On the global dynamics of a three-dimensional forced-damped differential system

Llibre, Jaume; Martínez, Y. Paulina; Valls, Clàudìa

doi:10.1080/14029251.2020.1757232

Cited by 4 publications

(5 citation statements)

References 7 publications

(12 reference statements)

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“…Another tool that deals with the integrability of 3D differential systems is the Darboux theory of integrability [15,16], which is a useful tool to find first integrals for polynomial ordinary differential equations, and has been successfully applied in many nonlinear models [24][25][26][27]. This theory can also help us make a more precise analysis of the global dynamics of a system topologically (see [28,29] and the references therein). In the framework of the Darboux theory of integrability, we will give a complete classification of the irreducible Darboux polynomials, of the polynomial first integrals, of the proper rational first integrals, and of the algebraic integrability for the SIR model.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the Integrability of the SIR Epidemic Model with Vital Dynamics

Ding

2020

Advances in Mathematical Physics

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In this paper, we study the SIR epidemic model with vital dynamics Ṡ=−βSI+μN−S,İ=βSI−γ+μI,Ṙ=γI−μR, from the point of view of integrability. In the case of the death/birth rate μ=0, the SIR model is integrable, and we provide its general solutions by implicit functions, two Lax formulations and infinitely many Hamilton-Poisson realizations. In the case of μ≠0, we prove that the SIR model has no polynomial or proper rational first integrals by studying the invariant algebraic surfaces. Moreover, although the SIR model with μ≠0 is not integrable and we cannot get its exact solution, based on the existence of an invariant algebraic surface, we give the global dynamics of the SIR model with μ≠0.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the Integrability of the SIR Epidemic Model with Vital Dynamics

Ding

2020

Advances in Mathematical Physics

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“…Generally, a system of differential equations is integrable if it possesses a sufficient number of first integrals (and/or other tensor invariants) such that we can solve this system explicitly. Hence we could obtain its global information and understand its topological structure [18,26]. Furthermore, non-integrability of the system also seems necessary for better understanding of the complex phenomenon [13,41].…”

Section: Homothetic Transformation Between the Gd Model And Other Qua...mentioning

confidence: 99%

“…Darboux integrability theory plays an important role in the integrability of the polynomial differential systems [5,6,8,29,30], which helps us find first integrals by knowing a sufficient number of algebraic invariant surfaces (the Darboux polynomials) and of exponential factors, see [3,28,35,36,37] for instance. Moreover, it can also help us make a more precise analysis of the global dynamics of the considered system topologically [26,38].…”

Section: Homothetic Transformation Between the Gd Model And Other Qua...mentioning

confidence: 99%

On a simple model for describing convection of the rotating fluid: Integrability, bifurcations and global dynamics

Jia

Yang

Zhou

et al. 2023

DCDS-B

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The Glukhovsky-Dolzhansky (GD) model arises naturally from geophysical science, which describes rotating fluid convection inside the ellipsoid. This work aims to provide some new insights into the GD model. (i) We first show that, under some conditions there are homothetic transformations which covert the GD model into other similar quadric physical models, therefore, our results on the GD model can be naturally applied to the investigation of these models. (ii) We propose a complete classification of Darboux polynomials and exponent factors for the GD model, which implies that the GD model has no polynomial, rational, or Darboux first integrals. In addition, some integrable cases of the GD model are also given when the physical parameters are allowed to be non-positive. (iii) The existence of global attractor is proved. The stability and local bifurcations of all co-dimension one and two are investigated. Particularly, we show that the GD model undergoes two dynamical transitions as the Rayleigh number increases. (iv) To understand the asymptotic behavior of the orbits for the GD model, we use the Poincaré compactification method to study its dynamical behavior at infinity. More precisely, we prove that the phase portraits of the GD model at infinity consist of an infinite sequence of periodic solutions and two heteroclinic loops. Our results may help us better understand the complex and rich dynamics of rotating fluid convection.

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“…Generally, a system of differential equations is integrable if it possesses a sufficient number of first integrals (and/or other tensor invariants) such that we can solve this system explicitly. Hence we could obtain its global information and understand its topological structure [8,9]. Furthermore, non-integrability of the system also seems necessary for better understanding of the complex phenomenon [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…Darboux integrability theory plays an important role in the integrability of the polynomial differential systems [10,11,12,13,14], which helps us find first integrals by knowing a sufficient number of algebraic invariant surfaces (the Darboux polynomials) and of the exponential factors, see [15,16,17,18,19] for instance. Moreover, it can also help us make a more precise analysis of the global dynamics of the considered system topologically [20,8]. In addition, let us mention that the Darboux integrability of the Rabinovich system, 3D forced-damped system and D2 vector field, which are similar but different from the Glukhovsky-Dolzhansky system (1), have been studied in [28,29,30,31].…”

Section: Introductionmentioning

confidence: 99%

Integrability analysis of a simple model for describing convection of a rotating fluid

Jiao,

Zhou,

Yang

2021

Preprint

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We study the Darboux integrability of a simple system of three ordinary differential equations called the Glukhovsky-Dolzhansky system, which describes a three-mode model of rotating fluid convection inside the ellipsoid. (1) Our results show that it has no polynomial, rational, or Darboux first integrals for any value of parameters in the physical sense, that is, positive parameters. (2) We also provide some integrable cases of this model when parameters are allowed to be non-positive.(3) We finally give some links between the Glukhovsky-Dolzhansky system and other similar systems in R 3 , which admits rotational symmetry and has three nonlinear cross terms.

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On the global dynamics of a three-dimensional forced-damped differential system

Cited by 4 publications

References 7 publications

On the Integrability of the SIR Epidemic Model with Vital Dynamics

On the Integrability of the SIR Epidemic Model with Vital Dynamics

On a simple model for describing convection of the rotating fluid: Integrability, bifurcations and global dynamics

Integrability analysis of a simple model for describing convection of a rotating fluid

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