Oscillations and stability in quasiautonomous system. II. Critical point of the one-parameter family of periodic motions

2011 ◽  
Vol 72 (7) ◽  
pp. 1450-1457 ◽  
Author(s):  
V. N. Tkhai
Keyword(s):  
Critical Point ◽  
Parameter Family ◽  
Periodic Motions ◽  
The One

Scaling Near the Critical Point in Mixture

2005 ◽  
Author(s):  
Eldred H. Chimowitz
Keyword(s):  
Critical Point ◽  
Binary Systems ◽  
Critical Line ◽  
Phase Region ◽  
Unique Point ◽  
Pure Fluid ◽  
High Volatility ◽  
Two Phases ◽  
The One ◽  
Dew Line

The critical point of mixtures requires a more intricate set of conditions to hold than those at a pure-fluid critical point. In contrast to the pure-fluid case, in which the critical point occurs at a unique point, mixtures have additional thermodynamic degrees of freedom. They, therefore, possess a critical line which defines a locus of critical points for the mixture. At each point along this locus, the mixture exhibits a critical point with its own composition, temperature, and pressure. In this chapter we investigate the critical behavior of binary mixtures, since higher-order systems do not bring significant new considerations beyond those found in binaries. We deal first with mixtures at finite compositions along the critical locus, followed by consideration of the technologically important case involving dilute mixtures near the solvent’s critical point. Before taking up this discussion, however, we briefly describe some of the main topographic features of the critical line of systems of significant interest: those for which nonvolatile solutes are dissolved in a solvent near its critical point. The critical line divides the P–T plane into two distinctive regions. The area above the line is a one-phase region, while below this line, phase transitions can occur. For example, a mixture of overall composition xc will have a loop associated with it, like the one shown in figure 4.1, which just touches the critical line of the mixture at a unique point. The leg of the curve to the “left” of the critical point is referred to as the bubble line; while that to the right is termed the dew line. Phase equilibrium occurs between two phases at the point where the bubble line at one composition intersects the dew line; this requires two loops to be drawn of the sort shown in figure 4.1. A question naturally arises as to whether or not all binary systems exhibit continuous critical lines like that shown. In particular we are interested in the situation involving a nonvolatile solute dissolved in a supercritical fluid of high volatility.

scholarly journals K-theory for Cuntz-Krieger algebras arising from real quadratic maps

2003 ◽  
Vol 2003 (34) ◽  
pp. 2139-2146 ◽  
Author(s):  
Nuno Martins ◽  
Ricardo Severino ◽  
J. Sousa Ramos
Keyword(s):  
Transition Matrix ◽  
Parameter Family ◽  
Quadratic Maps ◽  
Markov Transition Matrix ◽  
Kneading Sequence ◽  
Markov Transition ◽  
K Theory ◽  
The One ◽  
Real Quadratic

We compute theK-groups for the Cuntz-Krieger algebras𝒪A𝒦(fμ), whereA𝒦(fμ)is the Markov transition matrix arising from the kneading sequence𝒦(fμ)of the one-parameter family of real quadratic mapsfμ.

DYNAMICAL INTEGRATION

1993 ◽  
Vol 03 (01) ◽  
pp. 217-222 ◽  
Author(s):  
RAY BROWN ◽  
LEON O. CHUA
Keyword(s):  
Numerical Methods ◽  
Transformation Group ◽  
Forthcoming Paper ◽  
Parameter Family ◽  
Continuous Transformation ◽  
Highly Nonlinear ◽  
One Step ◽  
Function Approximations ◽  
The One ◽  
Parameter Values

