Fractal analysis of Hopf bifurcation for a class of completely integrable nonlinear Schrödinger Cauchy problems

We study the complexity of solutions for a class of completely integrable, nonlinear integro-differential Schrödinger initial-boundary value problems on a bounded domain, depending on a real bifurcation parameter. The considered Schrödinger problem is a natural extension of the classical Hopf bifurcation model for planar systems into an infinite-dimensional phase space. Namely, the change in the sign of the bifurcation parameter has a consequence that an attracting (or repelling) invariant subset of the sphere in $L^2(\Omega)$ is born. We measure the complexity of trajectories near the origin by considering the Minkowski content and the box dimension of their finite-dimensional projections. Moreover we consider the compactness and rectifiability of trajectories, and box dimension of multiple spirals and spiral chirps. Finally, we are able to obtain the box dimension of trajectories of some nonintegrable Schrödinger evolution problems using their reformulation in terms of the corresponding (not explicitly solvable) dynamical systems in $\mathbb{R}^n$

Fractal analysis of Hopf bifurcation for a class of completely integrable nonlinear Schrödinger Cauchy problems

We study the complexity of solutions for a class of completely integrable, nonlinear integro-differential Schrödinger initial-boundary value problems on a bounded domain, depending on a real bifurcation parameter. The considered Schrödinger problem is a natural extension of the classical Hopf bifurcation model for planar systems into an infinite-dimensional phase space. Namely, the change in the sign of the bifurcation parameter has a consequence that an attracting (or repelling) invariant subset of the sphere in $L^2(\Omega)$ is born. We measure the complexity of trajectories near the origin by considering the Minkowski content and the box dimension of their finite-dimensional projections. Moreover we consider the compactness and rectifiability of trajectories, and box dimension of multiple spirals and spiral chirps. Finally, we are able to obtain the box dimension of trajectories of some nonintegrable Schrödinger evolution problems using their reformulation in terms of the corresponding (not explicitly solvable) dynamical systems in $\mathbb{R}^n$

Contrasting life history and reproductive traits in two populations of Scyliorhinus canicula

The role of natural and sexual selection in generating variability in biological traits between populations represents an intriguing issue in evolutionary biology. Considering their occurrence in different environments and the extensive incidence of post-copulatory sexual selection, elasmobranchs represent an interesting, yet still poorly investigated group. In this study, the life history and reproductive traits of two populations of Scyliorhinus canicula from the northern Adriatic Sea and the Strait of Sicily were compared. Differences in maximum size and size at sexual maturity were observed. The two populations also displayed differences in male and female genitalia. Males in the northern Adriatic Sea presented heavier testes, longer epididymis, seminal vesicles and claspers compared with those in the Strait of Sicily, suggesting the occurrence of stronger sperm competition at the former site. Similarly, females in the northern Adriatic Sea showed heavier oviducal glands and longer reproductive tracts compared with those in the Strait of Sicily. The coevolution between male and female genitalia suggests the occurrence of stronger sexual conflict and/or cryptic female choice in the population from the northern Adriatic Sea. Therefore, natural selection, represented by the different selective pressures occurring at different latitudes, and sexual selection, represented by potentially differing strengths of post-copulatory sexual selection and sexual conflict, may act individually in driving divergence in life history and reproductive traits in these two populations of S. canicula