Maoan Han

TWELVE LIMIT CYCLES IN A CUBIC CASE OF THE 16th HILBERT PROBLEM

In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This system was studied by the authors and shown to exhibit ten small limit cycles: five around each of the focus points. It will be proved in this paper that the system can have twelve small limit cycles. The major tasks involved in the proof are to compute the focus values and solve coupled enormous large polynomial equations. A computationally efficient perturbation technique based on multiple scales is employed to calculate the focus values. Moreover, the focus values are perturbed to show that the system can exactly have twelve small limit cycles.

Periodic boundary value problems for the first order impulsive functional differential equations

This paper is concerned with the existence of extreme solutions of the periodic boundary value problems for a class of first order functional differential equations. We introduce new concept of lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid. Meanwhile, we extend previous results.

HOPF BIFURCATIONS FOR NEAR-HAMILTONIAN SYSTEMS

In this paper, we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian function of the system depends on parameters, in addition to the coefficients involved in perturbations, which generates more limit cycles in the neighborhood of the origin. The results are applied to a quadratic system with cubic perturbations to show that the system can have five limit cycles in the vicinity of the origin.

Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles

Dynamical system theory has developed rapidly over the past fifty years. It is a subject upon which the theory of limit cycles has a significant impact for both theoretical advances and practical solutions to problems. Hopf bifurcation from a center or a focus is integral to the theory of bifurcation of limit cycles, for which normal form theory is a central tool. Although Hopf bifurcation has been studied for more than half a century, and normal form theory for over 100 years, efficient computation in this area is still a challenge with implications for Hilberts 16th problem. This book introduces the most recent developments in this field and provides major advances in fundamental theory of limit cycles. Split into two parts, the first focuses on the study of limit cycles bifurcating from Hopf singularity using normal form theory with later application to Hilberts 16th problem, while the second considers near Hamiltonian systems using Melnikov function as the main mathematical tool. Classic topics with new results are presented in a clear and concise manner and are accompanied by the liberal use of illustrations throughout. Containing a wealth of examples and structured algorithms that are treated in detail, a good balance between theoretical and applied topics is demonstrated. By including complete Maple programs within the text, this book also enables the reader to reconstruct the majority of formulas provided, facilitating the use of concrete models for study. Through the adoption of an elementary and practical approach, this book will be of use to graduate mathematics students wishing to study the theory of limit cycles as well as scientists, across a number of disciplines, with an interest in the applications of periodic behavior.

Dynamics of an SIS reaction-diffusion epidemic model for disease transmission.

Recently an SIS epidemic reaction-diffusion model with Neumann (or no-flux) boundary condition has been proposed and studied by several authors to understand the dynamics of disease transmission in a spatially heterogeneous environment in which the individuals are subject to a random movement. Many important and interesting properties have been obtained: such as the role of diffusion coefficients in defining the reproductive number; the global stability of disease-free equilibrium; the existence and uniqueness of a positive endemic steady; global stability of endemic steady for some particular cases; and the asymptotical profiles of the endemic steady states as the diffusion coefficient for susceptible individuals is sufficiently small. In this research we will study two modified SIS diffusion models with the Dirichlet boundary condition that reflects a hostile environment in the boundary. The reproductive number is defined which plays an essential role in determining whether the disease will extinct or persist. We have showed that the disease will die out when the reproductive number is less than one and that the endemic equilibrium occurs when the reproductive number is exceeds one. Partial result on the global stability of the endemic equilibrium is also obtained.

Bifurcation of limit cycles and separatrix loops in singular Lienard systems

Abstract We give sufficient conditions for the existence of one or two limit cycles of singular Lienard systems through the construction of a Poincare–Bendixson domain. With the help of the theory of rotated vector fields,we develop a method to compute bifurcation value at Saddle-node bifurcation of limit cycles and homoclinic or symmetric heteroclinic bifurcations. We also present application examples and prove the existence of duck cycles.

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

ON THE NUMBER AND DISTRIBUTION OF LIMIT CYCLES IN A CUBIC SYSTEM

This paper concerns the number of limit cycles in a cubic system. Eleven limit cycles are found and two different distributions are given by using the methods of bifurcation theory and qualitative analysis.

Travelling wavefronts in the diffusive single species model with Allee effect and distributed delay

In this paper, we consider the diffusive single species model with Allee effect and distributed delay time. Special attention is paid to the existence of travelling wavefront solutions. First, we shall show that such fronts exist when the convolution kernel assumes the strong generic delay kernel and the delay is sufficiently small. Then, in the non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times, we shall show that such fronts still exist for the weak generic delay kernel and small delay. The approach used in this paper is the geometric singular perturbation theory.

GLOBAL STABILITY OF A STAGE-STRUCTURED PREDATOR–PREY MODEL WITH MODIFIED LESLIE–GOWER AND HOLLING-TYPE II SCHEMES

In this paper, we consider a stage-structured predator–prey model with modified Leslie–Gower and Holling-type II schemes. Using an iterative technique, we investigate the global stability of the positive equilibrium of the system. Finally, some examples are presented to verify our main result.