Nonautonomous difference equations with applications

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.

The possibility of primordial black hole direct detection

This thesis explores the possibility of directly detecting blackbody emission from Primordial Black Holes (PBHs). A PBH might form when a cosmological density uctuation with wavenumber k, that was once stretched to scales much larger than the Hubble radius during ination, reenters inside the Hubble radius at some later epoch. By modeling these uctuations with a running{tilt power{law spectrum (n(k) = n0 + a1(k)n1 + a2(k)n2 + a3(k)n3; n0 = 0:951; n1 = ????0:055; n2 and n3 unknown) each pair (n2,n3) gives a di erent n(k) curve with a maximum value (n+) located at some instant (t+). The (n+,t+) parameter space [(1:20,10????23 s) to (2:00,109 s)] has t+ = 10????23 s{109 s and n+ = 1:20{2:00 in order to encompass the formation of PBHs in the mass range 1015 g{1010M (from the ones exploding at present to the most massive known). It was evenly sampled: n+ every 0.02; t+ every order of magnitude. We thus have 41 33 = 1353 di erent cases. However, 820 of these ( 61%) are excluded (because they would provide a PBH population large enough to close the Universe) and we are left with 533 cases for further study. Although only sub{stellar PBHs ( 1M ) are hot enough to be detected at large distances we studied PBHs with 1015 g{1010M and determined how many might have formed and still exist in the Universe. Thus, for each of the 533 (n+,t+) pairs we determined the fraction of the Universe going into PBHs at each epoch ( ), the PBH density parameter (PBH), the PBH number density (nPBH), the total number of PBHs in the Universe (N), and the distance to the nearest one (d). As a rst result, 14% of these (72 cases) give, at least, one PBH within the observable Universe, one{third being sub{stellar and the remaining evenly spliting into stellar, intermediate mass and supermassive. Secondly, we found that the nearest stellar mass PBH might be at 32 pc, while the nearest intermediate mass and supermassive PBHs might be 100 and 1000 times farther, respectively. Finally, for 6% of the cases (four in 72) we might have substellar mass PBHs within 1 pc. One of these cases implies a population of 105 PBHs, with a mass of 1018 g(similar to Halley's comet), within the Oort cloud, which means that the nearest PBH might be as close as 103 AU. Such a PBH could be directly detected with a probability of 10????21 (cf. 10????32 for low{energy neutrinos). We speculate in this possibility.