scholarly journals Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Jinki Kim ◽  
R. L. Harne ◽  
K. W. Wang
Keyword(s):  
Normal Form ◽  
Analog Circuit ◽  
Duffing Oscillator ◽  
Critical Parameters ◽  
Escape Time ◽  
Estimation Of Parameters ◽  
Saddle Node Bifurcation ◽  
Analytical Approximations ◽  
Saddle Node ◽  
Two Factors

Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and microscale mass sensing. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors toward the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept toward a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. The stochastic normal form of the saddle-node bifurcation is derived from the governing equation of this oscillator to formulate the probability distribution of escape events. Non-stationarity is accounted for using a time-dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function (PDF) to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verifies that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation

2016 ◽  
Author(s):  
Jinki Kim ◽  
R. L. Harne ◽  
K. W. Wang
Keyword(s):  
Normal Form ◽  
Analog Circuit ◽  
Critical Parameters ◽  
Material Structure ◽  
Escape Time ◽  
Estimation Of Parameters ◽  
Saddle Node Bifurcation ◽  
Analytical Approximations ◽  
Piezoelectric Energy ◽  
Saddle Node

Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and adaptive material/structure applications such as structural health monitoring and piezoelectric energy harvesting. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors towards the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept towards a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. Using this archetype example, the stochastic normal form of the saddle-node bifurcation is derived from which expressions of the escape statistics are formulated. Non-stationarity is accounted for using a time dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verify that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

TRAVELLING WAVES WITH SPATIALLY RESONANT FORCING: BIFURCATIONS OF A MODIFIED LANDAU EQUATION

1993 ◽  
Vol 03 (06) ◽  
pp. 1447-1455 ◽  
Author(s):  
PAUL GLENDINNING ◽  
MICHAEL PROCTOR
Keyword(s):  
Normal Form ◽  
Landau Equation ◽  
Travelling Waves ◽  
Duffing Equation ◽  
Invariant Curve ◽  
Invariant System ◽  
Van Der Pol ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Resonant Forcing

Normal form equations are derived representing the effects of spatially resonant forcing on a bifurcation to travelling waves in a nonreflexionally invariant system. The equations also describe the effects of forcing on the van der Pol–Duffing equation. It is found that the forcing prevents the propagation of the wave if it is sufficiently strong: the transition to a nonuniformly propagating solution can occur either via an oscillatory bifurcation or due to a saddle node bifurcation on a closed invariant curve.

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 $.

scholarly journals Saddle Node Bifurcation Point Analysis of Voltage Stability of IEEE 30-BUS Power System for Removal of Generation through Bacteria Foraging Optimization Algorithm

2018 ◽  
Vol 7 (3.31) ◽  
pp. 36
Author(s):  
Srikanth B. Venkata ◽  
Lakshmi Devi Ai
Keyword(s):  
Power System ◽  
Optimization Algorithm ◽  
Reactive Power ◽  
Reactive Power Compensation ◽  
Bacterial Foraging Optimization ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Point Analysis ◽  
Bacteria Foraging Optimization Algorithm ◽  
Bacterial Foraging Optimization Algorithm

This paper deals with the identification of instability nodes of IEEE 30 BUS power system to generation removal. Optimal sizing and locations of reactive power compensations are obtained. Firstly one of the generators is assumed to be removed from service and the saddle node bifurcation (SNB) point voltages are evaluated without reactive power compensation. Secondly two generators are assumed to be removed from service and the saddle node point voltage magnitudes are obtained without reactive power compensation. For both cases the study is conducted by placing optimal reactive power compensations at optimal locations using Bacterial Foraging Optimization Algorithm (BFOA).  

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