Multi-clustered solutions for a singularly perturbed forced pendulum equation

In this paper, we are concerned with unbounded solutions of the singularly perturbed forced pendulum equation in the presence of friction, namely $$\varepsilon ^2u_\varepsilon ^{{\prime}{\prime}} + \sin u_\varepsilon = \varepsilon ^2\alpha (t)u_\varepsilon + \varepsilon ^2\beta (t)u_\varepsilon ^{\prime} \quad {\rm in}\;(-L,L).{\rm }$$Using a limiting energy function, we describe the behaviour of the solutions as the parameter ε approaches zero. We also prove the existence of a family of solutions having a prescribed asymptotic profile and exhibiting a highly rotatory behaviour alternated with a highly oscillatory behaviour in some open subsets of the domain. The proof relies on a combination of the Nehari finite dimensional reduction with the topological degree theory.

Statistical complexity of the quasiperiodical damped systems

We consider the concept of statistical complexity to write the quasiperiodical damped systems applying the snapshot attractors. This allows us to understand the behaviour of these dynamical systems by the probability distribution of the time series making a difference between the regular, random and structural complexity on finite measurements. We interpreted the statistical complexity on snapshot attractor and determined it on the quasiperiodical forced pendulum.

Ratchet baryogenesis and an analogy with the forced pendulum

A new scenario of baryogenesis via the ratchet mechanism is proposed based on an analogy with the forced pendulum. The oscillation of the inflaton field during the reheating epoch after inflation plays the role of the driving force, while the phase [Formula: see text] of a scalar baryon field (a complex scalar field with baryon number) plays the role of the angle of the pendulum. When the inflaton is coupled to the scalar baryon, the behavior of the phase [Formula: see text] can be analogous to that of the angle of the forced pendulum. If the oscillation of the driving force is adjusted to the pendulum’s motion, a directed rotation of the pendulum is obtained with a nonvanishing value of [Formula: see text], which models successful baryogenesis since [Formula: see text] is proportional to the baryon number density. Similar ratchet models which lead to directed motion have been used in the study of molecular motors in biology. There, the driving force is supplied by chemical reactions, while in our scenario this role is played by the inflaton during the reheating epoch.