Dynamics at Infinity and Existence of Singularly Degenerate Heteroclinic Cycles in Maxwell–Bloch System

2020 ◽  
Vol 15 (10) ◽  
Author(s):  
Haimei Chen ◽  
Yongjian Liu ◽  
Chunsheng Feng ◽  
Aimin Liu ◽  
Xiezhen Huang
Keyword(s):  
Dynamic Behavior ◽  
Numerical Analyses ◽  
Suitable Choice ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
Poincaré Compactification ◽  
Heteroclinic Cycles ◽  
Physical Essence ◽  
Infinite Set ◽  
Special Parameter

Abstract In this paper, global dynamics of the Maxwell–Bloch system is discussed. First, the complete description of its dynamic behavior on the sphere at infinity is presented by using the Poincaré compactification in R3. Second, the existence of singularly degenerate heteroclinic cycles is investigated. It is proved that for a suitable choice of the parameters, there is an infinite set of singularly degenerate heteroclinic cycles in Maxwell–Bloch system. Specially, the chaotic attractors are found nearby singularly degenerate heteroclinic cycles in Maxwell–Bloch system by combining theoretical and numerical analyses for a special parameter value. It is hoped that these theoretical and numerical value results are given a contribution in an understanding of the physical essence for chaos in the Maxwell–Bloch system.

DYNAMICS OF THE LÜ SYSTEM ON THE INVARIANT ALGEBRAIC SURFACE AND AT INFINITY

2011 ◽  
Vol 21 (09) ◽  
pp. 2559-2582 ◽  
Author(s):  
YONGJIAN LIU ◽  
QIGUI YANG
Keyword(s):  
Algebraic Surface ◽  
Global Analysis ◽  
Chaotic Attractors ◽  
Numerical Techniques ◽  
Poincaré Compactification ◽  
Heteroclinic Cycles ◽  
Lü System ◽  
Invariant Algebraic Surface ◽  
Infinite Set ◽  
Lu System

Firstly, the dynamics of the Lü system having an invariant algebraic surface are analyzed. Secondly, by using the Poincaré compactification in ℝ3, a global analysis of the system is presented, including the complete description of its dynamic behavior on the sphere at infinity. Lastly, combining analytical and numerical techniques, it is shown that for the parameter value b = 0, the system presents an infinite set of singularly degenerate heteroclinic cycles. The chaotic attractors for the Lü system in the case of small b > 0 are found numerically, hence the singularly degenerate heteroclinic cycles.

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.

scholarly journals Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

2007 ◽  
Vol 79 (4) ◽  
pp. 563-575 ◽  
Author(s):  
Jaume Llibre ◽  
Marcelo Messias
Keyword(s):  
Large Amplitude ◽  
Symmetric Polynomial ◽  
The Other ◽  
Poincaré Compactification ◽  
Heteroclinic Loop ◽  
Differential Systems ◽  
Heteroclinic Cycles ◽  
Polynomial Differential Systems ◽  
Global Study ◽  
Straight Lines

In this paper we study a class of symmetric polynomial differential systems in R³, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n <FONT FACE=Symbol>Î</FONT> N there is epsilonn > 0 such that for 0 < epsilon < epsilonn the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y > 0 and the other in y < 0.

scholarly journals SINGULAR ORBITS AND DYNAMICS AT INFINITY OF A CONJUGATE LORENZ-LIKE SYSTEM

2015 ◽  
Vol 20 (2) ◽  
pp. 148-167 ◽  
Author(s):  
Fengjie Geng ◽  
Xianyi Li
Keyword(s):  
Theoretical Analysis ◽  
Lyapunov Exponents ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Basic Properties ◽  
Homoclinic And Heteroclinic Orbits ◽  
Quadratic Nonlinearities ◽  
Singular Orbits

A conjugate Lorenz-like system which includes only two quadratic nonlinearities is proposed in this paper. Some basic properties of this system, such as the distribution of its equilibria and their stabilities, the Lyapunov exponents, the bifurcations are investigated by some numerical and theoretical analysis. The forming mechanisms of compound structures of its new chaotic attractors obtained by merging together two simple attractors after performing one mirror operation are also presented. Furthermore, some of its other complex dynamical behaviours, which include the existence of singularly degenerate heteroclinic cycles, the existence of homoclinic and heteroclinic orbits and the dynamics at infinity, etc, are formulated in detail. In the meantime, some problems deserving further investigations are presented.

