Evolution of the L1 halo family in the radial solar sail CRTBP

We present a detailed investigation of the dramatic changes that occur in the $\mathcal{L}_1$ halo family when radiation pressure is introduced into the Sun-Earth circular restricted three-body problem (CRTBP). This photo-gravitational CRTBP can be used to model the motion of a solar sail orientated perpendicular to the Sun-line. The problem is then parameterized by the sail lightness number, the ratio of solar radiation pressure acceleration to solar gravitational acceleration. Using boundary-value problem numerical continuation methods and the AUTO software package (Doedel et al. 1991) the families can be fully mapped out as the parameter $\beta$ is increased. Interestingly, the emergence of a branch point in the retrograde satellite family around the Earth at $\beta\approx0.0387$ acts to split the halo family into two new families. As radiation pressure is further increased one of these new families subsequently merges with another non-planar family at $\beta\approx0.289$, resulting in a third new family. The linear stability of the families changes rapidly at low values of $\beta$, with several small regions of neutral stability appearing and disappearing. By using existing methods within AUTO to continue branch points and period-doubling bifurcations, and deriving a new boundary-value problem formulation to continue the folds and Krein collisions, we track bifurcations and changes in the linear stability of the families in the parameter $\beta$ and provide a comprehensive overview of the halo family in the presence of radiation pressure. The results demonstrate that even at small values of $\beta$ there is significant difference to the classical CRTBP, providing opportunity for novel solar sail trajectories. Further, we also find that the branch points between families in the solar sail CRTBP provide a simple means of generating certain families in the classical case.Comment: 31 pages, 17 figures, accepted by Celestial Mechanics and Dynamical Astronom

Numerical computation of nonlinear normal modes in mechanical engineering

This paper reviews the recent advances in computational methods for nonlinear normal modes (NNMs). Different algorithms for the computation of undamped and damped NNMs are presented, and their respective advantages and limitations are discussed. The methods are illustrated using various applications ranging from low-dimensional weakly nonlinear systems to strongly nonlinear industrial structures. © 2015 Elsevier Ltd

Critical homoclinics in a restricted four body problem: numerical continuation and center manifold computations

The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four body problem. The problem can be viewed as a two parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for computing normal forms and center stable/unstable manifold parameterizations. After careful numerical study of a number of special cases we formulate several conjectures about the global bifurcations of the homoclinic families.Comment: 38 pages, 21 figures, fixed several typos, expanded the introduction and added a new appendix about numerical continuation of orbit

Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields

We consider a homoclinic bifurcation of a vector field in [\R^3] , where a one-dimensional unstable manifold of an equilibrium is contained in the two-dimensional stable manifold of this same equilibrium. How such one-dimensional connecting orbits arise is well understood, and software packages exist to detect and follow them in parameters. In this paper we address an issue that it is far less well understood: how does the associated two-dimensional stable manifold change geometrically during the given homoclinic bifurcation? This question can be answered with the help of advanced numerical techniques. More specifically, we compute two-dimensional manifolds, and their one-dimensional intersection curves with a suitable cross-section, via the numerical continuation of orbit segments as solutions of a boundary value problem. In this way, we are able to explain how homoclinic bifurcations may lead to quite dramatic changes of the overall dynamics. This is demonstrated with two examples. We first consider a Shilnikov bifurcation in a semiconductor laser model, and show how the associated change of the two-dimensional stable manifold results in the creation of a new basin of attraction. We then investigate how the basins of the two symmetrically related attracting equilibria change to give rise to preturbulence in the first homoclinic explosion of the Lorenz system

Pattern formation in nanoparticle suspensions: a Kinetic Monte Carlo approach

Various experimental settings that involve drying solutions or suspensions of nanoparticles often called nano-fluids have recently been used to produce structured nanoparticle layers. In addition to the formation of polygonal networks and spinodal-like patterns, the occurrence of branched structures has been reported. After reviewing the experimental results, the work presented in this thesis relies only on simulations. Using a modified version of the Monte Carlo model first introduced by Rabani et al. [95] the study of structure formation in evaporating films of nanoparticle solutions for the case that all structuring is driven by the interplay of evaporating solvent and diffusing nanoparticles is presented. The model has first been used to analyse the influence of the nanoparticles on the basic dewetting behaviour, i.e., on spinodal dewetting and on dewetting by nucleation and growth of holes. We focus, as well, on receding dewetting fronts which are initially straight but develop a transverse fingering instability. One can analyse the dependence of the characteristics of the resulting branching patterns on the driving effective chemical potential, the mobility and concentration of the nanoparticles, and the interaction strength between liquid and nanoparticles. This allows to understand the underlying fingering instability mechanism. We describe briefly how the combination of a Monte Carlo model with a Genetic Algorithm (GA) can be developed and used to tune the evolution of a simulated self-organizing nanoscale system toward a predefined nonequilibrium morphology. This work has presented evolutionary computation as a method for designing target morphologies of self-organising nano-structured systems. Finally, highly localised control of 2D pattern formation in colloidal nanoparticle arrays via surface inhomogeneities created by atomic force microscope (AFM) induced oxidation is presented and some simulations are shown. Furthermore, the model can be extended further, and by including the second type of nanoparticle, the binary mixture behaviour can be captured by simulations. We conclude that Kinetic Monte Carlo simulations have allowed the study of the processes that lead to the production of particular nanoparticle morphologies