Perturbation Theory for Approximation of Lyapunov Exponents by QR Methods

2006 ◽  
Vol 18 (3) ◽  
pp. 815-840 ◽  
Author(s):  
Luca Dieci ◽  
Erik S. Van Vleck
Keyword(s):  
Perturbation Theory ◽  
Lyapunov Exponents ◽  
Qr Methods

Codes for Approximating Lyapunov Exponents

2009 ◽  
Author(s):  
Luca Dieci ◽  
Michael S. Jolly ◽  
Erik S. Van Vleck
Keyword(s):  
Lyapunov Exponents ◽  
Jacobian Matrix ◽  
Alternative Strategy ◽  
Differential Systems ◽  
Small Systems ◽  
Large Systems ◽  
Linear Problems ◽  
Qr Methods ◽  
Nonlinear Differential Systems

We present a suite of codes for approximating Lyapunov exponents of nonlinear differential systems by so-called QR methods. The basic solvers perform integration of the trajectory and approximation of the Lyapunov exponents simultaneously. That is, they integrate for the trajectory at the same time, and with the same underlying schemes, as integration for the Lyapunov exponents is carried out. Separate codes solve small systems for which we can compute and store the Jacobian matrix, and for large systems for which the Jacobian matrix cannot be stored, and it may not even be explicitly known. If it is known, the user has the option to provide its action on a vector. An alternative strategy is also presented in which one may want to approximate the trajectory with a specialized solver, linearize around the computed trajectory, and then carry out the approximation of the Lyapunov exponents using codes for linear problems.

On the error in computing Lyapunov exponents by QR Methods

2005 ◽  
Vol 101 (4) ◽  
pp. 619-642 ◽  
Author(s):  
Luca Dieci ◽  
Erik S. Van Vleck
Keyword(s):  
Lyapunov Exponents ◽  
Qr Methods

scholarly journals Numerical Techniques for Approximating Lyapunov Exponents and Their Implementation

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Luca Dieci ◽  
Michael S. Jolly ◽  
Erik S. Van Vleck
Keyword(s):  
Lyapunov Exponents ◽  
Jacobian Matrix ◽  
Alternative Strategy ◽  
Numerical Techniques ◽  
Differential Systems ◽  
Small Systems ◽  
Large Systems ◽  
Linear Problems ◽  
Computational Procedures ◽  
Qr Methods

The algorithms behind a toolbox for approximating Lyapunov exponents of nonlinear differential systems by QR methods are described. The basic solvers perform integration of the trajectory and approximation of the Lyapunov exponents simultaneously. That is, they integrate for the trajectory at the same time, and with the same underlying schemes, as is carried out for integration of the Lyapunov exponents. Separate computational procedures solve small systems for which the Jacobian matrix can be computed and stored, and for large systems for which the Jacobian cannot be stored, and may not even be explicitly known. If it is known, the user has the option to provide the action of the Jacobian on a vector. An alternative strategy is also presented in which one may want to approximate the trajectory with a specialized solver, linearize around the computed trajectory, and then carry out the approximation of the Lyapunov exponents using techniques for linear problems.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

Lyapunov Exponents

2016 ◽  
Author(s):  
Arkady Pikovsky ◽  
Antonio Politi
Keyword(s):  
Lyapunov Exponents
Sign in / Sign up

Export Citation Format

Share Document