In this letter we show how to use a new form of integration, called dynamical integration, that utilizes the dynamics of a system defined by an ODE to construct a map that is in effect a one-step integrator. This method contrasts sharply with classical numerical methods that utilize polynomial or rational function approximations to construct integrators. The advantages of this integrator is that it uses only one step while preserving important dynamical properties of the solution of the ODE: First, if the ODE is conservative, then the one-step integrator is measure preserving. This is significant for a system having a highly nonlinear component. Second, the one-step integrator is actually a one-parameter family of one-step maps and is derived from a continuous transformation group as is the set of solutions of the ODE. If each element of the continuous transformation group of the ODE is topologically conjugate to its inverse, then so is each member of the one-parameter family of one-step integrators. If the solutions of the ODE are elliptic, then for sufficiently small values of the parameter, the one-step integrator is also elliptic. In the limit as the parameter of the one-step family of maps goes to zero, the one-step integrator satisfies the ODE exactly. Further, it can be experimentally verified that if the ODE is chaotic, then so is the one-step integrator. In effect, the one-step integrator retains the dynamical characteristics of the solutions of the ODE, even with relatively large step sizes, while in the limit as the parameter goes to zero, it solves the ODE exactly. We illustrate the dynamical, in contrast to numerical, accuracy of this integrator with two distinctly different examples: First we use it to integrate the unforced Van der Pol equation for large ∊, ∊≥10 which corresponds to an almost continuous square-wave solution. Second, we use it to obtain the Poincaré map for two different versions of the periodically forced Duffing equation for parameter values where the solutions are chaotic. The dynamical accuracy of the integrator is illustrated by the reproduction of well-known strange attractors. The production of these attractors is eleven times longer when using a conventional fourth-order predictor-corrector method. The theory presented here extends to higher dimensions and will be discussed in detail in a forthcoming paper. However, we caution that the theory we present here is not intended as a line of research in numerical methods for ODEs.

ON A ONE-PARAMETER FAMILY OF EXOTIC SUPERSPACES IN TWO DIMENSIONS

1991 ◽  
Vol 06 (35) ◽  
pp. 3239-3250 ◽  
Author(s):  
MURAT GÜNAYDIN
Keyword(s):  
Two Dimensions ◽  
Parameter Family ◽  
Oscillator Algebra ◽  
Jordan Superalgebras ◽  
Special Values ◽  
Finite Dimensional ◽  
Operator Realization ◽  
The One ◽  
Conformal Superalgebras ◽  
Bosonic Oscillator

Using Jordan algebraic techniques we define and study a family of exotic superspaces in two dimensions with two bosonic and two fermionic coordinates. They are defined by the one-parameter family of Jordan superalgebras JD (2/2)α. For two special values of α the JD (2/2)α can be realized in terms of a single fermionic or a single bosonic oscillator, respectively. For other values of α it can be interpreted as defining an exotic oscillator algebra. The derivation, reduced structure and Möbius superalgebras of JD (2/2)α are identified with the rotation, Lorentz and finite-dimensional conformal superalgebras of the corresponding superspaces. The conformal superalgebras turn out to be the superalgebras D(2,1;α) with the even subgroup SO(2,2)×SU(2) . We give an explicit differential operator realization of the actions of D(2,1;α) on these superspaces.

scholarly journals Optimization of one-parameter family of integration formulae for solving stiff chemical-kinetic ODEs

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Youhi Morii ◽  
Eiji Shima
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Chemical Kinetic ◽  
Parameter Family ◽  
Minimum Error ◽  
Time Stepping ◽  
Time Integration Method ◽  
Dual Time Stepping ◽  
Stiff Odes ◽  
The One

AbstractA fast and robust Jacobian-free time-integration method—called Minimum-error Adaptation of a Chemical-Kinetic ODE Solver (MACKS)—for solving stiff ODEs pertaining to chemical-kinetics is proposed herein. The MACKS formulation is based on optimization of the one-parameter family of integration formulae coupled with a dual time-stepping method to facilitate error minimization. The proposed method demonstrates higher accuracy compared to the method—Extended Robustness-enhanced numerical algorithm (ERENA)—previously proposed by the authors. Additionally, when this method is employed in homogeneous-ignition simulations, it facilitates realization of faster performance compared to CVODE.

The one-way information deficit for a class of two-qubit states

2015 ◽  
Vol 13 (02) ◽  
pp. 1550012
Author(s):  
H. Eftekhari ◽  
E. Faizi
Keyword(s):  
Dynamic Behavior ◽  
Quantum States ◽  
Parameter Family ◽  
General Quantum ◽  
The One ◽  
Qubit States ◽  
Additional Assumptions ◽  
Explicit Expressions

So far, one-way information deficit (OWID) has been calculated explicitly only for Bell-diagonal states and the four-parameter family of X-states with additional assumptions and expressions for more general quantum states are not known. In this paper, we derive explicit expressions for OWID for a larger class of two-qubit states, namely, a five-parameter family of two-qubit states. The dynamic behavior of the OWID under decoherence channel is investigated and it is shown that the OWID is more robust against the decoherence than the entanglement.

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