Research on Bifurcation of Asymmetric System Based on Second-Order Averaging Method

2015 ◽  
Vol 713-715 ◽  
pp. 1835-1838
Author(s):  
An Kang Hu ◽  
Ya Chong Liu ◽  
Yu Lu ◽  
Feng Lei Han
Keyword(s):  
Nonlinear System ◽  
Averaging Method ◽  
Second Order ◽  
Local Bifurcation ◽  
Saddle Node Bifurcation ◽  
Asymmetric System ◽  
Saddle Node ◽  
Basic Characteristics

Bifurcation which may lead to chaos is the typical character of nonlinear system, and an asymmetric system with asymmetric parameter is adopted in this paper. The basic characteristics which vary with the asymmetric parameter are investigated firstly, and then, the second-order averaging method is used to investigate the local bifurcation of the asymmetric system. The super and sub critical saddle-node bifurcation curves of both left center and right center of the system are solved analytically. The results show that the degree of asymmetric is influenced by the value of asymmetric parameter and the two bifurcation curves of the same center are intersected at the point which also depends on the asymmetric parameter value.

Attractor-preserving control to avoid saddle-node bifurcation

2014 ◽  
Vol 2 ◽  
pp. 150-153
Author(s):  
Daisuke Ito ◽  
Tetsushi Ueta ◽  
Shigeki Tsuji ◽  
Kazuyuki Aihara
Keyword(s):  
Saddle Node Bifurcation ◽  
Saddle Node

BASIN ORGANIZATION PRIOR TO A TANGLED SADDLE-NODE BIFURCATION

1991 ◽  
Vol 01 (01) ◽  
pp. 107-118 ◽  
Author(s):  
MOHAMED S. SOLIMAN ◽  
J. M. T. THOMPSON
Keyword(s):  
Fractal Structure ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Homoclinic Connections

Heteroclinic and homoclinic connections of saddle cycles play an important role in basin organization. In this study, we outline how these events can lead to an indeterminate jump to resonance from a saddle-node bifurcation. Here, due to the fractal structure of the basins in the vicinity of the saddle-node, we cannot predict to which available attractor the system will jump in the presence of even infinitesimal noise.

Bifurcation analysis of steady natural convection in a tilted cubical cavity with adiabatic sidewalls

2014 ◽  
Vol 756 ◽  
pp. 650-688 ◽  
Author(s):  
J. F. Torres ◽  
D. Henry ◽  
A. Komiya ◽  
S. Maruyama
Keyword(s):  
Natural Convection ◽  
Rotation Axis ◽  
Numerical Technique ◽  
Continuation Method ◽  
Rotation Vector ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Stable Solutions ◽  
The Stability ◽  
Cubical Cavity

AbstractNatural convection in an inclined cubical cavity heated from two opposite walls maintained at different temperatures and with adiabatic sidewalls is investigated numerically. The cavity is inclined by an angle $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\theta $ around a lower horizontal edge and the isothermal wall set at the higher temperature is the lower wall in the horizontal situation ($\theta = 0^\circ $). A continuation method developed from a three-dimensional spectral finite-element code is applied to determine the bifurcation diagrams for steady flow solutions. The numerical technique is used to study the influence of ${\theta }$ on the stability of the flow for moderate Rayleigh numbers in the range $\mathit{Ra} \leq 150\, 000$, focusing on the Prandtl number $\mathit{Pr} = 5.9$. The results show that the inclination breaks the degeneracy of the stable solutions obtained at the first bifurcation point in the horizontal cubic cavity: (i) the transverse stable rolls, whose rotation vector is in the same direction as the inclination vector ${\boldsymbol{\Theta}}$, start from $\mathit{Ra} \to 0$ forming a leading branch and becoming more predominant with increasing $\theta $; (ii) a disconnected branch consisting of transverse rolls, whose rotation vector is opposite to ${\boldsymbol{\Theta}}$, develops from a saddle-node bifurcation, is stabilized at a pitchfork bifurcation, but globally disappears at a critical inclination angle; (iii) the semi-transverse stable rolls, whose rotation axis is perpendicular to ${\boldsymbol{\Theta}}$ at $\theta \to 0^\circ $, develop from another saddle-node bifurcation, but the branch also disappears at a critical angle. We also found the stability thresholds for the stable diagonal-roll and four-roll solutions, which increase with $\theta $ until they disappear at other critical angles. Finally, the families of stable solutions are presented in the $\mathit{Ra}-\theta $ parameter space by determining the locus of the primary, secondary, saddle-node, and Hopf bifurcation points as a function of $\mathit{Ra}$ and $\theta $.

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