First integrals of the Maxwell–Bloch system

Huang, Kaiyin; Shi, Shaoyun; Li, Wenlei

doi:10.5802/crmath.6

Cited by 6 publications

(4 citation statements)

References 14 publications

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“…In particular, the existence of first integrals plays a crucial role in the integrability of ordinary differential equations [17][18][19][20]. If the considered system of ordinary differential equations admits a straight line solution, by the differential Galois method, one can usually prove that this system has no rational first integrals for almost all the parameters [21][22][23]. However, this method cannot tell whether this system is integrable for the remaining parameters.…”

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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“…are elliptic integrals of the first and second kind, by (17) and (24), in order for A 2m−1 (x, y, z) = Ā2m−1 (u, v, w) to be a homogeneous polynomial of degree 2m − 1, we must have…”

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

confidence: 99%

“…Another tool to study the non-integrability of non-Hamiltonian systems is the differential Galois theory [1,14,33]. Observing system (1) has a straight line solution (x(t), y(t), z(t)) = (0, R a −e −t , 0), we can analyze the differential Galois group of the normal variational equations along this solution, and show that system (1) is not rationally integrable in Bogoyavlenskij sense for almost all parameter values, see [15,16,17,31] for more details.…”

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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“…Another tool to study the non-integrability of non-Hamiltonian systems is the differential Galois theory [21,22,23]. Observing system (1) has a straight line solution (x(t), y(t), z(t)) = (0, Ra − e −t , 0), we can analyze the differential Galois group of the normal variational equations along this solution, and show that system (1) is not rationally integrable in Bogoyavlenskij sense for almost all parameter values, see [24,25,26,27] for more details.…”

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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First integrals of the Maxwell–Bloch system

Cited by 6 publications

References 14 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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