| This thesis contains two separate topics. The first topic concerns proof of a theorem that justifies the method of reconstruction of dynamics using inter-event time intervals. In particular, we prove that the function from an invariant set of a typical dynamical system into R d, defined by successive inter-event time intervals from integrate-and-fire dynamics, is prevalently a topological embedding. This allows topological information about a dynamical attractor to be inferred from spike trains.;The second topic is the active processes of particles advected by chaotic flows. While previous studies focused on the active processes of massless point particles, or an ideal tracer, we discuss the particles with finite mass and size. Their equations of motion are inherently dissipative, due to the Stokes drag. The dynamics of the advected particles can be chaotic even with a flow field that is simply time-periodic. Similarly to the case of ideal tracers, whose dynamics is Hamiltonian, chemical or biological activity involving such particles advected by fluid flows can be analyzed using the theory of chaotic dynamics. We choose the cellular vortex flow field with periodically varying vorticity as an example, and analyze the dynamics of the reaction of autocatalytic type, A + B → 2 B, and of coalescence type, B + B → B.;Another assumption that the previous studies on the active processes had, was that the reaction of all particles in the system occurs simultaneously. Here we investigate the effect of asynchronism of the autocatalytic reaction taking place in an open hydrodynamical flow, by assigning each particle in the system with a phase to differentiate the timing of their reactions, but not their periodicity. The chaotic saddle in the flow dynamics acts as a catalyst and enhances the reaction in the same fashion as in the case of synchronous reaction that was studied previously. However, we show that, in certain range of a parameter, the group of particles with a particular phase can be favored over the others, thus occupying a larger part of the available space, or eventually, leading to extinction of the unfavored phase. |
