Spectra of Stretching Numbers and Helicity Angles in Dynamical Systems

Details Spectra of Stretching Numbers and Helicity Angles in Dynamical Systems Spectra of Stretching Numbers and Helicity Angles in Dynamical Systems

Mar 1996

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1996cemda..64....1c&link_type=abstract

Celestial Mechanics and Dynamical Astronomy, Volume 64, Issue 1-2, pp. 1-20

Astronomy and Astrophysics

Astronomy

23

Lyapunov Characteristic Numbers, Stretching Numbers, Helicity Angles, Conservative And Dissipative Mappings

Scientific paper

We define a “stretching number” (or “Lyapunov characteristic number for one period”) (or “stretching number”) a = In % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada% Wcaaqaaiabe67a4jaadshacqGHRaWkcaaIXaaabaGaeqOVdGNaamiD% aaaaaiaawEa7caGLiWoaaaa!3F1E![left| {{ξ t + 1}/{ξ t}} right|]as the logarithm of the ratio of deviations from a given orbit at times t and t + 1. Similarly we define a “helicity angle” as the angle between the deviation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaam% iDaaaa!3793![ξ t]and a fixed direction. The distributions of the stretching numbers and helicity angles (spectra) are invariant with respect to initial conditions in a connected chaotic domain. We study such spectra in conservative and dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees of freedom. In 2-D conservative systems we found that the lines of constant stretching number have a fractal form.

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