Topological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle

Agraïments: the second author is is partially supported by CNPq grant "Projeto Universal" 472796/2013-5, by CAPES CSF-PVE-88881.030454/2013-01, by Projeto Temático FAPESP number 2014/00304-2. The third author is supported by CNPq-PDE 232336/2014-8.The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family QTS of all real quadratic polynomial differential systems which have a finite semi-elemental triple saddle (triple saddle with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this normal form. This bifur- cation diagram yields 27 phase portraits for systems in QTS counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set and we present the phase portraits on the Poincar ́e disk. The bifurcation set is not just algebraic due to the presence of a surface found numerically, whose points correspond to connections of separatrices

Spanning trees of 3-uniform hypergraphs

Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs -- one of these is given by a forbidden subgraph characterization analogous to Little's for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples. We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.Comment: 34 pages, 9 figure

The effects of external planets on inner systems: multiplicities, inclinations, and pathways to eccentric warm Jupiters

We study how close-in systems such as those detected by Kepler are affected by the dynamics of bodies in the outer system. We consider two scenarios: outer systems of giant planets potentially unstable to planet--planet scattering, and wide binaries that may be capable of driving Kozai or other secular variations of outer planets' eccentricities. Dynamical excitation of planets in the outer system reduces the multiplicity of Kepler-detectable planets in the inner system in $\sim20-25\%$ of our systems. Accounting for the occurrence rates of wide-orbit planets and binary stars, $\approx18\%$ of close-in systems could be destabilised by their outer companions in this way. This provides some contribution to the apparent excess of systems with a single transiting planet compared to multiple, however, it only contributes at most $25\%$ of the excess. The effects of the outer dynamics can generate systems similar to Kepler-56 (two coplanar planets significantly misaligned with the host star) and Kepler-108 (two significantly non-coplanar planets in a binary). We also identify three pathways to the formation of eccentric warm Jupiters resulting from the interaction between outer and inner systems: direct inelastic collision between an eccentric outer and an inner planet, secular eccentricity oscillations that may "freeze out" when scattering resolves in the outer system; and scattering in the inner system followed by "uplift", where inner planets are removed by interaction with the outer planets. In these scenarios, the formation of eccentric warm Jupiters is a signature of a past history of violent dynamics among massive planets beyond $\sim1$ au.Comment: 24 pages, 19 figures. Accepted to MNRA

The Mass-Loss Induced Eccentric Kozai Mechanism: A New Channel for the Production of Close Compact Object-Stellar Binaries

Over a broad range of initial inclinations and eccentricities an appreciable fraction of hierarchical triple star systems with similar masses are essentially unaffected by the Kozai-Lidov mechanism (KM) until the primary in the central binary evolves into a compact object. Once it does, it may be much less massive than the other components in the ternary, enabling the "eccentric Kozai mechanism (EKM):" the mutual inclination between the inner and outer binary can flip signs driving the inner binary to very high eccentricity, leading to a close binary or collision. We demonstrate this "Mass-loss Induced Eccentric Kozai" (MIEK) mechanism by considering an example system and defining an ad-hoc minimal separation between the inner two members at which tidal affects become important. For fixed initial masses and semi-major axes, but uniform distributions of eccentricity and cosine of the mutual inclination, ~10% of systems interact tidally or collide while the primary is on the MS due to the KM or EKM. Those affected by the EKM are not captured by earlier quadrupole-order secular calculations. We show that fully ~30% of systems interact tidally or collide for the first time as the primary swells to AU scales, mostly as a result of the KM. Finally, ~2% of systems interact tidally or collide for the first time after the primary sheds most of its mass and becomes a WD, mostly as a result of the MIEK mechanism. These findings motivate a more detailed study of mass-loss in triple systems and the formation of close NS/WD-MS and NS/WD-NS/WD binaries without an initial common envelope phase.Comment: 12 pages, 6 figures, 1 table. Accepted for publication in ApJ. For a brief video explaining this paper, see http://youtu.be/4CdTOF17q5

Dynamical Formation of Close Binaries During the Pre-main-sequence Phase

Solar-type binaries with short orbital periods ($P_{\rm close}$ $\equiv$ 1 - 10 days; $a$ $\lesssim$ 0.1 AU) cannot form directly via fragmentation of molecular clouds or protostellar disks, yet their component masses are highly correlated, suggesting interaction during the pre-main-sequence (pre-MS) phase. Moreover, the close binary fraction of pre-MS stars is consistent with that of their MS counterparts in the field ($F_{\rm close}$ = 2.1%). Thus we can infer that some migration mechanism operates during the early pre-MS phase ($\tau$ $\lesssim$ 5 Myr) that reshapes the primordial separation distribution. We test the feasibility of this hypothesis by carrying out a population synthesis calculation which accounts for two formation channels: Kozai-Lidov (KL) oscillations and dynamical instability in triple systems. Our models incorporate (1) more realistic initial conditions compared to previous studies, (2) octupole-level effects in the secular evolution, (3) tidal energy dissipation via weak-friction equilibrium tides at small eccentricities and via non-radial dynamical oscillations at large eccentricities, and (4) the larger tidal radius of a pre-MS primary. Given a 15% triple star fraction, we simulate a close binary fraction from KL oscillations alone of $F_{\rm close}$ $\approx$ 0.4% after $\tau$ = 5 Myr, which increases to $F_{\rm close}$ $\approx$ 0.8% by $\tau$ = 5 Gyr. Dynamical ejections and disruptions of unstable coplanar triples in the disk produce solitary binaries with slightly longer periods $P$ $\approx$ 10 - 100 days. The remaining $\approx$60% of close binaries with outer tertiaries, particularly those in compact coplanar configurations with log $P_{\rm out}$ (days) $\approx$ 2 - 5 ($a_{\rm out}$ $<$ 50 AU), can be explained only with substantial extra energy dissipation due to interactions with primordial gas.Comment: Accepted by ApJ; 23 pages; 8 figures; this version incorporates changes made to address comments by refere