Regina Martínez

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

Abstract The restricted three-body problem is considered for values of the Jacobi constant C near the value C 2 associated to the Euler critical point L 2 . A Lyapunov family of periodic orbits near L 2 , the so-called family ( c ), is born for C = C 2 and exists for values of C less than C 2 . These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ≲ C 2 and μ ≳ 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.

On the optimal station keeping control of halo orbits

Abstract Techniques for computing and controlling a halo orbit are considered in this paper. It presents a semi-analytical theory for the halo orbits in the Restricted Three Bodies Problem (RTBP), that is valid and amenable to computation to any order. Results are presented up to order 11. The Floquet modes of the monodromy matrix are used to define a local optimal control procedure through the concepts of projection and gain functions.

Qualitative study of the planar isosceles three-body problem

We consider the particular case of the planar three body problem obtained when the masses form an isosceles triangle for all time. Various authors [1, 2, 12, 8, 9, 13, 10] have contributed in the knowledge of the triple collision and of several families of periodic orbits in this problem. We study the flow on a fixed level of negative energy. First we obtain a topological representation of the energy manifold including the triple collision and infinity as boundaries of that manifold. The existence of orbits connecting the triple collision and infinity gives some homoclinic and heteroclinic orbits. Using these orbits and the homothetic solutions of the problem we can characterize orbits which pass near triple collision and near infinity by pairs of sequences. One of the sequences describes the regions visited by the orbit, the other refers to the behaviour of the orbit between two consecutive passages by a suitable surface of section. This symbolic dynamics which has a topological character is given in an abstract form and after it is applied to the isosceles problem. We try to keep globality as far as possible. This strongly relies on the fact that the intersection of some invariant manifolds with an equatorial plane (v=0) have nice spiraling properties. This can be proved by analytical means in some local cases. Numerical simulations given in Appendix A make clear that these properties hold globally.

On the stability of tetrahedral relative equilibria in the positively curved 4-body problem

Abstract We consider the motion of point masses given by a natural extension of Newtonian gravitation to spaces of constant positive curvature, in which the gravitational attraction between the bodies acts along geodesics. We aim to explore the spectral stability of tetrahedral orbits of the corresponding 4-body problem in the 2-dimensional case, a situation that can be reduced to studying the motion of the bodies on the unit sphere. We first perform some extensive and highly precise numerical experiments to find the likely regions of stability and instability, relative to the values of the masses and to the latitude of the position of the three equal masses. Then we support the numerical evidence with rigorous analytic proofs in the vicinity of some limit cases in which certain masses are either very large or negligible, or the latitude is close to zero.

Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples

This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order k variational equations, for arbitrary values of k, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear spring-pendulum problem for the values of the parameter that can not be decided using first order variational equations.

Variability in mutational fitness effects prevents full lethal transitions in large quasispecies populations

The distribution of mutational fitness effects (DMFE) is crucial to the evolutionary fate of quasispecies. In this article we analyze the effect of the DMFE on the dynamics of a large quasispecies by means of a phenotypic version of the classic Eigens model that incorporates beneficial, neutral, deleterious, and lethal mutations. By parameterizing the model with available experimental data on the DMFE of Vesicular stomatitis virus (VSV) and Tobacco etch virus (TEV), we found that increasing mutation does not totally push the entire viral quasispecies towards deleterious or lethal regions of the phenotypic sequence space. The probability of finding regions in the parameter space of the general model that results in a quasispecies only composed by lethal phenotypes is extremely small at equilibrium and in transient times. The implications of our findings can be extended to other scenarios, such as lethal mutagenesis or genomically unstable cancer, where increased mutagenesis has been suggested as a potential therapy.

The degree of differentiability of the regularization of simultaneous binary collisions in some N-body problems

Simultaneous binary collisions in the four-body problem are studied in the cases which reduce to one-dimensional problems: the collinear, bi-isosceles and trapezoidal cases and the tetrahedron problem, some of them requiring certain symmetries of the masses. It is shown that in all of these cases the regularization is differentiable but the map passing from initial to final conditions (in some suitable transversal sections) is exactly C8/3. Then the result is extended to other symmetric N-body problems.

Simultaneous binary collisions in the planar four-body problem

We consider simultaneous binary collisions in the general planar four-body problem. We prove they are regularizable in the sense of continuity with respect to initial conditions using a blow-up of the singularity. Furthermore, numerical evidence is given that the differentiability of the regularization should be, in general, less than C8/3 . As a simple example, the double isosceles four-body problem, displays that kind of behaviour.

Abrupt transitions to tumor extinction: a phenotypic quasispecies model

The dynamics of heterogeneous tumor cell populations competing with healthy cells is an important topic in cancer research with deep implications in biomedicine. Multitude of theoretical and computational models have addressed this issue, especially focusing on the nature of the transitions governing tumor clearance as some relevant model parameters are tuned. In this contribution, we analyze a mathematical model of unstable tumor progression using the quasispecies framework. Our aim is to define a minimal model incorporating the dynamics of competition between healthy cells and a heterogeneous population of cancer cell phenotypes involving changes in replication-related genes (i.e., proto-oncogenes and tumor suppressor genes), in genes responsible for genomic stability, and in house-keeping genes. Such mutations or loss of genes result into different phenotypes with increased proliferation rates and/or increased genomic instabilities. Despite bifurcations in the classical deterministic quasispecies model are typically given by smooth, continuous shifts (i.e., transcritical bifurcations), we here identify a novel type of bifurcation causing an abrupt transition to tumor extinction. Such a bifurcation, named as trans-heteroclinic, is characterized by the exchange of stability between two distant fixed points (that do not collide) involving tumor persistence and tumor clearance. The increase of mutation and/or the decrease of the replication rate of tumor cells involves this catastrophic shift of tumor cell populations. The transient times near bifurcation thresholds are also characterized, showing a power law dependence of exponent