Comments on ‘Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces’

2014 ◽  
Vol 266 ◽  
pp. 80-82 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Algebraic Surfaces ◽  
Global Dynamics ◽  
Invariant Algebraic Surfaces

scholarly journals Global dynamics of a virus model with invariant algebraic surfaces

2019 ◽  
Vol 69 (2) ◽  
pp. 535-546
Author(s):  
Fabio Scalco Dias ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Algebraic Surfaces ◽  
Global Dynamics ◽  
Invariant Algebraic Surfaces ◽  
Virus Model

scholarly journals On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550122 ◽  
Author(s):  
Jaume Llibre ◽  
Ana Rodrigues
Keyword(s):  
Chaotic System ◽  
Parameter Family ◽  
Algebraic Surfaces ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Differential Systems ◽  
Special Values ◽  
Unified Chaotic System ◽  
Invariant Algebraic Surfaces ◽  
The Family

A one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.

GLOBAL DYNAMICS OF THE LORENZ SYSTEM WITH INVARIANT ALGEBRAIC SURFACES

2010 ◽  
Vol 20 (10) ◽  
pp. 3137-3155 ◽  
Author(s):  
JAUME LLIBRE ◽  
MARCELO MESSIAS ◽  
PAULO RICARDO DA SILVA
Keyword(s):  
Phase Portrait ◽  
Lorenz System ◽  
Algebraic Surfaces ◽  
Global Dynamics ◽  
State Variables ◽  
Poincaré Compactification ◽  
Invariant Algebraic Surfaces ◽  
The Lorenz System ◽  
Global Phase Portrait ◽  
Parameter Values

In this paper by using the Poincaré compactification of ℝ3 we describe the global dynamics of the Lorenz system [Formula: see text] having some invariant algebraic surfaces. Of course (x, y, z) ∈ ℝ3 are the state variables and (s, r, b) ∈ ℝ3 are the parameters. For six sets of the parameter values, the Lorenz system has invariant algebraic surfaces. For these six sets, we provide the global phase portrait of the system in the Poincaré ball (i.e. in the compactification of ℝ3 with the sphere 𝕊2 of the infinity).

scholarly journals GLOBAL DYNAMICS IN THE POINCARÉ BALL OF THE CHEN SYSTEM HAVING INVARIANT ALGEBRAIC SURFACES

2012 ◽  
Vol 22 (06) ◽  
pp. 1250154 ◽  
Author(s):  
JAUME LLIBRE ◽  
MARCELO MESSIAS ◽  
PAULO RICARDO DA SILVA
Keyword(s):  
Global Analysis ◽  
Dynamical Behavior ◽  
First Integral ◽  
Algebraic Surfaces ◽  
Heteroclinic Orbits ◽  
Global Dynamics ◽  
Chen System ◽  
Invariant Algebraic Surfaces ◽  
Global Phase Portrait ◽  
Parameter Values

In this paper, we perform a global analysis of the dynamics of the Chen system [Formula: see text] where (x, y, z) ∈ ℝ3 and (a, b, c) ∈ ℝ3. 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 𝕊2, i.e. in the compactification of ℝ3 with the sphere 𝕊2 of 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.

scholarly journals On the Integrability of the SIR Epidemic Model with Vital Dynamics

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Ding Chen
Keyword(s):  
Epidemic Model ◽  
Birth Rate ◽  
Sir Model ◽  
Point Of View ◽  
Algebraic Surfaces ◽  
Global Dynamics ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Invariant Algebraic Surfaces ◽  
Invariant Algebraic Surface

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.

scholarly journals Coexistence of chaotic attractor and unstable limit cycles in a 3D dynamical system

2021 ◽  
Vol 1 ◽  
pp. 50
Author(s):  
Dana Constantinescu ◽  
Gheorghe Tigan ◽  
Xiang Zhang
Keyword(s):  
Dynamical System ◽  
Limit Cycles ◽  
Algebraic Surfaces ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
Stable Limit ◽  
Invariant Algebraic Surfaces ◽  
Stable Limit Cycles ◽  
T System

The coexistence of stable limit cycles and chaotic attractors has already been observed in some 3D dynamical systems. In this paper we show, using the T-system, that unstable limit cycles and chaotic attractors can also coexist. Moreover, by completing the characterization of the existence of invariant algebraic surfaces and their associated global dynamics, we give a better understanding on the disappearance of the strange attractor and the limit cycles of the studied system.

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