Normal forms of polynomial differential systems in $${\mathbb{R}}^3$$ having at least three invariant algebraic surfaces

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418300069 ◽

2018 ◽

Vol 28 (03) ◽

pp. 1830006 ◽

Cited By ~ 1

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415501229 ◽

2015 ◽

Vol 25 (09) ◽

pp. 1550122 ◽

Cited By ~ 3

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.

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Quadratic systems with an invariant conic having Darboux invariants

Communications in Contemporary Mathematics ◽

10.1142/s021919971750033x ◽

2018 ◽

Vol 20 (04) ◽

pp. 1750033

Author(s):

Jaume Llibre ◽

Regilene Oliveira

Keyword(s):

Vector Fields ◽

Normal Forms ◽

Complete Characterization ◽

Phase Portraits ◽

Affine Transformations ◽

Differential Systems ◽

Quadratic Systems ◽

Quadratic Vector Fields ◽

Polynomial Differential Systems

The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real affine transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these normal forms, we complete the characterization of the phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic [Formula: see text]: [Formula: see text], and a Darboux invariant of the form [Formula: see text] with [Formula: see text].

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Isochronicity of bi-centers for symmetric quartic differential systems

Discrete and Continuous Dynamical Systems - Series B ◽

10.3934/dcdsb.2021215 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Wilker Fernandes ◽

Viviane Pardini Valério ◽

Patricia Tempesta

Keyword(s):

Normal Forms ◽

Phase Portraits ◽

Differential Systems ◽

Polynomial Differential Systems ◽

Quartic Polynomial ◽

Different Types ◽

Simultaneous Existence ◽

Isochronous Centers

<p style='text-indent:20px;'>In this paper we investigate the simultaneous existence of isochronous centers for a family of quartic polynomial differential systems under four different types of symmetry. Firstly, we find the normal forms for the system under each type of symmetry. Next, the conditions for the existence of isochronous bi-centers are presented. Finally, we study the global phase portraits of the systems possessing isochronous bi-centers. The study shows the existence of seven non topological equivalent global phase portraits, where three of them are exclusive for quartic systems under such conditions.</p>

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Normal Forms for Polynomial Differential Systems in ℝ3 Having an Invariant Quadric and a Darboux Invariant

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415500157 ◽

2015 ◽

Vol 25 (01) ◽

pp. 1550015 ◽

Cited By ~ 7

Author(s):

Jaume Llibre ◽

Marcelo Messias ◽

Alisson de Carvalho Reinol

Keyword(s):

Algebraic Surface ◽

Normal Forms ◽

Chen System ◽

Differential Systems ◽

Polynomial Differential Systems ◽

Invariant Algebraic Surface

We give the normal forms of all polynomial differential systems in ℝ3 which have a nondegenerate or degenerate quadric as an invariant algebraic surface. We also characterize among these systems those which have a Darboux invariant constructed uniquely using the invariant quadric, giving explicitly their expressions. As an example, we apply the obtained results in the determination of the Darboux invariants for the Chen system with an invariant quadric.

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