University of Maribor, Slovenia

Recently the interest in time-dependent dynamical systems has increased a lot. In this talk I shall present most recent results on time-dependent one-dimensional Hamiltonian oscillators. The time-dependence describes the interaction of an oscillator with its neigbourhood. While the Liouville theorem still applies (the phase space volume is preserved), the energy of the system changes with time. We are interested in the statistical properties of the energy of an initial microcanonical ensemble with sharply defiend initial energy, but uniform distribution of the initial conditions with respect to the canonical angle. We are in particular interested in the change of the action at the average energy, which is also adiabatic invariant, and is conserved in the ideal adibatic limit, but otherwise changes with time. It will be shown that in the linear oscillator the value of the adiabatic invariant always increases, implying the increase of the Gibbs entropy in the mean (at the average energy). The energy is universally described by the arcsine distribution, independent of the driving law. In nonlinear oscillators things are different. For slow but not yet ideal adiabatic drivings the adiabatic invariant at the mean energy can decrease, just due to the nonlinearity and nonisochronicity, but nevertheless increases at faster drivings, including the limiting fastest possible driving, namely parametric kick (jump of the parameter). This is so-called PR property, following Papamikos and Robnik {\em J. Phys. A: Math. Theor. {\bf 45} (2011) 015206}, proven rigorously to be satisfied in a number of model potentials, such as homogeneous power law potential, and many others, giving evidence that the PR property is always sastified in a parametric kick, except if we are too close to a separatrix or if the potential is not smooth enough. The local analysis is possible and the PR property is formulated in terms of a geometrical criterion for the underlying potential. We also study the periodic kicking and the strong (nonadiabatic) linear driving of the quartic oscillator. In the latter case we employ the nonlinear WKB method following Papamikos and Robnik {\em J. Phys. A: Math. Theor. {\bf 44} (2012) 315102} and calculate the mean energy and the variance of the energy distribution, and also the adiabatic invariant which is asymptotically constant, but slightly higher than its initial value. The key references for the most recent work are Andresas et al (2014)