Polynomial First Integrals for the Chen and Lü Systems

Llibre, Jaume; Valls, Clàudìa

doi:10.1142/s0218127412502628

Cited by 14 publications

(11 citation statements)

References 13 publications

(8 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the recent work [Llibre and Valls, 2011] the authors characterized all the the parameter values for which the Chen system (1) has a polynomial first integral. More precisely they proved that system (1) with February 13, 2012 11:42 LMS˙ChenSystem˙IJBC a = 0 has a first integral if and only if b = c = 0 and, in this case, the first integral is a polynomial in the variable y 2 + z 2 + 2az, corresponding to the case (f) of Table 1.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Global Dynamics in the Poincaré Ball of the Chen System Having Invariant Algebraic Surfaces

Llibre

Messias

Silva

2012

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

In this paper we perform a global analysis of the dynamics of the Chen systeṁwhere (x, y, z) ∈ R 3 and (a, b, c) ∈ R 3 . We give the complete description of its dynamics on the sphere at infinity. For six sets of the parameter values the system has invariant algebraic surfaces. In these cases we provide the global phase portrait of the Chen system and give a complete description of the α-and ω-limit sets of its orbits in the Poincaré ball, including its boundary S 2 , i.e. in the compactification of R 3 with the sphere S 2 of the infinity. Moreover, combining the analytical results obtained with an accurate numerical analysis, we prove the existence of a family with infinitely many heteroclinic orbits contained on invariant cylinders when the Chen system has a line of singularities and a first integral, which indicates the complicated dynamical behavior of the Chen system solutions even in the absence of chaotic dynamics.

show abstract

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Global Dynamics in the Poincaré Ball of the Chen System Having Invariant Algebraic Surfaces

Llibre

Messias

Silva

2012

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is well known that (15) admits a chaotic attractor for a = 36, b = 3 and c = 20. So far there have been a large number of related studies on the Lü system such as dynamical behaviors, chaotic control, chaos synchronization (see [38][39][40] and the references therein).…”

Section: Chaotic Systemsmentioning

confidence: 99%

“…Llibre et al [38,39] consider the Darboux first integrals and polynomial first integrals. Here, we investigate the first integrals of the Lü system in the category of meromorphic functions.…”

Section: Chaotic Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Meromorphic Non-Integrability of Several 3D Dynamical Systems

Huang

Shi

2017

Entropy

View full text Add to dashboard Cite

Abstract:In this paper, we apply the differential Galoisian approach to investigate the meromorphic non-integrability of a class of 3D equations in mathematical physics, including Nosé-Hoover equations, the Lü system, the Rikitake-like system and Rucklidge equations, which are well known in the fields of molecular dynamics, chaotic theory and fluid mechanics, respectively. Our main results show that all these considered systems are, in fact, non-integrable in nearly all parameters.

show abstract

“…For suitable choices of the values of parameters α, β and γ this system exhibits chaotic phenomena. The following result on system (20) was proved in [Lu & Zhang, 2007;Llibre & Valls, 2012] (see also for the analysis of system (20) having invariant algebraic surfaces).…”

Section: Darboux Invariants Of the Chen Systemmentioning

confidence: 99%

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

Llibre

Messias

Reinol

2015

Int. J. Bifurcation Chaos

Self Cite

View full text Add to dashboard Cite

We give the normal forms of all polynomial differential systems in R 3 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.

show abstract

Polynomial First Integrals for the Chen and Lü Systems

Abstract: We characterize all the values of the parameters for which the Chen and Lü systems have polynomial first integrals by using weight homogeneous polynomials and the method of characteristics for solving partial differential equations. We improve previous results which were not complete.

Cited by 14 publications

References 13 publications

Global Dynamics in the Poincaré Ball of the Chen System Having Invariant Algebraic Surfaces

Global Dynamics in the Poincaré Ball of the Chen System Having Invariant Algebraic Surfaces

Meromorphic Non-Integrability of Several 3D Dynamical Systems

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

Contact Info

Product

Resources

About

Polynomial First Integrals for the Chen and Lü Systems

Abstract: We characterize all the values of the parameters for which the Chen and Lü systems have polynomial first integrals by using weight homogeneous polynomials and the method of characteristics for solving partial differential equations. We improve previous results which were not complete.

Cited by 14 publications

References 13 publications

Global Dynamics in the Poincaré Ball of the Chen System Having Invariant Algebraic Surfaces

Global Dynamics in the Poincaré Ball of the Chen System Having Invariant Algebraic Surfaces

Meromorphic Non-Integrability of Several 3D Dynamical Systems

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

Contact Info

Product

Resources

About

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