Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system

Gouveia, Márcio Ricardo Alves; Messias, Marcelo; Pessoa, Claudio

doi:10.1007/s11071-015-2520-4

Cited by 6 publications

(4 citation statements)

References 20 publications

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“…is a polynomial first integral of system (1), where f i are the homogeneous polynomials of degree i and f n ̸ = 0. Firstly, substituting (11) into (10) and identifying the homogeneous components of degree n + 1, we get…”

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

confidence: 99%

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

confidence: 99%

“…To this end, in this section we will analyze the Poincaré compactification of the GD model in the local charts U i and V i (i = 1, 2, 3). We refer to [10], [7], [11], [25] for the detailed theory of Poincaré compactification.…”

Section: Co-dimension Two Bifurcationsmentioning

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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“…To do this, in the next three subsections we shall analyze the Poincaré compactification of system (6) in the local charts U i and V i (i = 1, 2, 3). The detailed theory of Poincaré compactification can be found in [10], [11], [12], [13].…”

Section: Theorem 23 (Existence Of Hopf Bifurcation) Assume That (Hmentioning

confidence: 99%

“…In Section 4, we contribute to investigating the global dynamics of the QPP model by studying its behavior at infinity. Since the QPP model is a polynomial vector field in R 3 , by the Poincaré compactification one can extend the QPP model into an analytic system defined on a closed ball of radius one, called the Poincaré ball, whose interior is diffeomorphic to R 3 and its invariant boundary, the twodimensional spherical shell S 2 = {(x, y, z)|x 2 + y 2 + z 2 = 1} plays the role of the infinity, called the Poincaré sphere [10], [11], [12], [13]. Using this compactification technique, we give a complete description of the dynamical behavior of the QPP model on the sphere at infinity (Theorem 4.1).…”

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confidence: 99%

Complex dynamics in a quasi-periodic plasma perturbations model

Zhang¹,

Yang²

2021

DCDS-B

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In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [Phys. Plasmas, 18(2011):1-7]. In particular, assuming that there exists a finite time lag τ between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power h and the relaxation of the instability δ do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient η just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.

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New Insights on Non-integrability and Dynamics in a Simple Quadratic Differential System

Qu,

Yang

2024

J Nonlinear Math Phys

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This study focuses on the integrability and qualitative behaviors of a quadratic differential system $$\dot{x}=a+yz,\quad\dot{y}=-y + x^{2},\quad\dot{z}=b-4x.$$ x ˙ = a + y z , y ˙ = - y + x 2 , z ˙ = b - 4 x . We provide some new perspectives on the system and reveal its diverse properties, including non-integrability in the sense of absence of first integrals, bifurcations of co-dimension one or two, Jacobi instability and dynamics at infinity.

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Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system

Cited by 6 publications

References 20 publications

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

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

Complex dynamics in a quasi-periodic plasma perturbations model

New Insights on Non-integrability and Dynamics in a Simple Quadratic Differential System

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