Integrability of 3-dim polynomial systems with three invariant planes

Hu, Zhaoping; Aldazharova, M. M.; Aldibekov, T. М.; Romanovski, Valery G.

doi:10.1007/s11071-013-1025-2

Cited by 11 publications

(4 citation statements)

References 25 publications

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“…They used mainly Darboux theory of integrability and other methods to prove the sufficiency of their conditions. Romanovski and collaborators in [17,18] investigated the integrability problem with (0 : −1 : 1) resonance at the origin. Local integrability and linearizability of a quadratic three dimensional family with (1 : −1 : 1) resonance is considered in [2].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Integrability and linearizability of a family of three-dimensional quadratic systems

2021

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We provide necessary and sufficient conditions for both integrability and linearizability of a three dimensional vector field with quadratic nonlinearities. For our investigation we consider the case of (1 : −2 : 1)-resonance at the origin and in general non of the axes planes is invariant.Hence, we deal with a nine parametric family of quadratic systems. Some techniques like Darboux method are used to prove the sufficiency of the obtained conditions. For a particular three parametric subfamily we provide conditions to guarantee the non existence of a polynomial first integral.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Integrability and linearizability of a family of three-dimensional quadratic systems

2021

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“…These have been investigated in the last decades from different dynamical points of view. Despite their simplicity quadratic systems are not completely understood from the view point of the integrability, see for instance [8].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Integrability of the Rucklidge system

2014

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“…Llibre et al in [19] studied the Darboux integrability of a three-dimensional, forced-damped differential system. The integrability of three-dimensional quadratic systems in a neighborhood of a (0:−1:1) resonant equilibrium was investigated in [20,21].…”

Section: Integrability Of 3d Persistent and Semi-persistent Systemsmentioning

confidence: 99%

On the Integrability of Persistent Quadratic Three-Dimensional Systems

Ferčec,

Žulj,

Mencinger

2024

Mathematics

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We consider a nine-parameter familiy of 3D quadratic systems, x˙=x+P2(x,y,z), y˙=−y+Q2(x,y,z), z˙=−z+R2(x,y,z), where P2,Q2,R2 are quadratic polynomials, in terms of integrability. We find necessary and sufficient conditions for the existence of two independent first integrals of corresponding semi-persistent, weakly persistent, and persistent systems. Unlike some of the earlier works, which primarily focus on planar systems, our research covers three-dimensional spaces, offering new insights into the complex dynamics that are not typically apparent in lower dimensions.

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Integrability of 3-dim polynomial systems with three invariant planes

Cited by 11 publications

References 25 publications

Integrability and linearizability of a family of three-dimensional quadratic systems

Integrability and linearizability of a family of three-dimensional quadratic systems

Integrability of the Rucklidge system

On the Integrability of Persistent Quadratic Three-Dimensional Systems

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