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.

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.

Structural refinement

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Diffraction Experiment ◽  
Structural Refinement ◽  
X Ray Scattering ◽  
Density Map ◽  
Linear Problems ◽  
Refinement Procedure ◽  
Computational Procedures ◽  
Electron Density Map ◽  
Best Fit

At this stage, we have derived a model from an electron-density map and have interpreted it as closely as we can in terms of molecular structure. Provided the job has been done well enough, the next task of improving that interpretation can be left to computational procedures known as structural refinement. If further uninterpreted features of the structure are revealed, it will be necessary to go back to the methods of Chapter 11 to improve the interpretation. The purpose of structural refinement is to adjust a structure to give the best possible fit to the crystallographic observations. The intensities of the Bragg reflections constitute the observations, and the various quantities that define the structure are adjusted to give the best fit. Box 12.1 gives an outline of what is meant by refinement of quantities to fit observations. In structural refinement, a measure of the discrepancies between the calculated X-ray scattering by the model structure and the observed intensities is defined: this is called the refinement parameter. The purpose of the refinement procedure is to alter the model to give the lowest possible refinement parameter. Box 12.1 uses a simple example to bring out some important general points: 1. My model will be specified by a number of variables. In a diffraction experiment, they are usually the coordinates and B factor of every atom. If the number of observed quantities is less than the number of variables, the results can have no validity. 2. If the number of observations equals the number of variables, a perfect fit can be obtained, irrespective of the accuracy of the observations or of the model. (This is true of so-called linear problems, and approximately so in non-linear cases.) 3. If the number of observations exceeds the number of variables by only a small quantity, the estimate of the reliability of the model is questionable. In practice, refinement procedures can only work when there is a sufficient number of observations which are sufficiently accurate. Also, the model must already be good enough to make the refinement procedure meaningful.

Nonlinear Dynamics of a Pendulum Excited by a Crank-Shaft-Slider Mechanism

2014 ◽  
Author(s):  
Rafael H. Avanço ◽  
Hélio A. Navarro ◽  
Reyolando M. L. R. F. Brasil ◽  
José M. Balthazar
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponents ◽  
Nonlinear Model ◽  
Special Interest ◽  
Initial Conditions ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Numerical Techniques ◽  
Vertical Excitation ◽  
Time Histories

In this analysis, we consider the dynamics of a pendulum under vertical excitation of a crank-shaft-slider mechanism. The nonlinear model approaches that of a classical parametrically excited pendulum when the ratio of the length of the shaft to the radius of the crank is very large. Numerical techniques are employed to investigate the results for different parameters and initial conditions. Lyapunov exponents, bifurcation diagrams, time histories and phase portraits are presented to explore conditions when the pendulum performs or not full rotations. Of special interest are the resonance regions. Rotations together with oscillations and chaos were observed in some resonance zones.

Normal forms for almost periodic differential systems

2009 ◽  
Vol 29 (2) ◽  
pp. 637-656 ◽  
Author(s):  
WEIGU LI ◽  
JAUME LLIBRE ◽  
HAO WU
Keyword(s):  
Lyapunov Exponents ◽  
Normal Forms ◽  
Almost Periodic ◽  
Differential Systems

AbstractIn this paper we prove smooth conjugate theorems of Sternberg type for almost periodic differential systems, based on the Lyapunov exponents of the corresponding reduced systems.

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