Duffingo lygtis – tai antro laipsnio netiesinė diferencialinė lygtis. Tai yra dinaminė sistema, kurioje nagrinėjamas chaotinis elgesys. Magistro darbe mes išnagrinėjome Duffingo lygtį su pradinėmis sąlygomis ir tyrėme lygties sprendinių sinchronizavimą. Buvo sukurta programa, kuri sprendžia duotą uždavinį laiko t intervale ir apskaičiuoja sprendinių skirtumus. Šią programą modifikavome, priklausomai nuo reikalaujamų rezultatų. Tai padėjo greičiau surasti reikalingus duomenis ir jas panaudoti brėžiniuose ir išvadose. Pirmoje dalyje mes išnagrinėjome Duffingo lygties sprendinius ir jų skirtumus. Skaitinio modeliavimo rezultatus analizavome naudodami grafikus. Buvo nustatyti tokie parametrai, kada visi sprendiniai artėja į vieną sprendinį. Buvo išsiaiškinta sprendinių konvergavimo priklausomybė nuo šių parametrų. Antroje dalyje mus domino laikas, kai sprendiniai sutampa. Priklausomai nuo skirtingų parametrų sutapimo laikas elgiasi skirtingai. Trečioje dalyje nagrinėjome Duffingo lygtį, kai ir . Apskaičiavome sprendinių periodus bei išanalizavome sprendinių energijos keitimą. Patikrinome energijos tvermės dėsnį. / The Duffing equation is a non-linear second-order differential equation. It is an example of a dynamical system that exhibits chaotic behavior. In my final degree project we have investigated Duffing‘s equation with initial conditions and researched synchrony of solutions. A special program was created, it solves deteminated problem in t time range and finds the difference between answers. Depending on results, this program was modified in order to get the results needed. In first part we have looked at Duffing equation‘s answers and their difference. Numeric modeling results were studied using graphs. Parameters were chosen for all results be converged to one solution only. As well, we studied their convergence depending on those parameters. In the second part, we were interested in the time period when results match. The third part was dedicated to Duffing‘s equation, when and . We found periods for our solutions and researched their energy. Law of the energy conservation was double-checked as well.

This job is considered about mobile technologies, and mobile devices data synchronization problem. Presented trading agent‘s mobile work places and server data synchronization solution, wich was developed using free software.

Information synchronization problems are analyzed in this thesis. Two aspects are being surveyed – clock synchronization, algorithms and their use, and data synchronization and maintaining the functionality of software at the times, when connection with database is broken. Existing products, their uses, cons and pros are overviewed. There are suggested models, how to solve these problems, which were implemented in “Distributed basketball competition registration and analysis software system”, and other theoretical solutions. Synchronization models are compared with other available solutions, the detail algorithms and methods are reviewed in this document.