Constructing a Chaotic System with Any Number of Attractors

2017 ◽  
Vol 27 (08) ◽  
pp. 1750118 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Vector Field ◽  
Geometric Structure ◽  
Autonomous System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Original System ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
The Basin Of Attraction ◽  
New System

Applying some transformation to an autonomous system, we obtain a new system, which might keep the dynamical behavior of the original system or generate different dynamics. But this is often accompanied by the appearance of discontinuous points, where the vector field for the new system is not continuous at these points. We discuss the effects of the discontinuous points, and provide two methods to construct systems with any preassigned number of chaotic attractors via some transformation. The first one does not change the geometric structure of the attractors, since the discontinuous points are out of the basin of attraction. The second one might make the new systems have different dynamics, like multiscroll chaotic attractors, or infinitely many chaotic attractors. These results illustrate that both the equilibria and the discontinuous points affect the global dynamics.

Dynamics of SVEIS epidemic model with distinct incidence

2015 ◽  
Vol 08 (06) ◽  
pp. 1550076 ◽  
Author(s):  
N. Nyamoradi ◽  
M. Javidi ◽  
B. Ahmad
Keyword(s):  
Numerical Simulations ◽  
Dynamic Behavior ◽  
Epidemic Model ◽  
Stability Theory ◽  
Global Dynamics ◽  
Turing Bifurcation ◽  
Diffusive Model ◽  
Stability And Bifurcation ◽  
Matrix Stability ◽  
Bifurcation Behavior

In this paper, we study the global dynamics of a SVEIS epidemic model with distinct incidence for exposed and infectives. The model is analyzed for stability and bifurcation behavior. To account for the realistic phenomenon of non-homogeneous mixing, the effect of diffusion on different population subclasses is considered. The diffusive model is analyzed using matrix stability theory and conditions for Turing bifurcation are derived. Numerical simulations support our analytical results on the dynamic behavior of the model.

On Singular Orbits and Global Exponential Attractive Set of a Lorenz-Type System

2019 ◽  
Vol 29 (06) ◽  
pp. 1950082
Author(s):  
Haijun Wang
Keyword(s):  
Lyapunov Functions ◽  
Three Dimensional ◽  
Type System ◽  
Homoclinic Orbits ◽  
Global Dynamics ◽  
Heteroclinic Cycles ◽  
Mathematical Expressions ◽  
Lasalle Theorem ◽  
Singular Orbits ◽  
Attractive Sets

This paper deals with some unsolved problems of the global dynamics of a three-dimensional (3D) Lorenz-type system: [Formula: see text], [Formula: see text], [Formula: see text] by constructing a series of Lyapunov functions. The main contribution of the present work is that one not only proves the existence of singularly degenerate heteroclinic cycles, existence and nonexistence of homoclinic orbits for a certain range of the parameters according to some known results and LaSalle theorem but also gives a family of mathematical expressions of global exponential attractive sets for that system with respect to its parameters, which is available only in very few papers as far as one knows. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.

scholarly journals Global Dynamics of the Hastings-Powell System

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis N. Coria
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
State Variables ◽  
System Parameters ◽  
First Order ◽  
Extremum Conditions

This paper studies the problem of bounding a domain that contains all compact invariant sets of the Hastings-Powell system. The results were obtained using the first-order extremum conditions and the iterative theorem to a biologically meaningful model. As a result, we calculate the bounds given by a tetrahedron with excisions, described by several inequalities of the state variables and system parameters. Therefore, a region is identified where all the system dynamics are located, that is, its compact invariant sets: equilibrium points, periodic-homoclinic-heteroclinic orbits, and chaotic attractors. It was also possible to formulate a nonexistence condition of the compact invariant sets. Additionally, numerical simulations provide examples of the calculated boundaries for the chaotic attractors or periodic orbits. The results provide insights regarding the global dynamics of the system.

scholarly journals Dynamics at Infinity, Degenerate Hopf and Zero-Hopf Bifurcation for Kingni–Jafari System with Hidden Attractors

2016 ◽  
Vol 26 (07) ◽  
pp. 1650125 ◽  
Author(s):  
Zhouchao Wei ◽  
Irene Moroz ◽  
Zhen Wang ◽  
Julien Clinton Sprott ◽  
Tomasz Kapitaniak
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Small Amplitude ◽  
Complex Dynamics ◽  
Global Dynamics ◽  
Polynomial Vector ◽  
Hidden Attractors ◽  
Poincaré Compactification ◽  
Existence Of Periodic Solutions ◽  
The Rich

To understand the complex dynamics of Kingni–Jafari system with hidden attractors, the first objective of this paper is to study the global dynamics, and give a complete description of the dynamics of Kingni–Jafari system at infinity by using the Poincaré compactification of a polynomial vector field in [Formula: see text]. The second objective of this paper is to prove the existence of periodic solutions in the Kingni–Jafari system by classical Hopf bifurcation and degenerate Hopf bifurcation. Moreover, based on averaging theory, a small amplitude periodic solution that bifurcates from a zero-Hopf equilibrium was derived in the Kingni–Jafari system. The theoretical analysis and simulations demonstrate the rich dynamics of the system.

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