Invariant algebraic surfaces of the generalized Lorenz system

2013 ◽  
Vol 64 (5) ◽  
pp. 1443-1449 ◽  
Author(s):  
Xijun Deng
Keyword(s):  
Lorenz System ◽  
Algebraic Surfaces ◽  
Generalized Lorenz System ◽  
Invariant Algebraic Surfaces

Comments on “Invariant algebraic surfaces of the generalized Lorenz system”

2014 ◽  
Vol 66 (3) ◽  
pp. 1295-1297 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Lorenz System ◽  
Algebraic Surfaces ◽  
Generalized Lorenz System ◽  
Invariant Algebraic Surfaces

The invariant algebraic surfaces of the Lorenz system

2002 ◽  
Vol 132 (3) ◽  
pp. 385-393 ◽  
Author(s):  
SIR PETER SWINNERTON-DYER
Keyword(s):  
Lorenz System ◽  
Algebraic Surfaces ◽  
Special Values ◽  
Invariant Algebraic Surfaces ◽  
The Lorenz System

The object of this paper is to find all the irreducible algebraic surfaces which (for special values of the parameters b, r, s) are invariant under the Lorenz systemx˙ = X(x, y, z) = s(y−x), y˙ = Y(x, y, z) = rx−y−xz, ż = Z(x, y, z) =−bz+xy. (1)It is customary in considering the Lorenz system to require the parameters b, r, s to be all strictly positive; however for this particular problem we shall follow previous practice in only imposing the condition s ≠ 0. (If s = 0 the equations are trivially integrable and x is constant on any trajectory; thus x should be regarded as a parameter and the question discussed in this paper ceases to be a natural one.)

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).

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

Hopf Bifurcations, Periodic Windows and Intermittency in the Generalized Lorenz Model

2019 ◽  
Vol 29 (14) ◽  
pp. 1930042
Author(s):  
Anna Wawrzaszek ◽  
Agata Krasińska
Keyword(s):  
Lorenz System ◽  
Dimensional System ◽  
Type I ◽  
Control Parameters ◽  
System Behavior ◽  
Lorenz Model ◽  
Generalized Lorenz System ◽  
Subcritical Hopf Bifurcation ◽  
Bifurcation Structures ◽  
The Magnetic Field

In the present study, we analyze the dynamics of a four-dimensional generalized Lorenz system with one variable describing the profile of the magnetic field induced in a convected magnetized fluid. In particular, we identify the subcritical Hopf bifurcation, at which the dimension of the unstable manifold is increased or reduced by two. Moreover, the new four-dimensional system behavior depending on the control parameters is considered and bidirectional bifurcation structures are revealed. The results show the existence of several windows of nonchaotic variation (windows of order), in particular period-3 windows at the edge of which type I intermittency is observed.

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.

Dynamical Analysis on a Finance System with Nonconstant Elasticity of Demand

2020 ◽  
Vol 30 (10) ◽  
pp. 2050148
Author(s):  
Ting Yang
Keyword(s):  
Chaotic Attractor ◽  
Period Doubling ◽  
Approximate Expression ◽  
Algebraic Surfaces ◽  
Dynamical Analysis ◽  
Periodic Attractor ◽  
Normal Form Theory ◽  
Finance System ◽  
Elasticity Of Demand ◽  
Invariant Algebraic Surfaces

This paper investigates a finance system with nonconstant elasticity of demand. First, under some conditions, the system has invariant algebraic surfaces and the analytic expressions of the surfaces are given. Furthermore, when the two surfaces coincide and become one surface, the dynamics on the surface are analyzed and a globally stable equilibrium is found. Second, by using the normal form theory, the Hopf bifurcation is studied and the approximate expression and stability of the bifurcating periodic orbit are obtained. Third, the chaotic behaviors are investigated and the route to chaos is period-doubling bifurcations. Moreover, it is found that the system has coexisting attractors, including periodic attractor and periodic attractor, chaotic attractor and chaotic attractor. With the change of parameter, the two chaotic attractors coincide and then a symmetrical chaotic attractor arises.

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