ΑΝΑΛΥΤΙΚΕΣ ΚΑΙ ΑΡΙΘΜΗΤΙΚΕΣ ΜΕΘΟΔΟΙ ΧΑΟΤΙΚΗΣ ΔΥΝΑΜΙΚΗΣ

Abstract

THE MAIN GOAL OF THIS THESIS IS TO DEVELOP AND USE ANALYTICAL AS WELL AS NUMERICAL METHODS STUDYING THE SOLUTIONS OF SYSTEMS OF NON LINEAR O.D.E'S WHICHDESCRIBE DYNAMICAL SYSTEMS OF PHYSICAL INTEREST. THE STUDY OF SYSTEMS PRESERVING A "DENSITY" OR "MEASURE" QUANTITY IN TIME, ESPECIALLY THE STUDY OF PERIOD DOUBLING SENARIO, SHOWS THAT SUCH SYSTEMS HAVE THE SAME PROPERTIES WITH HAMILTONIAN ONES. ALSO THE MEASURE PRESERVATION PROPERTY IS DIRECTLY CONNECTED WITH THE REVERSIBILITY ONE, WHERE REVERSING THE TIME, THE VECTOR FIELD OF THE SYSTEM REVERSES. IN THE STUDY OF NONLINEAR OSCILLATIONS THE ACURATE COMPUTATION, STABILITY ANALYSIS AND BIFURCATION PROPERTIES OF PERIODIC SOLUTIONS,ARE OF PRIMITIVE ROLE. GENERALLY SPEAKING, THE SOLUTIONS (OR ORBITS) OF THESE SYSTEMS ARE COMPUTED BY NUMERICAL EXPLORATION OF THE PHASE SPACE, SO THATTHE INITIAL CONDITIONS OF EVERY ORBIT ARE ACCURATELY COMPUTED. BUT IF AN ORBIT IS UNSTABLE, SUCH AN EXPLORATION FAILS OR HAS A BIG NUMERICAL COST, ESPECIALLY IN THE CASE OF THE HIGH PERIOD ORBITS. HERE THE COMPUTATION OF PERIODIC ORBITS USING FOURIER SERIES EXPANSIONS, COMPUTED SOLVING, WITH REPEATIVE METHODS, THE CORRESPONDING ALGEBRAIC SYSTEM WITH (COMPLEX) COEFFICIENTS OF THE SERIES AS UNKNOWNS, SHOWS WHERE THESE METHODS ARE BETTER OR NOT, COMPAREDWITH THE CLASSICAL METHODS SOLVING ALGEBRAIC SYSTEMS AS WELL AS COMPUTING PERIODIC ORBITS. ALSO THE STABILITY ANALYSIS IS MADE USING THE COEFFICIENT OF FOURIER EXPANSIONS USING HILL'S METHOD OF ANALYSIS. (ABSTRACT TRUNCATED)