The topological structure of the Rabinovich system having an invariant algebraic surface

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.

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

Nonlinear Dynamics ◽

10.1007/s11071-014-1395-0 ◽

2014 ◽

Vol 77 (4) ◽

pp. 1503-1518 ◽

Cited By ~ 3

Author(s):

Zhen Wang ◽

Zhouchao Wei ◽

Xiaojian Xi ◽

Yongxin Li

Keyword(s):

Algebraic Surface ◽

Quadratic System ◽

Invariant Algebraic Surface

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THE CHEN SYSTEM HAVING AN INVARIANT ALGEBRAIC SURFACE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408022706 ◽

2008 ◽

Vol 18 (12) ◽

pp. 3753-3758 ◽

Cited By ~ 9

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415500157 ◽

2015 ◽

Vol 25 (01) ◽

pp. 1550015 ◽

Cited By ~ 7

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.

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New Heteroclinic Orbits Coined

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501947 ◽

2016 ◽

Vol 26 (12) ◽

pp. 1650194 ◽

Cited By ~ 4

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].

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Dynamics of the Lorenz system having an invariant algebraic surface

Journal of Mathematical Physics ◽

10.1063/1.2767007 ◽

2007 ◽

Vol 48 (8) ◽

pp. 082702 ◽

Cited By ~ 14

Author(s):

Jinlong Cao ◽

Xiang Zhang

Keyword(s):

Algebraic Surface ◽

Lorenz System ◽

Invariant Algebraic Surface ◽

The Lorenz System

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Uncovering Atherosclerotic Risk Disease Gene Based on Expression and Network Topological Structure*

PROGRESS IN BIOCHEMISTRY AND BIOPHYSICS ◽

10.3724/sp.j.1206.2010.00687 ◽

2010 ◽

Vol 37 (8) ◽

pp. 916-922

Author(s):

Hong WANG ◽

Xiao-Li QU ◽

Yan ZHAO ◽

Jing ZHANG ◽

Li-Na CHEN

Keyword(s):

Topological Structure ◽

Disease Gene

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Brief Survey of Biological Network Alignment and a Variant with Incorporation of Functional Annotations

Current Bioinformatics ◽

10.2174/1574893612666171020103747 ◽

2018 ◽

Vol 14 (1) ◽

pp. 4-10

Author(s):

Fang Jing ◽

Shao-Wu Zhang ◽

Shihua Zhang

Keyword(s):

Gene Ontology ◽

Topological Structure ◽

Metabolic Networks ◽

Biological Network ◽

Genomic Sequence ◽

Network Alignment ◽

Future Directions ◽

Functional Annotations ◽

Protein Protein Interaction ◽

Ppi Networks

Background:Biological network alignment has been widely studied in the context of protein-protein interaction (PPI) networks, metabolic networks and others in bioinformatics. The topological structure of networks and genomic sequence are generally used by existing methods for achieving this task.Objective and Method:Here we briefly survey the methods generally used for this task and introduce a variant with incorporation of functional annotations based on similarity in Gene Ontology (GO). Making full use of GO information is beneficial to provide insights into precise biological network alignment.Results and Conclusion:We analyze the effect of incorporation of GO information to network alignment. Finally, we make a brief summary and discuss future directions about this topic.

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Information on evolutionary age in redundancy of complex network

Modern Physics Letters B ◽

10.1142/s0217984919503317 ◽

2019 ◽

Vol 33 (27) ◽

pp. 1950331

Author(s):

Shiguo Deng ◽

Henggang Ren ◽

Tongfeng Weng ◽

Changgui Gu ◽

Huijie Yang

Keyword(s):

Principal Component Analysis ◽

Complex Networks ◽

Metabolic Network ◽

Complex Network ◽

Cellular Networks ◽

Topological Structure ◽

Quantitative Measure ◽

Principal Component ◽

Evolutionary Information ◽

Local Patterns

Evolutionary processes of many complex networks in reality are dominated by duplication and divergence. This mechanism leads to redundant structures, i.e. some nodes share most of their neighbors and some local patterns are similar, called redundancy of network. An interesting reverse problem is to discover evolutionary information from the present topological structure. We propose a quantitative measure of redundancy of network from the perspective of principal component analysis. The redundancy of a community in the empirical human metabolic network is negatively and closely related with its evolutionary age, which is consistent with that for the communities in the modeling protein–protein network. This behavior can be used to find the evolutionary difference stored in cellular networks.

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