Relaxation oscillations and canard explosion in a predator–prey system of Holling and Leslie types

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.     

Relaxation oscillations in a class of predator-prey systems

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.     

Hyperbolic-like solutions for singular Hamiltonian systems

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     

奇异二阶微分系统的正解

利用锥上的不动点定理,讨论奇异的二阶微分系统正解的存在性。     

A remark on periodic solutions of singular Hamiltonian systems

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.     

Families of regular solutions of singular systems

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     

Non-collision periodic solutions for singular Hamiltonian systems

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.     

Computing singular solutions to nonlinear analytic systems

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.     

Hardy space of solutions of elliptic systems on the plane

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.     

计算代数方程组孤立奇异解的符号数值方法

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.     

Singular points of solutions of elliptic systems in the plane

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.     

Conditions for existence of relaxation oscillations in singular systems of low dimension

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.     

Periodic solutions of second order singular coupled systems

We establish the existence of periodic solutions for the second order non-autonomous singular coupled systems

     

Singular points of solutions of some elliptic systems on the plane

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.     

Periodic solutions of singular systems without the strong force condition

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.

     

Relaxation oscillations and diffusion chaos in the Belousov reaction

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.