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Bifurcations of limit cycles in a Z_6-equivariant planar vector field of degree 5

science China Mathematics 2002年第7期45卷 817-826页

作者：李继彬 H.s.Y.Chan K.W.Chung school of science Kunming University of Science and Technology Kunming 650093 China Department of Mathematics City University of Hong Kong Hong Kong Hong Kong

A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. there is reason to conjecture that the hilbert number H(2k + 1) ? (2k + I)2 - 1 for the perturbed Hamiltonian systems.

关键词： hilbert’s 16th problem limit cycle equivariant vector field method of detection function polynomial system

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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” ... 详细信息

Let us consider higher dimensional canards in a sow-fast system Rsup>2+2sup> 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

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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

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