Assessment of Critical Parameter Values for H2O and D2O

1985 ◽  
Vol 14 (1) ◽  
pp. 193-207 ◽  
Author(s):  
J. M. H. Levelt Sengers ◽  
J. Straub ◽  
K. Watanabe ◽  
P. G. Hill
Keyword(s):  
Critical Parameter ◽  
Parameter Values

ANALYTICAL PREDICTION OF THE TWO FIRST PERIOD-DOUBLINGS IN A THREE-DIMENSIONAL SYSTEM

2000 ◽  
Vol 10 (06) ◽  
pp. 1497-1508 ◽  
Author(s):  
M. BELHAQ ◽  
M. HOUSSNI ◽  
E. FREIRE ◽  
A. J. RODRÍGUEZ-LUIS
Keyword(s):  
Periodic Orbit ◽  
Critical Parameter ◽  
Order Approximation ◽  
Three Dimensional ◽  
Period Doubling ◽  
Dimensional System ◽  
Analytical Prediction ◽  
Parameter Values ◽  
Three Dimensional System ◽  
Period Doubling Bifurcations

Analytical study of the two first period-doubling bifurcations in a three-dimensional system is reported. The multiple scales method is first applied to construct a higher-order approximation of the periodic orbit following Hopf bifurcation. The stability analysis of this periodic orbit is then performed in terms of Floquet theory to derive the critical parameter values corresponding to the first and second period-doubling bifurcations. By introducing suitable subharmonic components in the first order of the multiple scale analysis the two critical parameter values are obtained simultaneously solving analytically the resulting system of two algebraic equations. Comparisons of analytic predictions to numerical simulations are also provided.

Dynamic bifurcations in non-stationary systems: transitions with an unpredictable outcome

1995 ◽  
Vol 451 (1942) ◽  
pp. 471-485 ◽  
Keyword(s):  
Critical Parameter ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Important Concept ◽  
New Form ◽  
Parameter Values ◽  
Dynamic Bifurcations

In nonlinear non-stationary systems, dynamic bifurcations result in a transition to a qualitatively new state. In this paper we examine how the dynamics of transition of such systems may be assessed using the concept of transient basins of attraction. We delineate the phenomenon of indeterminate dynamic bifurcations, where it is shown that the response, after the system passes through critical parameter values, may be extremely sensitive to the choice of initial conditions or parameter states. This new form of unpredictability in systems whose parameters vary with time, is clearly an important concept to be assimilated in the theory of non-stationary dynamics.

Analytics of Bifurcation

1998 ◽  
Vol 08 (03) ◽  
pp. 471-482 ◽  
Author(s):  
Paul E. Phillipson ◽  
Peter Schuster
Keyword(s):  
Differential Equation ◽  
Critical Parameter ◽  
Three Dimensional ◽  
System Control ◽  
Control Parameters ◽  
Single Period ◽  
Period Oscillation ◽  
Asymptotic Averaging ◽  
Parameter Values

Oscillations described by autonomous three-dimensional differential equation systems display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario, the key instability is usually an initial bifurcation from a single period oscillation to its subharmonic of period two, or the reverse. An asymptotic averaging formalism is introduced by means of an example which permits an analytic development of the bifurcation dynamics, and in particular, prediction of the onset of periods 1 ↔ 2 bifurcations in terms of the system control parameters.

DETERMINATION OF CRISIS PARAMETER VALUES BY DIRECT OBSERVATION OF MANIFOLD TANGENCIES

1992 ◽  
Vol 02 (02) ◽  
pp. 383-396 ◽  
Author(s):  
JOHN C. SOMMERER ◽  
CELSO GREBOGI
Keyword(s):  
Periodic Point ◽  
Chaotic System ◽  
Direct Observation ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Critical Parameter ◽  
Scaling Relation ◽  
Chaotic Transients ◽  
Parameter Values

We discuss an algorithm to find the parameter value at which a nonlinear, dissipative, chaotic system undergoes crisis. The algorithm is based on the observation that, at crisis, the unstable manifold of an unstable periodic point becomes tangent to the stable manifold of the same or another, related unstable periodic point. This geometric algorithm uses much less computation (or data) than estimating the critical parameter value by using the scaling relation for chaotic transients, τ~(p−pc)−γ. We demonstrate the algorithm in both numerical and experimental contexts.

scholarly journals BIFURCATION DYNAMICS OF THREE-DIMENSIONAL SYSTEMS

2000 ◽  
Vol 10 (08) ◽  
pp. 1787-1804 ◽  
Author(s):  
PAUL E. PHILLIPSON ◽  
PETER SCHUSTER
Keyword(s):  
Differential Equation ◽  
Critical Parameter ◽  
Three Dimensional ◽  
System Control ◽  
Control Parameters ◽  
Lorenz Equations ◽  
Third Order ◽  
Period Oscillation ◽  
Asymptotic Averaging ◽  
Parameter Values

Oscillations described by autonomous three-dimensional differential equations display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario the key instability is often an initial bifurcation from a single period oscillation to either its subharmonic of period two, or a symmetry breaking bifurcation. A generalized third-order nonlinear differential equation is developed which embraces the dynamics vicinal to these bifurcation events. Subsequently, an asymptotic averaging formalism is applied which permits an analytic development of the bifurcation dynamics, and, within quantifiable limits, prediction of the instability of the period one orbit in terms of the system control parameters. Illustrative applications of the general formalism, are made to the Rössler equations, Lorenz equations, three-dimensional replicator equations and Chua's circuit equations. The results provide the basis for discussion of the class of systems which fall within the framework of the formalism.

Pulsation Energy in the Suction Manifold of a Reciprocating Compressor as a Measure for Parameter Sensitivity

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Nasir Bilal ◽  
Douglas E. Adams
Keyword(s):  
Design Parameter ◽  
Critical Parameter ◽  
Design Parameters ◽  
Reciprocating Compressor ◽  
Input Design ◽  
Parameter Variations ◽  
Pulsation Model ◽  
Unique Method ◽  
Parameter Values ◽  
Gas Pulsations

Gas pulsations in a compressor suction manifold radiate noise and reduce the efficiency of the compressor. The objective of this paper is to identify and quantify the effects of modeling assumptions and uncertainties in input parameters on the pulsation model output predictions and to estimate the sensitivity of the model to changes in the input design parameters. A unique method of sensitivity analysis is presented that uses the total pulsation energy in the suction manifold of a compressor as a measure of gas pulsations. This method is used to determine the sensitivity of the gas pulsations in the suction manifold to input design parameters. First, the gas pulsations in the suction manifold are calculated using linear acoustic theory. Second, the effects of varying several different design parameters of the suction manifold on gas pulsations are analyzed, and the three most important parameters are selected. Next, energy due to gas pulsations in the suction manifold due to these design parameter variations is calculated. Suction manifold radius was identified as the most critical parameter, followed by width and depth. The optimized values of manifold radius resulted in an overall reduction of up to 24% in the gas pulsation energy compared to the pulsation energy at the nominal design parameter values in the suction manifold.

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