Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.Comment: 22 page, 16 figure

Spin-Spin Coupling in the Solar System

The richness of dynamical behavior exhibited by the rotational states of various solar system objects has driven significant advances in the theoretical understanding of their evolutionary histories. An important factor that determines whether a given object is prone to exhibiting non-trivial rotational evolution is the extent to which such an object can maintain a permanent aspheroidal shape, meaning that exotic behavior is far more common among the small body populations of the solar system. Gravitationally bound binary objects constitute a substantial fraction of asteroidal and TNO populations, comprising systems of triaxial satellites that orbit permanently deformed central bodies. In this work, we explore the rotational evolution of such systems with specific emphasis on quadrupole-quadrupole interactions, and show that for closely orbiting, highly deformed objects, both prograde and retrograde spin-spin resonances naturally arise. Subsequently, we derive capture probabilities for leading order commensurabilities and apply our results to the illustrative examples of (87) Sylvia and (216) Kleopatra asteroid systems. Cumulatively, our results suggest that spin-spin coupling may be consequential for highly elongated, tightly orbiting binary objects.Comment: 9 pages, 4 figures, accepted to Ap

Chaotic quasi-collision trajectories in the 3-centre problem

We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin and Negrini, J. Diff. Eq. 190 (2003), 539--558): this result has been obtained by the use of the Poincar\'e-Melnikov theory. Here we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the perturbing centre on the real plane, we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use of a general result of Bolotin and MacKay (see Cel. Mech. & Dyn. Astr. 77 (2000), 49--75). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic orbits of the regularised problem passing through the third centre.Comment: 22 pages, 6 figure

Periodic solutions for the N-vortex problem via a superposition principle

We examine the $N$-vortex problem on general domains $\Omega\subset\mathbb{R}^2$ concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form $$\Gamma_k\dot{z}_k=J\nabla_{z_k}H(z_1,\ldots,z_N),\quad k=1,\ldots,N,$$ where $\Gamma_k\in\mathbb{R}\setminus\{0\}$ is the strength of the $k$th vortex at position $z_k(t)\in\Omega$, $J\in\mathbb{R}^{2\times 2}$ is the standard symplectic matrix and $$H(z_1,\ldots,z_N)=-\frac{1}{2\pi}\sum_{\underset{k\neq j}{k,j=1}}^N\Gamma_j\Gamma_k\log|z_k-z_j|-\sum_{k,j=1}^N\Gamma_j\Gamma_k g(z_k,z_j)$$ with some regular and symmetric, but in general not explicitely known function $g:\Omega\times\Omega\rightarrow \mathbb{R}$. The investigation relies on the idea to superpose a stationary solution of a system of less than $N$ vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple $T$-periodic solutions are shown to exist for every $T>0$ small enough. The crucial condition holds in generic bounded domains and is explicitely verified for an example in the unit disc $\Omega=B_1(0)$. In particular we therefore obtain various examples of periodic solutions in $B_1(0)$ that are not rigidly rotating configurations.Comment: 27 pages, 1 figur

In Situ Formation and Dynamical Evolution of Hot Jupiter Systems

Hot Jupiters, giant extrasolar planets with orbital periods shorter than ~10 days, have long been thought to form at large radial distances, only to subsequently experience long-range inward migration. Here, we propose that in contrast with this picture, a substantial fraction of the hot Jupiter population formed in situ via the core accretion process. We show that under conditions appropriate to the inner regions of protoplanetary disks, rapid gas accretion can be initiated by Super-Earth type planets, comprising 10-20 Earth masses of refractory composition material. An in situ formation scenario leads to testable consequences, including the expectation that hot Jupiters should frequently be accompanied by additional low-mass planets with periods shorter than ~100 days. Our calculations further demonstrate that dynamical interactions during the early stages of planetary systems' lifetimes should increase the inclinations of such companions, rendering transits rare. High-precision radial velocity monitoring provides the best prospect for their detection.Comment: 19 pages, 10 figures, accepted to Ap

Quantum spin models with electrons in Penning traps

We propose a scheme to engineer an effective spin Hamiltonian starting from a system of electrons confined in micro-Penning traps. By means of appropriate sequences of electromagnetic pulses, alternated to periods of free evolution, we control the shape and strength of the spin-spin interaction. Moreover, we can modify the effective magnetic field experienced by the particle spin. This procedure enables us to reproduce notable quantum spin systems, such as Ising and XY models. Thanks to its scalability, our scheme can be applied to a fairly large number of trapped particles within the reach of near future technology.Comment: 22 pages, 1 figure, added minor changes and typos, accepted for publication in PR