THE CHEN SYSTEM HAVING AN INVARIANT ALGEBRAIC SURFACE

2008 ◽  
Vol 18 (12) ◽  
pp. 3753-3758 ◽  
Author(s):  
JINLONG CAO ◽  
CHENG CHEN ◽  
XIANG ZHANG
Keyword(s):  
Algebraic Surface ◽  
Chen System ◽  
Invariant Algebraic Surface

In this paper, we characterize the dynamics of the Chen system ẋ = a(y - x), ẏ = (c - a)x - xz + cy, ż = xy - bz which has an invariant algebraic surface.

scholarly journals Normal Forms for Polynomial Differential Systems in ℝ3 Having an Invariant Quadric and a Darboux Invariant

2015 ◽  
Vol 25 (01) ◽  
pp. 1550015 ◽  
Author(s):  
Jaume Llibre ◽  
Marcelo Messias ◽  
Alisson de Carvalho Reinol
Keyword(s):  
Algebraic Surface ◽  
Normal Forms ◽  
Chen System ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Surface

We give the normal forms of all polynomial differential systems in ℝ3 which have a nondegenerate or degenerate quadric as an invariant algebraic surface. We also characterize among these systems those which have a Darboux invariant constructed uniquely using the invariant quadric, giving explicitly their expressions. As an example, we apply the obtained results in the determination of the Darboux invariants for the Chen system with an invariant quadric.

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.

Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface

2014 ◽  
Vol 77 (4) ◽  
pp. 1503-1518 ◽  
Author(s):  
Zhen Wang ◽  
Zhouchao Wei ◽  
Xiaojian Xi ◽  
Yongxin Li
Keyword(s):  
Algebraic Surface ◽  
Quadratic System ◽  
Invariant Algebraic Surface

New Heteroclinic Orbits Coined

2016 ◽  
Vol 26 (12) ◽  
pp. 1650194 ◽  
Author(s):  
Haijun Wang ◽  
Chang Li ◽  
Xianyi Li
Keyword(s):  
Homoclinic Orbit ◽  
Algebraic Surface ◽  
Type System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Homoclinic And Heteroclinic Orbits ◽  
Invariant Algebraic Surface

We devote to studying the problem for the existence of homoclinic and heteroclinic orbits of Unified Lorenz-Type System (ULTS). Other than the known results that the ULTS has two homoclinic orbits to [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and two heteroclinic orbits to [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] on its invariant algebraic surface [Formula: see text], formulated in the literature by Yang and Chen [2014], we seize two new heteroclinic orbits of this Unified Lorenz-Type System. Namely, we rigorously prove that this system has another two heteroclinic orbits to [Formula: see text] and [Formula: see text] while no homoclinic orbit when [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text].

scholarly journals Isotopic triangulation of a real algebraic surface

2009 ◽  
Vol 44 (9) ◽  
pp. 1291-1310 ◽  
Author(s):  
Lionel Alberti ◽  
Bernard Mourrain ◽  
Jean-Pierre Técourt
Keyword(s):  
Algebraic Surface ◽  
Real Algebraic Surface

scholarly journals On the Reconstruction of the Center of a Projection by Distances and Incidence Relations

2020 ◽  
Author(s):  
András Pongrácz ◽  
Csaba Vincze
Keyword(s):  
Algebraic Surface ◽  
Basic Problem ◽  
Rotational Symmetry ◽  
Image Plane ◽  
Equivalent Condition ◽  
Central Projection ◽  
Profile Curve ◽  
Reconstruction Process ◽  
Restricted Area ◽  
Photographic Images

AbstractUp to an orientation-preserving symmetry, photographic images are produced by a central projection of a restricted area in the space into the image plane. To obtain reliable information about physical objects and the environment through the process of recording is the basic problem of photogrammetry. We present a reconstruction process based on distances from the center of projection and incidence relations among the points to be projected. For any triplet of collinear points in the space, we construct a surface of revolution containing the center of the projection. It is a generalized conic that can be represented as an algebraic surface. The rotational symmetry allows us to restrict the investigations to the defining polynomial of the profile curve in the image plane. An equivalent condition for the boundedness is given in terms of the input parameters, and it is shown that the defining polynomial of the profile curve is irreducible.

Chaos Anti-Synchronization between Chen System and Lu System

2014 ◽  
Vol 631-632 ◽  
pp. 710-713 ◽  
Author(s):  
Xian Yong Wu ◽  
Hao Wu ◽  
Hao Gong
Keyword(s):  
Chaotic Systems ◽  
Sufficient Conditions ◽  
Lyapunov Theory ◽  
State Variables ◽  
Chen System ◽  
System Parameters ◽  
Control Scheme ◽  
Error Dynamics ◽  
The Stability ◽  
Different Chaotic Systems

Anti-synchronization of two different chaotic systems is investigated. On the basis of Lyapunov theory, adaptive control scheme is proposed when system parameters are unknown, sufficient conditions for the stability of the error dynamics are derived, where the controllers are designed using the sum of the state variables in chaotic systems. Numerical simulations are performed for the Chen and Lu systems to demonstrate the effectiveness of the proposed control strategy.

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