scholarly journals A note on operator semigroups associated to chaotic flows

2015 ◽  
Vol 36 (5) ◽  
pp. 1396-1408 ◽  
Author(s):  
OLIVER BUTTERLEY
Keyword(s):  
Dynamical System ◽  
Laplace Transform ◽  
Continuous Time ◽  
Operator Theory ◽  
Transfer Operator ◽  
Operator Semigroups ◽  
Chaotic Flows ◽  
Rapid Mixing ◽  
The Laplace Transform ◽  
Theory Framework

The transfer operator associated to a flow (continuous time dynamical system) is a one-parameter operator semigroup. We consider the operator-valued Laplace transform of this one-parameter semigroup. Estimates on the Laplace transform have been used in various settings in order to show the rate at which the flow mixes. Here we consider the case of exponential mixing and the case of rapid mixing (superpolynomial). We develop the operator theory framework amenable to this setting and show that the same estimates may be used to produce results, in terms of the operators, which go beyond the results for the rate of mixing.

scholarly journals Descriptor fractional linear systems with regular pencils

2013 ◽  
Vol 23 (2) ◽  
pp. 309-315 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Laplace Transform ◽  
Linear Systems ◽  
Discrete Time ◽  
Continuous Time ◽  
The State ◽  
Transition Matrices ◽  
State Equations ◽  
Numerical Examples ◽  
The Laplace Transform ◽  
Time Linear

Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.

Symbolic Calculations

2021 ◽  
pp. 277-310
Keyword(s):  
Dynamical System ◽  
Shear Stress ◽  
Laplace Transform ◽  
Symbolic Calculation ◽  
Lubrication Film ◽  
Algebraic Expressions ◽  
Mathematical Expressions ◽  
Symbolic Calculations ◽  
The Laplace Transform ◽  
Scale Shear

The chapter is devoted to symbolic calculations in which the variables and commands operate on mathematical expressions containing symbolic variables. The representation of a symbolic expression, its simplification, the solution of algebraic expressions, symbolic differentiation and integration, and conversion of the symbolic numbers to their decimal form are described. ODEs solutions are also presented. The final sections of the chapter give examples of the symbolic calculation implementation for some mechanical and tribological problems that were solved numerically in previous chapters, namely lengthening a two-spring scale, shear stress in a lubrication film, a centroid of a certain plate, and two-way solutions of the ODE describing the second order dynamical system – traditional and using the Laplace transform.

Laplace transforms and the renewal equation

1997 ◽  
Vol 34 (02) ◽  
pp. 395-403 ◽  
Author(s):  
Y. Kebir
Keyword(s):  
Laplace Transform ◽  
Continuous Time ◽  
Laplace Transforms ◽  
Renewal Equation ◽  
The Laplace Transform ◽  
Life Distributions

Vinogradov (1973) used the Laplace transform to characterize the IFR class of life distributions and later Block and Savits (1980) extended the characterization to the main reliability classes. Here we use the same transform again to characterize the continuous time renewal equation and some properties of its solution.

Recurrence times of clusters of Poisson points

1969 ◽  
Vol 6 (02) ◽  
pp. 372-388 ◽  
Author(s):  
R.T. Leslie
Keyword(s):  
Laplace Transform ◽  
Continuous Time ◽  
Distribution Functions ◽  
Numerical Inversion ◽  
Bernoulli Trials ◽  
Recurrence Times ◽  
Failure Patterns ◽  
The Laplace Transform

In a previous paper (Leslie (1967)) the distribution of recurrence times for a particular set of success-failure patterns on a sequence of Bernoulli trials was investigated. We now consider the analogous events in continuous time and obtain the Laplace Transform (L.T.) of the distribution of recurrence times; numerical inversion yields the distribution functions.

Laplace transforms and the renewal equation

1997 ◽  
Vol 34 (2) ◽  
pp. 395-403 ◽  
Author(s):  
Y. Kebir
Keyword(s):  
Laplace Transform ◽  
Continuous Time ◽  
Laplace Transforms ◽  
Renewal Equation ◽  
The Laplace Transform ◽  
Life Distributions

Vinogradov (1973) used the Laplace transform to characterize the IFR class of life distributions and later Block and Savits (1980) extended the characterization to the main reliability classes. Here we use the same transform again to characterize the continuous time renewal equation and some properties of its solution.

State space solution of implicit fractional continuous time systems

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Djillali Bouagada ◽  
Paul Dooren
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
State Space ◽  
Caputo Fractional Derivative ◽  
Continuous Time ◽  
State Space Models ◽  
Continuous Time Systems ◽  
The Laplace Transform ◽  
Time Systems ◽  
Time Linear

AbstractIn this work we extend a result from the literature on fractional continuous-time linear systems to the case of implicit fractional continuous-time state space models, based on the Caputo fractional derivative. The solution of the problem is derived using the Laplace transform.

Recurrence times of clusters of Poisson points

1969 ◽  
Vol 6 (2) ◽  
pp. 372-388 ◽  
Author(s):  
R.T. Leslie
Keyword(s):  
Laplace Transform ◽  
Continuous Time ◽  
Distribution Functions ◽  
Numerical Inversion ◽  
Bernoulli Trials ◽  
Recurrence Times ◽  
Failure Patterns ◽  
The Laplace Transform

In a previous paper (Leslie (1967)) the distribution of recurrence times for a particular set of success-failure patterns on a sequence of Bernoulli trials was investigated. We now consider the analogous events in continuous time and obtain the Laplace Transform (L.T.) of the distribution of recurrence times; numerical inversion yields the distribution functions.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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