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25 pages, 365 KiB

Open AccessArticle

Orbital Reversibility of Planar Vector Fields

by Antonio Algaba, Cristóbal García and Jaume Giné

Mathematics 2021, 9(1), 14; https://doi.org/10.3390/math9010014 - 23 Dec 2020

Cited by 6 | Viewed by 1486

In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in [...] Read more.

In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally reversible. Moreover, using this algorithm is possible to find degenerate centers which are orbitally reversible. Full article

(This article belongs to the Special Issue Qualitative Theory for Ordinary Differential Equations)

6 pages, 240 KiB

Open AccessArticle

Topologically Stable Chain Recurrence Classes for Diffeomorphisms

by Manseob Lee

Mathematics 2020, 8(11), 1912; https://doi.org/10.3390/math8111912 - 1 Nov 2020

Cited by 1 | Viewed by 1106

Abstract

Let

f : M \to M

be a diffeomorphism of a finite dimension, smooth compact Riemannian manifold M. In this paper, we demonstrate that if a diffeomorphism f lies within the

C^{1}

interior of the set of all chain recurrence class-topologically [...] Read more.

Let

f : M \to M

be a diffeomorphism of a finite dimension, smooth compact Riemannian manifold M. In this paper, we demonstrate that if a diffeomorphism f lies within the

C^{1}

interior of the set of all chain recurrence class-topologically stable diffeomorphisms, then the chain recurrence class is hyperbolic. Full article

(This article belongs to the Special Issue Qualitative Theory for Ordinary Differential Equations)

16 pages, 2126 KiB

Open AccessFeature PaperArticle

Two Nested Limit Cycles in Two-Species Reactions

by Ilona Nagy, Valery G. Romanovski and János Tóth

Mathematics 2020, 8(10), 1658; https://doi.org/10.3390/math8101658 - 25 Sep 2020

Cited by 3 | Viewed by 2931

Abstract

We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential [...] Read more.

We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential system. Symbolic calculations were carried out using the Mathematica computer algebra system, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each have two limit cycles surrounding the stationary point of focus type in the positive quadrant. In the case of Model 1, the outer limit cycle is stable and the inner one is unstable, which appears in a supercritical Hopf bifurcation. Moreover, the oscillations in a neighborhood of the outer limit cycle are slow-fast oscillations. In the case of Model 2, the outer limit cycle is unstable and the inner one is stable. With another set of parameters, the outer limit cycle can be made stable and the inner one unstable. Full article

(This article belongs to the Special Issue Qualitative Theory for Ordinary Differential Equations)

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19 pages, 326 KiB

Open AccessArticle

Zero-Hopf Bifurcations of 3D Quadratic Jerk System

by Bo Sang and Bo Huang

Mathematics 2020, 8(9), 1454; https://doi.org/10.3390/math8091454 - 30 Aug 2020

Cited by 10 | Viewed by 2115

Abstract

This paper is devoted to local bifurcations of three-dimensional (3D) quadratic jerk system. First, we start by analysing the saddle-node bifurcation. Then we introduce the concept of canonical system. Next, we study the transcritial bifurcation of canonical system. Finally we study the zero-Hopf [...] Read more.

This paper is devoted to local bifurcations of three-dimensional (3D) quadratic jerk system. First, we start by analysing the saddle-node bifurcation. Then we introduce the concept of canonical system. Next, we study the transcritial bifurcation of canonical system. Finally we study the zero-Hopf bifurcations of canonical system, which constitutes the core contributions of this paper. By averaging theory of first order, we prove that, at most, one limit cycle bifurcates from the zero-Hopf equilibrium. By averaging theory of second order, third order, and fourth order, we show that, at most, two limit cycles bifurcate from the equilibrium. Overall, this paper can help to increase our understanding of local behaviour in the jerk dynamical system with quadratic non-linearity. Full article

(This article belongs to the Special Issue Qualitative Theory for Ordinary Differential Equations)

10 pages, 513 KiB

Open AccessArticle

Qualitative Study of a Well-Stirred Isothermal Reaction Model

by Barbara Arcet, Maša Dukarić and Zhibek Kadyrsizova

Mathematics 2020, 8(6), 938; https://doi.org/10.3390/math8060938 - 8 Jun 2020

Viewed by 1542

Abstract

We consider a two-dimensional system which is a mathematical model for a temporal evolution of a well-stirred isothermal reaction system. We give sufficient conditions for the existence of purely imaginary eigenvalues of the Jacobian matrix of the system at its fixed points. Moreover, [...] Read more.

We consider a two-dimensional system which is a mathematical model for a temporal evolution of a well-stirred isothermal reaction system. We give sufficient conditions for the existence of purely imaginary eigenvalues of the Jacobian matrix of the system at its fixed points. Moreover, we show that the system admits a supercritical Hopf bifurcation. Full article

(This article belongs to the Special Issue Qualitative Theory for Ordinary Differential Equations)

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9 pages, 302 KiB

Open AccessArticle

Time-Varying Vector Norm and Lower and Upper Bounds on the Solutions of Uniformly Asymptotically Stable Linear Systems

by Robert Vrabel

Mathematics 2020, 8(6), 915; https://doi.org/10.3390/math8060915 - 4 Jun 2020

Cited by 3 | Viewed by 1985

Abstract

Based on the eigenvalue idea and the time-varying weighted vector norm in the state space

R^{n}

we construct here the lower and upper bounds of the solutions of uniformly asymptotically stable linear systems. We generalize the known results for the linear time-invariant [...] Read more.

Based on the eigenvalue idea and the time-varying weighted vector norm in the state space

R^{n}

we construct here the lower and upper bounds of the solutions of uniformly asymptotically stable linear systems. We generalize the known results for the linear time-invariant systems to the linear time-varying ones. Full article

(This article belongs to the Special Issue Qualitative Theory for Ordinary Differential Equations)

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15 pages, 270 KiB

Open AccessArticle

Rational Limit Cycles on Abel Polynomial Equations

by Claudia Valls

Mathematics 2020, 8(6), 885; https://doi.org/10.3390/math8060885 - 1 Jun 2020

Cited by 5 | Viewed by 1611

Abstract

In this paper we deal with Abel equations of the form

d y / d x = A_{1} (x) y + A_{2} (x) y^{2} + A_{3} (x) y^{3}

, where [...] Read more.

In this paper we deal with Abel equations of the form

d y / d x = A_{1} (x) y + A_{2} (x) y^{2} + A_{3} (x) y^{3}

, where

A_{1} (x), A_{2} (x)

and

A_{3} (x)

are real polynomials and

A_{3} ≢ 0

. We prove that these Abel equations can have at most two rational (non-polynomial) limit cycles when

A_{1} ≢ 0

and three rational (non-polynomial) limit cycles when

A_{1} \equiv 0

. Moreover, we show that these upper bounds are sharp. We show that the general Abel equations can always be reduced to this one. Full article

(This article belongs to the Special Issue Qualitative Theory for Ordinary Differential Equations)

14 pages, 326 KiB

Open AccessArticle

Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

by Rebiha Benterki and Jaume LLibre

Mathematics 2020, 8(5), 755; https://doi.org/10.3390/math8050755 - 10 May 2020

Cited by 9 | Viewed by 2087

Abstract

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they [...] Read more.

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines. Full article

(This article belongs to the Special Issue Qualitative Theory for Ordinary Differential Equations)

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