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

2015 ◽  
Vol 84 (2) ◽  
pp. 703-713 ◽  
Author(s):  
Márcio R. A. Gouveia ◽  
Marcelo Messias ◽  
Claudio Pessoa
Keyword(s):  
Chaotic System ◽  
Algebraic Surfaces ◽  
Heteroclinic Orbits ◽  
Homoclinic And Heteroclinic Orbits ◽  
Invariant Algebraic Surfaces

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.

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.

New Results to a Three-Dimensional Chaotic System With Two Different Kinds of Nonisolated Equilibria

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic System ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Heteroclinic Cycle ◽  
Homoclinic And Heteroclinic Orbits ◽  
Singular Orbits ◽  
The Stability ◽  
Singularly Degenerate Heteroclinic Cycle

In the paper by Liu et al. (2009, “A Novel Three-Dimensional Autonomous Chaos System,” Chaos Solitons Fractals, 39(4), pp. 1950–1958), the three-dimensional (3D) chaotic system x·=-ax-ey2,y·=by-kxz,z·=-cz+mxy is investigated, and some of its dynamics according to theoretical and numerical analyses only for the parameters (a, e, b, k, c, m) = (1, 1, 2.5, 4, 5, 4) are discussed. In 2013, the same chaotic system x·1=-ax1- fx2x3,x·2=cx2-dx1x3,x·3=-bx3+ex22 by Li et al. (2013, “Analysis of a Novel Three-Dimensional Chaotic System,” Optik, 124(13), pp. 1516–1522) was mainly discussed by numerical simulation. In this article, by some deeper investigations, combining some numerical simulations, we formulate some new results of the system. First, after some problems in the first paper are pointed out, we display that its parameters e, k, and m may be kicked out by some homothetic transformations. Second, some of its rich nonlinear dynamics hiding and not found previously, such as the stability and Hopf bifurcation of its isolated equilibria, the behavior of its nonisolated equilibria, the existence of singular orbits (including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, etc.), and its dynamics at infinity, etc., are clearly formulated. What's more interesting, we find, this system has two different kinds of nonisolated equilibria Ex and Ez, and new chaotic attractors can be bifurcated out with the disappearance of Ex, but this system has no such properties at Ez. In the meantime, several problems about the existence of singular orbits deserving further investigations are presented. Our results better complement and improve the known ones.

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 A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Jingjing Feng ◽  
Qichang Zhang ◽  
Wei Wang ◽  
Shuying Hao
Keyword(s):  
Dynamic Systems ◽  
Approximation Method ◽  
Padé Approximation ◽  
Global Bifurcation ◽  
Heteroclinic Orbits ◽  
Symmetric System ◽  
Pade Approximation ◽  
New Approach ◽  
Homoclinic And Heteroclinic Orbits ◽  
Single Orbit

In dynamic systems, some nonlinearities generate special connection problems of non-Z2symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2symmetric nonlinear quintic systems (orbit with one cusp); and Z2symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.

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

More Dynamical Properties Revealed from a 3D Lorenz-like System

2014 ◽  
Vol 24 (10) ◽  
pp. 1450133 ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Mathematical Work ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Properties ◽  
Theoretical Results

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

SHIL'NIKOV HETEROCLINIC ORBITS IN A CHAOTIC SYSTEM

2007 ◽  
Vol 21 (25) ◽  
pp. 4429-4436 ◽  
Author(s):  
FENG-YUN SUN
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Heteroclinic Orbits ◽  
Geometric Structures ◽  
Undetermined Coefficient ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

In this paper, a chaotic system which exhibits a chaotic attractor with only three equilibria for some parameters is considered. The existence of heteroclinic orbits of the Shil'nikov type in a chaotic system has been proved using the undetermined coefficient method. As a result, the Shil'nikov criterion guarantees that the system has Smale horseshoes. Moreover, the geometric structures of the attractor are determined by these heteroclinic orbits.

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.

Sign in / Sign up

Export Citation Format

Share Document