Global Dynamical Behavior of FitzHugh–Nagumo Systems with Invariant Algebraic Surfaces

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Liwei Zhang ◽  
Jiang Yu ◽  
Xiang Zhang
Keyword(s):  
Dynamical Behavior ◽  
Algebraic Surfaces ◽  
Invariant Algebraic Surfaces

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.

Nonchaotic Behavior in Quadratic Three-Dimensional Differential Systems with a Symmetric Jacobian Matrix

2018 ◽  
Vol 28 (03) ◽  
pp. 1830006 ◽  
Author(s):  
Marcelo Messias ◽  
Rafael Paulino Silva
Keyword(s):  
Jacobian Matrix ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Positive Answer ◽  
Algebraic Surfaces ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Algebraic Criterion ◽  
Invariant Algebraic Surfaces

In this paper, we give an algebraic criterion to determine the nonchaotic behavior for polynomial differential systems defined in [Formula: see text] and, using this result, we give a partial positive answer for the conjecture about the nonchaotic dynamical behavior of quadratic three-dimensional differential systems having a symmetric Jacobian matrix. The algebraic criterion presented here is proved using some ideas from the Darboux theory of integrability, such as the existence of invariant algebraic surfaces and Darboux invariants, and is quite general, hence it can be used to study the nonchaotic behavior of other types of differential systems defined in [Formula: see text], including polynomial differential systems of any degree having (or not having) a symmetric Jacobian matrix.

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 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.

Sign in / Sign up

Export Citation Format

Share Document