Nonchaotic Behavior in Quadratic Three-Dimensional Differential Systems with a Symmetric Jacobian Matrix

2018 ◽  
Vol 28 (03) ◽  
pp. 1830006 ◽  
Author(s):  
Marcelo Messias ◽  
Rafael Paulino Silva
Keyword(s):  
Jacobian Matrix ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Positive Answer ◽  
Algebraic Surfaces ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Algebraic Criterion ◽  
Invariant Algebraic Surfaces

In this paper, we give an algebraic criterion to determine the nonchaotic behavior for polynomial differential systems defined in [Formula: see text] and, using this result, we give a partial positive answer for the conjecture about the nonchaotic dynamical behavior of quadratic three-dimensional differential systems having a symmetric Jacobian matrix. The algebraic criterion presented here is proved using some ideas from the Darboux theory of integrability, such as the existence of invariant algebraic surfaces and Darboux invariants, and is quite general, hence it can be used to study the nonchaotic behavior of other types of differential systems defined in [Formula: see text], including polynomial differential systems of any degree having (or not having) a symmetric Jacobian matrix.

Integrability and Dynamics of Quadratic Three-Dimensional Differential Systems Having an Invariant Paraboloid

2016 ◽  
Vol 26 (08) ◽  
pp. 1650134 ◽  
Author(s):  
Marcelo Messias ◽  
Alisson C. Reinol
Keyword(s):  
Three Dimensional ◽  
Algebraic Surfaces ◽  
Quadratic Polynomial ◽  
Poincaré Compactification ◽  
Differential Systems ◽  
Elliptic Paraboloid ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Surfaces ◽  
Exponential Factors ◽  
Parameter Values

Invariant algebraic surfaces are commonly observed in differential systems arising in mathematical modeling of natural phenomena. In this paper, we study the integrability and dynamics of quadratic polynomial differential systems defined in [Formula: see text] having an elliptic paraboloid as an invariant algebraic surface. We obtain the normal form for these kind of systems and, by using the invariant paraboloid, we prove the existence of first integrals, exponential factors, Darboux invariants and inverse Jacobi multipliers, for suitable choices of parameter values. We characterize all the possible configurations of invariant parallels and invariant meridians on the invariant paraboloid and give necessary conditions for the invariant parallel to be a limit cycle and for the invariant meridian to have two orbits heteroclinic to a point at infinity. We also study the dynamics of a particular class of the quadratic polynomial differential systems having an invariant paraboloid, giving information about localization and local stability of finite singular points and, by using the Poincaré compactification, we study their dynamics on the Poincaré sphere (at infinity). Finally, we study the well-known Rabinovich system in the case of invariant paraboloids, performing a detailed study of its dynamics restricted to these invariant algebraic surfaces.

Three-Dimensional Hopf Bifurcation for a Class of Cubic Kolmogorov Model

2014 ◽  
Vol 24 (03) ◽  
pp. 1450036 ◽  
Author(s):  
Chaoxiong Du ◽  
Qinlong Wang ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Three Dimensional ◽  
Singular Values ◽  
Bifurcation Problem ◽  
Disturbed Condition ◽  
Differential Systems ◽  
Kolmogorov Model ◽  
Polynomial Differential Systems ◽  
Fine Focus

We study the Hopf bifurcation for a class of three-dimensional cubic Kolmogorov model by making use of our method (i.e. singular values method). We show that the positive singular point (1, 1, 1) of an investigated model can become a fine focus of 5 order, and moreover, it can bifurcate five small limit cycles under certain coefficients with disturbed condition. In terms of three-dimensional cubic Kolmogorov model, published references can hardly be seen, and our results are new. At the same time, it is worth pointing out that our method is valid to study the Hopf bifurcation problem for other three-dimensional polynomial differential systems.

scholarly journals On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550122 ◽  
Author(s):  
Jaume Llibre ◽  
Ana Rodrigues
Keyword(s):  
Chaotic System ◽  
Parameter Family ◽  
Algebraic Surfaces ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Differential Systems ◽  
Special Values ◽  
Unified Chaotic System ◽  
Invariant Algebraic Surfaces ◽  
The Family

A one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.

Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1, 1)SN - (B)

2021 ◽  
Vol 31 (09) ◽  
pp. 2130026
Author(s):  
Joan C. Artés ◽  
Marcos C. Mota ◽  
Alex C. Rezende
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Saddle Node ◽  
Complete Study ◽  
Real Quadratic ◽  
Finite Singularity

This paper presents a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity. This class can be divided into two different families, namely, [Formula: see text] phase portraits possessing a finite saddle-node as the only finite singularity and [Formula: see text] phase portraits possessing a finite saddle-node and also a simple finite elemental singularity. Each one of these two families is given by a specific normal form. The study of family [Formula: see text] was reported in [Artés et al., 2020b] where the authors obtained [Formula: see text] topologically distinct phase portraits for systems in the closure [Formula: see text]. In this paper, we provide the complete study of the geometry of family [Formula: see text]. This family which modulo the action of the affine group and time homotheties is three-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the three-dimensional real projective space. The respective bifurcation diagram yields 631 subsets with 226 topologically distinct phase portraits for systems in the closure [Formula: see text] within the representatives of [Formula: see text] given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.

scholarly journals Topological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle

2016 ◽  
Vol 26 (11) ◽  
pp. 1650188 ◽  
Author(s):  
Joan C. Artés ◽  
Regilene D. S. Oliveira ◽  
Alex C. Rezende
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Structure Stability ◽  
Differential Systems ◽  
Bifurcation Set ◽  
Polynomial Differential Systems

The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilbert’s 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental triple saddle (triple saddle with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this normal form. This bifurcation diagram yields 27 phase portraits for systems in [Formula: see text] counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set and we present the phase portraits on the Poincaré disk. The bifurcation set is not just algebraic due to the presence of a surface found numerically, whose points correspond to connections of separatrices.

scholarly journals The Geometry of Quadratic Polynomial Differential Systems with a Finite and an Infinite Saddle-Node (C)

2015 ◽  
Vol 25 (03) ◽  
pp. 1530009 ◽  
Author(s):  
Joan C. Artés ◽  
Alex C. Rezende ◽  
Regilene D. S. Oliveira
Keyword(s):  
Bifurcation Diagram ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Horizontal Axis ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Differential Systems ◽  
Bifurcation Set ◽  
Polynomial Differential Systems ◽  
Saddle Node

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. Our goal is to make a global study of the family QsnSN of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the collision of two infinite singular points. This family can be divided into three different subfamilies, all of them with the finite saddle-node in the origin of the plane with the eigenvectors on the axes and with the eigenvector associated with the zero eigenvalue on the horizontal axis and (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and time homotheties are three-dimensional and we give the bifurcation diagram of their closure with respect to specific normal forms, in the three-dimensional real projective space. The subfamilies (A) and (B) have already been studied [Artés et al., 2013b] and in this paper we provide the complete study of the geometry of the last family (C). The bifurcation diagram for the subfamily (C) yields 371 topologically distinct phase portraits with and without limit cycles for systems in the closure [Formula: see text] within the representatives of QsnSN(C) given by a chosen normal form. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of [Formula: see text] is not only algebraic due to the presence of some surfaces found numerically. All points in these surfaces correspond to either connections of separatrices, or the presence of a double limit cycle.

scholarly journals Algebraic Limit Cycles on Quadratic Polynomial Differential Systems

2018 ◽  
Vol 61 (2) ◽  
pp. 499-512 ◽  
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Algebraic Curve ◽  
Positive Answer ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Curve ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractAlgebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

Sign in / Sign up

Export Citation Format

Share Document