Vol.4, No 16, 2004 pp. 69-75
UDC 534.14+530.13(045)
THE GOLDEN ROUTE TO CHAOS
Ljubiša M. Kocić1, Liljana R. Stefanovska2
1Faculty of Electronic Engineering, University of Niš,
P. O. Box 73, 18000 Niš, Serbia and Montenegro
e-mail: kocic@elfak.ni.ac.yu
2Faculty of Technology and Metallurgy,
SS Cyril and Methodius University, P. O. Box 580, 1000 Skopje, Rep. of Macedonia
e-mail: liljana@ereb1.mf.ukim.edu.mk
Abstract. A strict mathematical description of appearance of a deterministic chaos in the set of orbits of the critical circle map ?n+1?= ?n + ??– (1/2?) sin(2??n) | mod 1, is considered. All relevant terms such as coupled oscillators, self-similar winding numbers, mode-locking, Farey sequences, Arnold tongues are thoroughly discussed. The locked-ratios of Fibonacci numbers that lie on the zigzag path on the Farey-tree approaching the golden mean ? = (?5 – 1)/2, or its unitary complement, which is the stem of the expression "golden route to chaos". This pattern occurs in many dynamical systems such as negative resistance circuits, biochemical and chemical oscillators, thermofluid convections, lasers, cortical neural oscillators and so on.
ZLATNI PUT U HAOS
U radu se razmatra pojava deterministickog haosa na skupu orbita kritičnog kružnog preslikavanja ?n+1?= ?n + ??? (1/2?)sin(2??n) ? mod 1, i daje se njegova stroga matematička interpretacija. Detaljno se obrađuju svi pridruženi fenomeni, kao sto su spregnuti oscilatori, samo-slični zavojni brojevi, zaključavanja faze, Fareyevi nizovi i Arnoldovi jezici. Fibonaccievi brojni odnosi zaključavanja koji leže na cik-cak putanji Fareyevog drveta, i koji konvergiraju ka odnosu zlatnog preseka ? = (?5 ? 1)/2 ili ka njegovom jediničnom komplementu, objašnjavaju naziv "zlatni put u haos". Ovakva šema prelaska u haos je karakteristična za mnoge dinamičke sisteme, kao na primer za električna kola sa negativnom otpornošću, za biohemijske i hemijske oscilatore, za provodjenje toplote fluidima, za lasere, za kortikalne neuro-oscilatore itd.