Dynamics and bifurcations of a ratio-dependent predator-prey model

In this paper, we study a ratio-dependent predator-prey model with modied Holling-Tanner formalism, by using dynamical techniques and numerical continuation algorithms implemented in Matcont. We determine codim-1 and 2 bifurcation points and their corresponding normal form coecients. We also compute a curve of limit cycles of the system emanating from a Hopf point.

Analisis Dinamik Model Predator-Prey dengan Faktor Kanibalisme Pada Predator

Kajian dinamika populasi predator-prey di suatu ekosistem dengan adanya kanibalisme pada predator dilakukan pada penelitian ini. Ketika ada kanibalisme di tingkat predator dikhawatirkan populasi predator itu akan menurun atau terjadi kepunahan, sehingga populasi prey menjadi tidak terkontrol dan akan terjadi ketidakseimbangan ekosistem. Oleh karena itu, pada penelitian ini dibangunlah model matematika predator-prey dengan faktor kanibalisme pada predator berbentuk sistem persamaan diferensial biasa non linier dengan tiga persamaan. Pada model predator-prey tersebut ditemukan dua titik kesetimbangan yang memiliki kemungkinan stabil yaitu titik kesetimbangan ketika tidak ada prey dan titik kesetimbangan ketika kedua spesies eksis di ekosistem tersebut . Hasil sensitivitas analisis menunjukkan bahwa sifat kestabilan lokal dari titik maupun bergantung pada parameter kanibalisme yakni dan . Lebih lanjut, untuk titik telah dibuktikan sifat kestabilan global menggunakan fungsi lyapunov. Hasil simulasi numerik mengilustrasikan hasil analisa yang sudah diperoleh, sehingga ditemukan kemungkinan terjadinya limit cycles yang menandakan adanya bifurkasi hopf.

Limit Cycles Appearing From Piecewise Smooth Perturbations to a Reversible Nonlinear Center

This paper deals with the limit cycle bifurcation from a reversible differential center of degree [Formula: see text] due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

Limit Cycles from Hopf Bifurcation in Nongeneric Quadratic Reversible Systems with Piecewise Perturbations

In this paper, we consider a nongeneric quadratic reversible system with piecewise polynomial perturbations. We use the expansion of the first order Melnikov function to obtain the maximal number of small-amplitude limit cycles produced by Hopf bifurcation in the perturbed systems.

Simultaneous Bifurcation of Limit Cycles and Critical Periods

The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

The simplest amplitude-period formula for non-conservative oscillators

The simplest frequency formulation for conservative oscillators was proposed in 2019 (Results Phys 2019;15:102546). However, it becomes invalid for non-conservative oscillators. This work suggests the simplest amplitude-period formulation for non-conservative oscillators. The existence of a periodic solution of a second-order ordinary differential equation is given, and the periodic orbits are easily obtained. To the best of the authors’ knowledge, such a powerful result is not available in the literature, providing a tool to determining periodic orbits/limit cycles in the most general scenario.

The acyclicity of a quadratic differential system

Дан краткий обзор некоторых основных публикаций, посвященных исследованию вопроса о предельных циклах и сепаратрисах квадратичных дифференциальных систем. Рассмотрено наличие замкнутых траекторий для определенного класса автономных квадратичных систем на плоскости. Доказательство основано на применении теории прямых изоклин, признаков Дюлака и Бендиксона качественной теории дифференциальных уравнений. Предложенное доказательство покрывает результаты известной работы Л.А. Черкаса и Л.С. Жилевич. We now give a brief overview of some of the main publications devoted to the study of the question of limit cycles and separatrices of quadratic differential systems. In this paper, we consider the existence of closed trajectories for a certain class of autonomous quadratic systems on the plane. The proof is based on the application of the theory of straight line isoclines, Dulac and Bendixon criteria of the qualitative theory of differential equations. The proposed proof covers the results of the well-known work of L.A. Cherkas and L.S. Zhilevich.

On the Limit Cycles for a Class of Perturbed Fifth-Order Autonomous Differential Equations

We study the limit cycles of the fifth-order differential equation x ⋅ ⋅ ⋅ ⋅ ⋅ − e x ⃜ − d x ⃛ − c x ¨ − b x ˙ − a x = ε F x , x ˙ , x ¨ , x ⋯ , x ⃜ with a = λ μ δ , b = − λ μ + λ δ + μ δ , c = λ + μ + δ + λ μ δ , d = − 1 + λ μ + λ δ + μ δ , e = λ + μ + δ , where ε is a small enough real parameter, λ , μ , and δ are real parameters, and F ∈ C 2 is a nonlinear function. Using the averaging theory of first order, we provide sufficient conditions for the existence of limit cycles of this equation.

Limit Cycles on Piecewise Smooth Vector Fields with Coupled Rigid Centers

We provide an upper bound for the maximum number of limit cycles of the class of discontinuous piecewise differential systems formed by two differential systems separated by a straight line presenting rigid centers. These two rigid centers are polynomial differential systems with a linear part and a nonlinear homogeneous part. We study the maximum number of limit cycles that such a class of piecewise differential systems can exhibit.