20.500.12556/DKUM-47113 Dynamical and statistical properties of time-dependent one-dimensional nonlinear Hamilton systems Dinamične in statistične lastnosti časovno odvisnih enodimenzionalnih nelinearnih hamiltonskih sistemov We study the one-dimensional time-dependent Hamiltonian systems and their statistical behaviour, assuming the microcanonical ensemble of initial conditions and describing the evolution of the energy distribution in three characteristic cases: 1) parametric kick, which by definition means a discontinuous jump of a control parameter of the system, 2) linear driving, and 3) periodic driving. For the first case we specifically analyze the change of the adiabatic invariant (the canonical action) of the system under a parametric kick: A conjecture has been put forward by Papamikos and Robnik (2011) that the action at the mean energy always increases, which means, for the given statistical ensemble, that the Gibbs entropy in the mean increases (PR property). By means of a detailed rigorous analysis of a great number of case studies we show that the conjecture largely is satisfied, except if either the potential is not smooth enough (e.g. has discontinuous first derivative), or if the energy is too close to a stationary point of the potential (separatrix in the phase space). We formulate the conjecture in full generality, and perform the local theoretical analysis by introducing the ABR property. For the linear driving we study first 1D Hamilton systems with homogeneous power law potential and their statistical behaviour under monotonically increasing time-dependent function A(t) (prefactor of the potential). We used the nonlinear WKB-like method by Papamikos and Robnik J. Phys. A: Math. Theor., 44:315102, (2012) and following a previous work by Papamikos G and Robnik M J. Phys. A: Math. Theor., 45:015206, (2011) we specifically analyze the mean energy, the variance and the adiabatic invariant (action) of the system for large time t→∞. We also show analytically that the mean energy and the variance increase as powers of A(t), while the action oscillates and finally remains constant. By means of a number of detailed case studies we show that the theoretical prediction is correct. For the periodic driving cases we study the 1D periodic quartic oscillator and its statistical behaviour under periodic time-dependent function A(t) (prefactor of the potential). We compare the results for three different drivings, the periodic parametrically kicked case (discontinuous jumps of $A(t)$), the piecewise linear case (sawtooth), and the smooth case (harmonic). Considering the Floquet map and the energy distribution we perform careful numerical analysis using the 8th order symplectic integrator and present the phase portraits for each case, the evolution of the average energy and the distribution function of the final energies. In the case where we see a large region of chaos connected to infinity, we indeed find escape orbits going to infinity, meaning that the energy growth can be unbounded, and is typically exponential in time. The main results are published in two papers: Andresas, Batistić and Robnik Phys. Rev. E, 89:062927, (2014) and Andresas and Robnik J. Phys. A: Math. Theor., 47:355102, (2014). Preučujemo enodimenzionalne časovno odvisne Hamiltonove sisteme ter njihovo statistično vedenje, ob predpostavki mikrokanoničnega ansambla začetnih pogojev in opišemo razvoj porazdelitve energije v treh značilnih primerih: 1) parametrično brcanje, ki po definiciji pomeni nezvezne skoke kontrolnega parametra sistema, 2) linearna modulacija ter 3) periodična modulacija. V prvem primeru specifično analiziramo spremembo adiabatske invariante (kanonične akcije) sistema ob paramaterični brci: Papamikos in Robnik (2011) sta predlagala domnevo, da se akcija pri povprečni energiji vedno poveča, kar pomeni, za dani statistični ansambel, da se šrednjem Gibbsova entropija poveča (PR lastnost). Ob podrobni rigorozni analizi cele vrste sistemov pokažemo, da domneva večinoma velja, razen če potencial ni dovolj gladek (n.pr. ima nezvezen prvi odvod), ali če je energija preveč blizu stacionarni točki potenciala (separatrisi v faznem prostoru). Formuliramo domnevo šplošni obliki, izvedemo lokalno teoretično analizo ter uvedemo t.i. ABR lastnost. Za primer linearne modulacije študiramo najprej enodimenzionalne hamiltonske sisteme s homogenim potenčnim potencialom in njihove statistične lastnosti ob monotono naraščajoči časovno odvisni funkciji A(t) (predfaktor potenciala). Uporabimo nelinearno WKB podobno metodo Papamikosa in Robnika J. Phys. A: Math. Theor., 44:315102, (2012) in sledimo njunemu delu v članku J. Phys. A: Math. Theor., 45:015206, (2011), ter specifično analiziramo povprečno energijo, varianco in adiabatsko invarianto (akcijo) sistema za velike čase t→∞. Analitično pokažemo, da se povprečna energija in varianca povečujeta kot potenci A(t), medtem ko akcija oscilira and na koncu zavzame konstantno vrednost. Na osnovi študija posebnih primerov pokažemo, da je teoretična napoved točna. Za primer periodične modulacije preučujemo enodimenzionalni kvartični oscilator in njegovo statistično vedenje, ko je A(t) (predfaktor potenciala) periodična funkcija časa. Primerjamo rezultate za tri različne modulacije, periodično parametrično brcanje (nezvezni skoki parametra A(t)), odsekoma linearno modulacjo (žagasto), ter gladko (harmonično) modulacijo. Obravnavamo Floquetovo preslikavo in porazdelitev energije ter izvedemo temeljito numerično analizo uporabljajoč simplektične integratorje 8. reda ter predstavimo fazne portrete za vsak primer, razvoj povprečne energije ter porazdelitev končne energije. V primerih, ko vidimo veliko kaotično območje povezano do neskončnosti, najdemo ubežne orbite, ki pobegnejo v neskončnost, kar pomeni, da je lahko rast povprečne energije neomejena, in sicer je tipično eksponentna v času. Glavni rezultati so objavljeni v dveh člankih: Andresas, Batistić in Robnik Phys. Rev. E, 89:062927, (2014) in Andresas in Robnik J. Phys. A: Math. Theor., 47:355102, (2014). one-dimensional nonlinear Hamiltonian systems adiabatic invariant parametric kick periodic driving linear driving energy distribution WKB method action enodimenzionalni nelinearni Hamiltonski sistemi adiabatska invarianta parametrično brcanje periodična modulacija linearna modulacija energetska porazdelitev WKB metoda akcija true true false D. Andresas] Angleški jezik Slovenski jezik Doktorsko delo/naloga 2014-12-18 14:18:07 2015-03-02 15:08:12 2022-05-19 03:07:44 2021-05-05 03:51:18 2015 0 [Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 530.182(043.3) 21209096 URN:SI:UM:DK:EUJJCV1D DOK_Andresas_Dimitrios_2015.pdf DOK_Andresas_Dimitrios_2015.pdf 1 48ACC2F1EAE84578FD868977DB300727 5b4869c8c9fd45d0e6055dcec71c51745c9d3e1930f68731d48fdbf3f6b74a81 8edcd28c-7c0b-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=70206 Fakulteta za naravoslovje in matematiko 0 0 0