Observations of exoplanets over the last two decades have revealed a new class of Jupitersize planets with orbital periods of a few days, the so-called "hot Jupiters". Recent measurements using the Rossiter-McLaughlin effect have shown that many (∼ 50%) of these planets are misaligned; furthermore, some (∼ 15%) are even retrograde with respect to the stellar spin axis. Motivated by these observations, we explore the possibility of forming retrograde orbits in hierarchical triple configurations consisting of a star-planet inner pair with another giant planet, or brown dwarf, in a much wider orbit. Recently Naoz et al. (2011) showed that in such a system, the inner planet's orbit can flip back and forth from prograde to retrograde, and can also reach extremely high eccentricities. Here we map a significant part of the parameter space of dynamical outcomes for these systems. We derive strong constraints on the orbital configurations for the outer perturber (the tertiary) that could lead to the formation of hot Jupiters with misaligned or retrograde orbits. We focus only on the secular evolution, neglecting other dynamical effects such as mean-motion resonances, as well as all dissipative forces. For example, with an inner Jupiter-like planet initially on a nearly circular orbit at 5 AU, we show that a misaligned hot Jupiter is likely to be formed in the presence of a more massive planetary companion (> 2M J ) within ∼ 140 AU of the inner system, with mutual inclination > 50 • and eccentricity above ∼ 0.25. This is in striking contrast to the test-particle approximation, where an almost perpendicular configuration can still cause large eccentricity excitations, but flips of an inner Jupiter-like planet are much less likely to occur. The constraints we derive can be used to guide future observations, and, in particular, searches for more distant companions in systems containing a hot Jupiter.

The computational singular perturbation (CSP) method is an algorithm which iteratively approximates slow manifolds and fast fibers in multiple-timescale dynamical systems. Since its inception due to Lam and Goussis [17], the convergence of the CSP method has been explored in depth; however, rigorous applications have been confined to the standard framework, where the separation between 'slow' and 'fast' variables is made explicit in the dynamical system. This paper adapts the CSP method to nonstandard slow-fast systems having a normally hyperbolic attracting critical manifold. We give new formulas for the CSP method in this more general context, and provide the first concrete demonstrations of the method on genuinely nonstandard examples.

The Koper model is a three-dimensional vector field that was developed to study complex electrochemical oscillations arising in a diffusion process. Koper and Gaspard described paradoxical dynamics in the model: they discovered complicated, chaotic behavior consistent with a homoclinic orbit of Shil'nikov type, but were unable to locate the orbit itself. The Koper model has since served as a prototype to study the emergence of mixed-mode oscillations (MMOs) in slow-fast systems, but only in this paper is the existence of these elusive homoclinic orbits established. They are found first in a larger family that has been used to study singular Hopf bifurcation in multiple time scale systems with two slow variables and one fast variable. A curve of parameters with homoclinic orbits in this larger family is obtained by continuation and shown to cross the submanifold of the Koper system. The strategy used to compute the homoclinic orbits is based upon systematic investigation of intersections of invariant manifolds in this system with multiple time scales. Both canards and folded nodes are multiple time scale phenomena encountered in the analysis. Suitably chosen cross-sections and return maps illustrate the complexity of the resulting MMOs and yield a modified geometric model from the one Shil'nikov used to study spiraling homoclinic bifurcations.Comment: 23 pages, 13 figures (17 subfigures). To appear in SIAD

We consider a Kuramoto model of coupled oscillators that includes quenched random interactions of the type used by van Hemmen in his model of spin glasses. The phase diagram is obtained analytically for the case of zero noise and a Lorentzian distribution of the oscillators' natural frequencies. Depending on the size of the attractive and random coupling terms, the system displays four states: complete incoherence, partial synchronization, partial antiphase synchronization, and a mix of antiphase and ordinary synchronization.

No abstract

We present a novel method for computing slow manifolds and their fast fibre bundles in geometric singular perturbation problems. This coordinate-independent method is inspired by the parametrisation method introduced by Cabré, Fontich and de la Llave. By iteratively solving a so-called conjugacy equation, our method simultaneously computes parametrisations of slow manifolds and fast fibre bundles, as well as the dynamics on these objects, to arbitrarily high degrees of accuracy. We show the power of this top-down method for the study of systems with multiple (i.e. three or more) timescales. In particular, we highlight the emergence of hidden timescales and show how our method can uncover these surprising multiple timescale structures. We also apply our parametrisation method to several reaction network problems.

We develop the contact singularity theory for singularly-perturbed (or 'slow-fast') vector fields of the general form z ′ = H(z, ε), z ∈ R n and ε ≪ 1. Our main result is the derivation of computable, coordinate-independent defining equations for contact singularities under an assumption that the leading-order term of the vector field admits a suitable factorization. This factorization can in turn be computed explicitly in a wide variety of applications. We demonstrate these computable criteria by locating contact folds and, for the first time, contact cusps in some nonstandard models of biochemical oscillators.

No abstract