scholarly journals Dynamics of a plant-herbivore model with a chemically-mediated numerical response

2021 ◽  
pp. 1-16
Author(s):  
Lin Wang ◽  
James Watmough ◽  
Fang Yu
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Homoclinic Orbits ◽  
Chemical Defenses ◽  
Heteroclinic Orbits ◽  
Numerical Response ◽  
Homoclinic Bifurcations ◽  
Dimension 2 ◽  
Plant Herbivore ◽  
Growth And Reproduction

A system of two ordinary differential equations is proposed to model chemically-mediated interactions between plants and herbivores by incorporating a toxin-modifiednumerical response. This numerical response accounts for the reduction in the her-bivore's growth and reproduction due to chemical defenses from plants. It is shownthat the system exhibits very rich dynamics including saddle-node bifurcations, Hopfbifurcations, homoclinic bifurcations and co-dimension 2 bifurcations. Numerical sim-ulations are presented to illustrate the occurrence of multitype bistability, limit cycles,homoclinic orbits and heteroclinic orbits. We also discuss the ecological implicationsof the resulting dynamics.

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

scholarly journals On acyclicity of quadratic system with two non-rough foci

2021 ◽  
pp. 41-46
Author(s):  
Адам Дамирович Ушхо
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Phase Plane ◽  
Second Order ◽  
Quadratic System ◽  
Equilibrium States ◽  
System Of Differential Equations ◽  
Right Hand ◽  
The Right

Доказывается, что система дифференциальных уравнений, правые части которой представляют собой полиномы второй степени, не имеет предельных циклов, если в ограниченной части фазовой плоскости она имеет только два состояния равновесия и при этом они являются состояниями равновесия второй группы. It is proved that a system of differential equations, the right-hand sides of which are second-order polynomials, has no limit cycles if it has only two equilibrium states in the bounded part of the phase plane, and they are the equilibrium states of the second group.

scholarly journals ON POLYNOMIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON A CIRCLE WITHOUT SINGULAR POINTS

2020 ◽  
Vol 12 (4) ◽  
pp. 33-40
Author(s):  
V.Sh. Roitenberg ◽  
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Vector Fields ◽  
Second Order ◽  
Singular Points ◽  
Free Term ◽  
Small Perturbations ◽  
Cylindrical Phase ◽  
Autonomous Differential Equations ◽  
The Right

In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for n > 4 an equation of degree n can have arbitrarily many limit cycles. For n = 4, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.

Center-Focus Problem for Discontinuous Planar Differential Equations

2003 ◽  
Vol 13 (07) ◽  
pp. 1755-1765 ◽  
Author(s):  
Armengol Gasull ◽  
Joan Torregrosa
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Critical Point ◽  
Limit Cycles ◽  
New Method ◽  
Polar Coordinates ◽  
Mechanical Problem ◽  
Weak Focus ◽  
Elastic Walls

We study the center-focus problem as well as the number of limit cycles which bifurcate from a weak focus for several families of planar discontinuous ordinary differential equations. Our computations of the return map near the critical point are performed with a new method based on a suitable decomposition of certain one-forms associated with the expression of the system in polar coordinates. This decomposition simplifies all the expressions involved in the procedure. Finally, we apply our results to study a mathematical model of a mechanical problem, the movement of a ball between two elastic walls.

The number of limit cycles of certain polynomial differential equations

1984 ◽  
Vol 98 (3-4) ◽  
pp. 215-239 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Equations ◽  
Image Position ◽  
Cubic Systems

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

Bifurcation Analysis in a Class of Piecewise Nonlinear Systems with a Nonsmooth Heteroclinic Loop

2018 ◽  
Vol 28 (02) ◽  
pp. 1850026
Author(s):  
Yuanyuan Liu ◽  
Feng Li ◽  
Pei Dang
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
Heteroclinic Loop ◽  
First Order ◽  
Nilpotent Critical Point

We consider the bifurcation in a class of piecewise polynomial systems with piecewise polynomial perturbations. The corresponding unperturbed system is supposed to possess an elementary or nilpotent critical point. First, we present 17 cases of possible phase portraits and conditions with at least one nonsmooth periodic orbit for the unperturbed system. Then we focus on the two specific cases with two heteroclinic orbits and investigate the number of limit cycles near the loop by means of the first-order Melnikov function, respectively. Finally, we take a quartic piecewise system with quintic piecewise polynomial perturbation as an example and obtain that there can exist ten limit cycles near the heteroclinic loop.

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