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

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

Rendiconti del Circolo Matematico di Palermo (1952 -) ◽

10.1007/s12215-020-00537-y ◽

2020 ◽

Author(s):

Najmeh Khajoei ◽

Mohammad Reza Molaei

Keyword(s):

Normal Forms ◽

Algebraic Surfaces ◽

Differential Systems ◽

Polynomial Differential Systems ◽

Invariant Algebraic Surfaces

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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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THE CHEN SYSTEM HAVING AN INVARIANT ALGEBRAIC SURFACE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408022706 ◽

2008 ◽

Vol 18 (12) ◽

pp. 3753-3758 ◽

Cited By ~ 9

Author(s):

JINLONG CAO ◽

CHENG CHEN ◽

XIANG ZHANG

Keyword(s):

Algebraic Surface ◽

Chen System ◽

Invariant Algebraic Surface

In this paper, we characterize the dynamics of the Chen system ẋ = a(y - x), ẏ = (c - a)x - xz + cy, ż = xy - bz which has an invariant algebraic surface.

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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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