Exponential energy growth due to slow parameter oscillations in quantum mechanical systems

It is shown that a periodic emergence and destruction of an additional quantum number leads to an exponential growth of energy of a quantum mechanical system subjected to a slow periodic variation of parameters. The main example is given by systems (e.g., quantum billiards and quantum graphs) with periodically divided configuration space. In special cases, the process can also lead to a long period of cooling that precedes the acceleration, and to the desertion of the states with a particular value of the quantum number

Exponential energy growth in adiabatically changing Hamiltonian Systems

Fermi acceleration is the process of energy transfer from massive objects in slow motion to light objects that move fast. The model for such process is a time-dependent Hamiltonian system. As the parameters of the system change with time, the energy is no longer conserved, which makes the acceleration possible. One of the main problems is how to generate a sustained and robust energy growth. We show that the non-ergodicity of any chaotic Hamiltonian system must universally lead to the exponential growth of energy at a slow periodic variation of parameters. We build a model for this process in terms of a Geometric Brownian Motion with a positive drift, and relate it to the entropy increase

Fast Fermi Acceleration and Entropy Growth

Fermi acceleration is the process of energy transfer from massive objects in slow motion to light objects that move fast. The model for such process is a time-dependent Hamiltonian system. As the parameters of the system change with time, the energy is no longer conserved, which makes the acceleration possible. One of the main problems is how to generate a sustained and robust energy growth. We show that the non-ergodicity of any chaotic Hamiltonian system must universally lead to the exponential growth of energy at a slow periodic variation of parameters. We build a model for this process in terms of a Geometric Brownian Motion with a positive drift and relate it to the entropy increase

An example of a resonant homoclinic loop of infinite cyclicity

We describe a codimension-3 bifurcational surface in the space of 퐶푟-smooth (푟 ≥ 3) dynamical systems (with the dimension of the phase space equal to 4 or higher) which consists of systems which have an attractive two-dimensional invariant manifold with an infinite sequence of periodic orbits of alternating stability which converge to a homoclinic loop

Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps

It is shown that every symplectic map of $R^{2n}$ can be approximated, in the $C^\infty$-topology, on any compact set, by some iteration of some map of the form $(x,y)\mapsto (y+\eta, -x +\Phi(y))$ where $x\in R^{n}$, $y\in R^n$, and $\Phi$ is a polynomial $R^n\rightarrow R^n$ and $\eta\in R^n$ is a constant vector. For the case of area-preserving maps (i.e. $n=1$), it is shown how this result can be applied to prove that $C^r$-universal maps (a map is universal if its iterations approximate dynamics of all $C^r$-smooth area-preserving maps altogether) are dense (in the $C^r$-topology) in the Newhouse regions

Fundamental obstacles to self-pulsations in low-intensity lasers

We investigate most general properties of possible laser equations in the case where optics is linear. Exploiting the presence of a natural small parameter (the ratio of the photon lifetime in the laser device to the relaxation time of the population density) we establish the existence of an exponentially attracting invariant manifold which contains all bounded orbits, and show that only a small number of electromagnetic modes is sufficient to describe accurately the dynamics of the system. We give a general form of the reduced few-mode systems. We analyze the behavior of single-mode models and a double-mode model with a single optical frequency. We show that in the case where only one electromagnetic mode is excited, the rate equations are close to integrable ones, so the dynamics in this case can be understood by analytic means (by averaging method). In particular, it is shown that a non-stationary (periodic) output is possible only in relatively small (of order of some fractional powers of the small parameter) regions in the space of parameters of the system near some specially chosen parameter constellations. Estimates on the size of these regions and on the frequency of periodic self-pulsations are given for different situations