The Szekeres system with cosmological constant term describes the evolution of the kinematic quantities for Einstein field equations in $mathbb{R}^4$. In this study, we investigate the behavior of trajectories in the presence of cosmological constant
. It has been shown that the Szekeres system is a Hamiltonian dynamical system. It admits at least two conservation laws, $h$ and $I_{0}$ which indicate the integrability of the Hamiltonian system. We solve the Hamilton-Jacobi equation, and we reduce the Szekeres system from $mathbb{R}^4$ to an equivalent system defined in $mathbb{R}^2$. Global dynamics are studied where we find that there exists an attractor in the finite regime only for positive valued cosmological constant and $I_0<2.08$. Otherwise, trajectories reach infinity. For $I_ {0}>0$ the origin of trajectories in $mathbb{R}^2$ is also at infinity. Finally, we investigate the evolution of physical properties by using dimensionless variables different from that of Hubble-normalization conducing to a dynamical system in $mathbb{R}^5$. We see that the attractor at the finite regime in $mathbb{R}^5$ is related with the de Sitter universe for a positive cosmological constant.
Extending over three Hubble Space Telescope (HST) cycles, the Hubble Frontier Fields (HFF) initiative constitutes the largest commitment ever of HST time to the exploration of the distant Universe via gravitational lensing by massive galaxy clusters.
We here present models of the mass distribution in the six HFF cluster lenses, derived from a joint strong- and weak-lensing analysis anchored by a total of 88 multiple-image systems identified in existing HST data. The resulting maps of the projected mass distribution and of the gravitational magnification effectively calibrate the HFF clusters as gravitational telescopes. Allowing the computation of search areas in the source plane, these maps are provided to the community to facilitate the exploitation of forthcoming HFF data for quantitative studies of the gravitationally lensed population of background galaxies. Our models of the gravitational magnification afforded by the HFF clusters allow us to quantify the lensing-induced boost in sensitivity over blank-field observations and predict that galaxies at $z>10$ and as faint as m(AB)=32 will be detectable, up to 2 magnitudes fainter than the limit of the Hubble Ultra Deep Field.
