The paper presents results of numerical simulations on a model free boundary problem which is qualitatively equivalent to the free interface problems describing solid combustion and exothermic phase transitions. The model problem has been recently shown to exhibit transition to chaotic oscillations via a sequence of period doubling, assuming an Arrhenius type boundary kinetics. In the present paper we demonstrate that for a slightly different class of kinetics the behavior pattern, while retaining the above scenario, may undergo a drastic change. This behavior is characterized by slowly expanding oscillations followed by a powerful burst, after which the system returns to near equilibrium and the scenario is repeated periodically. As the bifurcation parameter approaches the stability threshold, the total period tends to infinity due to an increasingly prolonged “accumulation phase.” Additional scenarios corresponding to increasing supercriticality of the bifurcation parameter include finite period doubling sequences that return to simple periodic regime via a reversed cascade, infinite sequences followed by an interval of chaos and reversed sequences, or a combination of period doubling, chaos and “Shilnikov type” orbits. Our observations suggest that major dynamical patterns are qualitatively kinetics independent and, moreover, may be essentially finite-dimensional. We argue that for a deeper understanding of the mechanisms responsible for the formation of various patterns a further analytical study (including perhaps a reduction to a finite-dimensional model) of the free boundary problem is desirable.
Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition
