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1.
We give a geometric analysis of relaxation oscillations and canard cycles in a singularly perturbed predator–prey system of Holling and Leslie types. We discuss how the canard cycles are found near the Hopf bifurcation points. The transition from small Hopf-type cycles to large relaxation cycles is also discussed. Moreover, we outline one possibility for the global dynamics. Numerical simulations are also carried out to verify the theoretical results. 相似文献
2.
We consider a class of three-dimensional, singularly perturbed predator-prey systems having two predators competing exploitatively for the same prey in a constant environment. By using dynamical systems techniques and the geometric singular perturbation theory, we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class. Such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators. As an application, a well-known model will be discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameters values of the model. 相似文献
4.
Patricio Felmer Kazunaga Tanaka 《NoDEA : Nonlinear Differential Equations and Applications》2000,7(1):43-65
We study the existence of unbounded solutions of singular Hamiltonian systems: where is a potential with a singularity. For a class of singular potentials with a strong force , we show the existence of at least one hyperbolic-like solutions. More precisely, for given and , we find a solution q(t) of (*) satisfying
Received October 1998 相似文献
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Shinji Adachi Kazunaga Tanaka Masahito Terui 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(3):265-274
In this note we study the existence of non-collision periodic solutions for singular Hamiltonian systems with weak force.
In particular for potential
where D is a compact C3-surface in
we prove the existence of a non-collision periodic solution. 相似文献
8.
The seminal 1969 paper of W.A. Harris Jr., Y. Sibuya, and L. Weinberg provided new proofs for the Perron-Lettenmeyer theorem, as well as several other classical results, and has stimulated renewed consideration of families of regular solutions of certain singular problems. In this paper we give some further applications of the method developed there and, in addition, examine some connections between the Lettenmeyer theorem and an alternative theorem which addresses a problem posed by H.L. Turrittin that dates back to an 1845 example of Briot and Bouquet 相似文献
9.
We study the existence of classical (non-collision) T-periodic
solutions of the Hamiltonian system
where
and
is a T-periodic function in t which has a
singularity at
like
Under suitable conditions on H, we prove that if
then (HS) possesses at least one
non-collision solution and if
then the generalized solution of (HS) obtained in [5] has at most
one time of collision in its period. 相似文献
10.
Gurpreet Singh 《复变函数与椭圆型方程》2016,61(10):1389-1408
11.
Summary A method to generate an accurate approximation to a singular solution of a system of complex analytic equations is presented. Since manyreal systems extend naturally tocomplex analytic systems, this porvides a method for generating approximations to singular solutions to real systems. Examples include systems of polynomials and systems made up of trigonometric, exponential, and polynomial terms. The theorem on which the method is based is proven using results from several complex variables. No special conditions on the derivatives of the system, such as restrictions on the rank of the Jacobian matrix at the solution, are required. The numerical method itself is developed from techniques of homotopy continuation and 1-dimensional quadrature. A specific implementation is given, and the results of numerical experiments in solving five test problems are presented. 相似文献
12.
A. P. Soldatov 《Journal of Mathematical Sciences》2009,160(1):103-115
The space indicated in the title is introduced and studied.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal
Conference–2006, Part 3, 2008. 相似文献
13.
求解代数方程组是计算代数几何的最基本问题之一,孤立奇异解的计算则是其中最具挑战性的课题之一,在科学与工程计算中有着广泛的应用,如机器人、计算机视觉、机器学习、人工智能、运筹学、密码学和控制论等.本文结合作者的研究成果,综述了符号数值方法在计算代数系统孤立奇异解、特别是近似奇异解精化与验证方面的研究进展,并对未来的研究方向提出了展望. 相似文献
14.
We study the structure of solutions for some important classes of singular elliptic systems in the plane. In particular, it is proved that the solutions of such systems have principally nonanalytic behavior in neighborhoods of fixed singular points. These results enable one to correctly state certain boundary-value problems and make their complete analysis. 相似文献
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L. I. Kononenko 《Journal of Applied and Industrial Mathematics》2007,1(1):45-49
Relaxation oscillations are studied in a singularly perturbed system of ordinary differential equations with m slow and n fast variables for the case of m = 2 and n = 1. Necessary conditions and sufficient conditions for existence of relaxation oscillations are given. 相似文献
17.
We establish the existence of periodic solutions for the second order non-autonomous singular coupled systems
18.
G. Makatsaria 《Journal of Mathematical Sciences》2009,160(6):737-744
In the present paper, the structure of solutions of some important classes of singular elliptic systems on the plane are investigated.
In particular, it is proved that the solutions of such systems have in principle nonanalytic behavior in the neighborhood
of fixed singular points. These results make it possible to state correctly the boundary value problems and give their complete
analysis.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra
and Geometry, 2008. 相似文献
19.
Daniel Franco Pedro J. Torres 《Proceedings of the American Mathematical Society》2008,136(4):1229-1236
We present sufficient conditions for the existence of at least a non-collision periodic solution for singular systems under weak force conditions. We deal with two different types of systems. First, we assume that the system is generated by a potential, and then we consider systems without such hypothesis. In both cases we use the same technique based on Schauder fixed point theorem. Recent results in the literature are significantly improved.
20.
S. D. Glyzin A. Yu. Kolesov N. Kh. Rozov 《Computational Mathematics and Mathematical Physics》2011,51(8):1307-1324
Asymptotic and numerical analysis of relaxation self-oscillations in a three-dimensional system of Volterra ordinary differential
equations that models the well-known Belousov reaction is carried out. A numerical study of the corresponding distributed
model-the parabolic system obtained from the original system of ordinary differential equations with the diffusive terms taken
into account subject to the zero Neumann boundary conditions at the endpoints of a finite interval is attempted. It is shown
that, when the diffusion coefficients are proportionally decreased while the other parameters remain intact, the distributed
model exhibits the diffusion chaos phenomenon; that is, chaotic attractors of arbitrarily high dimension emerge. 相似文献