand, in R2+2, the fast vector field has combinatorial cases. It causes a complex state to analyze the system. First, we introduce a local model near the pseudo singular point on which we classify the fast vector field attractive(-,-), attractive-repulsive(-,+) or repulsive(+,+), simply as possible. We prove the existence of a 4-dimensional duck solution in the local model. Secondarily, we assume that the slow-fast system has an invariant manifold near the pseudo singular point. When the invariant manifold has a homoclinic point near the pseudo singular point, we show that the slow-fast sytem has a 4-dimensional duck solution having a relatively stable region.

关键词：相对稳定性鸭解不变流形局部模型矢量场吸引力奇异点系统