Accession Number : ADA258838


Title :   Extended Lyapunov Exponents for Second Order Systems


Descriptive Note : Master's thesis


Corporate Author : AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOOL OF ENGINEERING


Personal Author(s) : Horn, Janice M


Full Text : https://apps.dtic.mil/dtic/tr/fulltext/u2/a258838.pdf


Report Date : Dec 1992


Pagination or Media Count : 116


Abstract : It is shown that the concept of the Lyapunov exponent can be extended to include an imaginary part. A numerical technique used to calculate these extended Lyapunov exponents for second order systems is presented. There are two requirements to make this extension possible: the definition of a coordinate frame on the tangent space of the differential equation, and an extension of the classical limit, called the limit of the mean. An application of the technique to the van der Pol equation for the constant coefficient and periodic coefficient cases is given. The extended Lyapunov exponents found using this technique totally agree with the eigenvalues for the constant coefficient case. In the periodic coefficient case, not only do the extended Lyapunov exponents agree with the Poincare exponents calculated using standard Floquet theory, they confirm that the imaginary parts of the Poincare exponents are equal to the quotient + or - i(2pi) tau divided by tau where tau is the period of the trajectory. This imaginary part is uncertain when using standard Floquet theory. Additionally, fast Fourier transform (FFT) techniques are used to validate the existence of the extended Lyapunov exponent and the values obtained for its imaginary part. These techniques show that the power spectrum of relative motion is discrete for the trial cases presented, with the fundamental frequency almost exactly equal to the calculated imaginary part of the extended Lyapunov exponent. Coupled with the successful comparison of characteristic exponents for the constant coefficient and periodic coefficient cases, this power spectrum serves to decisively validate the existence of the extended Lyapunov exponent.


Descriptors :   *EXPONENTIAL FUNCTIONS , *LYAPUNOV FUNCTIONS , FREQUENCY , REQUIREMENTS , STABILITY , COMPARISON , THEORY , MOTION , THESES , EIGENVALUES , COEFFICIENTS , CONSTANTS , COORDINATES , STANDARDS , DIFFERENTIAL EQUATIONS , TANGENTS , FAST FOURIER TRANSFORMS , POWER , FRAMES , TRAJECTORIES , VALUE , MEAN , EQUATIONS


Subject Categories : Numerical Mathematics


Distribution Statement : APPROVED FOR PUBLIC RELEASE