Erratum: Stationary probability distribution near stable limit cycles far from Hopf bifurcation points

1995 ◽  
Vol 52 (6) ◽  
pp. 6916-6916 ◽  
Author(s):  
Mark Dykman ◽  
Xiaolin Chu ◽  
John Ross
Keyword(s):  
Hopf Bifurcation ◽  
Probability Distribution ◽  
Limit Cycles ◽  
Stable Limit ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Bifurcation Points ◽  
Stable Limit Cycles

Hopf Bifurcation to Limit Cycles in Fluid Film Bearings

1986 ◽  
Vol 108 (2) ◽  
pp. 184-189 ◽  
Author(s):  
P. Hollis ◽  
D. L. Taylor
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Critical Speed ◽  
Stability Boundary ◽  
Linear Stability Theory ◽  
Stable Limit ◽  
Fluid Film Bearings ◽  
Speed Stability ◽  
Stable Limit Cycles ◽  
Range Of Values

The nonlinear response of a cylindrical journal bearing operating close to the critical speed stability boundary is studied in this paper. Using linear stability theory, the value of the critical variable (usually speed) at the point of loss of stability is obtained and shown to agree with results of previous researchers. Using Hopf bifurcation analysis, parameters for determining the behavior close to this point are obtained. Analytically, these parameters prove that the system can exhibit stable limit cycles for speeds above the critical speed. Such supercritical limit cycles only exist for a narrow range of values of modified Sommerfeld number. In other cases, subcritical limit cycles are predicted. The results are supported by numerical simulation. The results show why it may be difficult to observe supercritical limit cycles in test rigs.

Response : Stable Limit Cycles in Prey-Predator Populations

Science ◽  
1973 ◽  
Vol 181 (4104) ◽  
pp. 1074-1074
Author(s):  
Robert M. May
Keyword(s):  
Limit Cycles ◽  
Stable Limit ◽  
Stable Limit Cycles

The censored Markov chain and the best augmentation

1996 ◽  
Vol 33 (03) ◽  
pp. 623-629 ◽  
Author(s):  
Y. Quennel Zhao ◽  
Danielle Liu
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
The Best Approximation ◽  
Finite Matrix ◽  
Countable State ◽  
Stationary Probabilities

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

STATISTICAL PROPERTIES OF INTENSITY FLUCTUATION OF SATURATION LASER MODEL DRIVEN BY CROSS-CORRELATED ADDITIVE AND MULTIPLICATIVE NOISES

2010 ◽  
Vol 24 (14) ◽  
pp. 2175-2188 ◽  
Author(s):  
PING ZHU ◽  
YI JIE ZHU
Keyword(s):  
Probability Distribution ◽  
Cross Correlation ◽  
Laser Intensity ◽  
Laser System ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Model Driven ◽  
Multiplicative Noises ◽  
Laser Model ◽  
The Stability

Statistical properties of the intensity fluctuation of a saturation laser model driven by cross-correlation additive and multiplicative noises are investigated. Using the Novikov theorem and the projection operator method, we obtain the analytic expressions of the stationary probability distribution Pst(I), the relaxation time Tc, and the normalized variance λ2(0) of the system. By numerical computation, we discussed the effects of the cross-correlation strength λ, the cross-correlation time τ, the quantum noise intensity D, and the pump noise intensity Q for the fluctuation of the laser intensity. Above the threshold, λ weakens the stationary probability distribution, speeds up the startup velocity of the laser system from start status to steady work, and attenuates the stability of laser intensity output; however, τ strengthens the stationary probability distribution and strengths the stability of laser intensity output; when λ < 0, τ speeds up the startup; on the contrast, when λ > 0, τ slows down the startup. D and Q make the relaxation time exhibit extremum structure, that is, the startup time possesses the least values. At the threshold, τ cannot generate the effects for the saturation laser system, λ expedites the startup velocity and weakens the stability of laser intensity output. Below threshold, the effects of λ and τ not only relate to λ and τ, but also relate to other parameters of the system.

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