Opis: 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). Ključne besede:one-dimensional nonlinear Hamiltonian systems, adiabatic invariant, parametric kick, periodic driving, linear driving, energy distribution, WKB method, action Objavljeno: 02.03.2015; Ogledov: 812; Prenosov: 15 Celotno besedilo (11,07 MB)

Opis: This doctoral dissertation is devoted to the studies of some qualitative properties of certain polynomial systems of ordinary differential equations. The main problems that are considered in this thesis are the problems of integrability and cyclicity. Some results on the classification of the global phase portraits of a family of cubic systems are presented as well. In the first chapter basic notions and results of the qualitative theory of systems of ODE's are introduced. Since one of important tools for our study of these problems is the commutative computational algebra, some main notions and properties of polynomial ideals and their varieties, including various algorithms related to them, are also presented in the introduction. In the second chapter methods for investigation of trajectories near degenerated singularities are presented. They are further used for the classification of global phase portraits of a family of cubic systems with the nilpotent center at the origin. In the third chapter the main problem of these thesis is studied, the problem of integrability. The problem of integrability which is connected to the problem of distinguishing between a center and a focus is studied for two different families of cubic polynomial systems of ODE's. With the computational algebra approach the necessary conditions for the existence of the first integral of these systems were obtained. For all but one condition was proven, using various approaches, the existence of the first integrals. The center problem for the real systems can be generalized to the complex systems. The origin of the system obtained after the complexification of the real system is the so-called 1:-1 resonant singular point, from which one additional generalization follows. This is the generalization to the p:-q resonant center. In the third chapter the :-3 resonant singular point of a quadratic family of complex systems is studied. The fourth chapter is devoted to the study of the problem of integrability of a three dimensional polynomial system with quadratic nonlinearities. The problem of existence of two independent first integrals and the existence of one first integral in the system was investigated. In the last chapter local bifurcations of limit cycles of a family of cubic systems are studied. Estimations for the number of limit cycles bifurcated from each components of the center variety are obtained. Ključne besede:planar systems of ODE's, higher dimensional systems of ODE's, phase portrait, nilpotent center, limit cylces, Poincaré compactification, center problem, Bautin ideal, focus quantities, time-reversibility, integrability problem, Darboux method, linearizability, limit cycle, cyclicity Objavljeno: 19.07.2016; Ogledov: 448; Prenosov: 35 Celotno besedilo (12,26 MB)

Opis: Based on the pseudo-division algorithm, we introduce a method for computing focal values of a class of 3-dimensional autonomous systems. Using the $Є^1$-order focal values computation, we determine the number of limit cycles bifurcating from each component of the center variety (obtained by Mahdi et al). It is shown that at most four limit cycles can be bifurcated from the center with identical quadratic perturbations and that the bound is sharp. Ključne besede:algorithms, three dimensional systems, focal value, limit cycle, Hopf bifurcation, center Objavljeno: 08.08.2017; Ogledov: 175; Prenosov: 9 Celotno besedilo (236,33 KB)