43 research outputs found
The speed of Arnold diffusion
A detailed numerical study is presented of the slow diffusion (Arnold
diffusion) taking place around resonance crossings in nearly integrable
Hamiltonian systems of three degrees of freedom in the so-called `Nekhoroshev
regime'. The aim is to construct estimates regarding the speed of diffusion
based on the numerical values of a truncated form of the so-called remainder of
a normalized Hamiltonian function, and to compare them with the outcomes of
direct numerical experiments using ensembles of orbits. In this comparison we
examine, one by one, the main steps of the so-called analytic and geometric
parts of the Nekhoroshev theorem. We are led to two main results: i) We
construct in our concrete example a convenient set of variables, proposed first
by Benettin and Gallavotti (1986), in which the phenomenon of Arnold diffusion
in doubly resonant domains can be clearly visualized. ii) We determine, by
numerical fitting of our data the dependence of the local diffusion coefficient
"D" on the size "||R_{opt}||" of the optimal remainder function, and we compare
this with a heuristic argument based on the assumption of normal diffusion. We
find a power law "D\propto ||R_{opt}||^{2(1+b)}", where the constant "b" has a
small positive value depending also on the multiplicity of the resonance
considered.Comment: 39 pages, 11 figure
Secondary resonances and the boundary of effective stability of Trojan motions
One of the most interesting features in the libration domain of co-orbital
motions is the existence of secondary resonances. For some combinations of
physical parameters, these resonances occupy a large fraction of the domain of
stability and rule the dynamics within the stable tadpole region. In this work,
we present an application of a recently introduced `basic Hamiltonian model' Hb
for Trojan dynamics, in Paez and Efthymiopoulos (2015), Paez, Locatelli and
Efthymiopoulos (2016): we show that the inner border of the secondary resonance
of lowermost order, as defined by Hb, provides a good estimation of the region
in phase-space for which the orbits remain regular regardless the orbital
parameters of the system. The computation of this boundary is straightforward
by combining a resonant normal form calculation in conjunction with an
`asymmetric expansion' of the Hamiltonian around the libration points, which
speeds up convergence. Applications to the determination of the effective
stability domain for exoplanetary Trojans (planet-sized objects or asteroids)
which may accompany giant exoplanets are discussed.Comment: 21 pages, 9 figures. Accepted for publication in Celestial Mechanics
and Dynamical Astronom
Bohmian trajectories in an entangled two-qubit system
In this paper we examine the evolution of Bohmian trajectories in the
presence of quantum entanglement. We study a simple two-qubit system composed
of two coherent states and investigate the impact of quantum entanglement on
chaotic and ordered trajectories via both numerical and analytical
calculations.Comment: 12 Figures, corrected typos, replaced figure 10 and revised captions
in figures 8 and 1
Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlev\'{e} Series
The Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to
identify new integrable nonlinear dynamical systems. For a system which passes
the Painlev\'{e} test, the calculation of the integrals relies on a variety of
methods which are independent from Painlev\'{e} analysis. The present paper
proposes an explicit algorithm to build first integrals of a dynamical system,
expressed as `quasi-polynomial' functions, from the information provided solely
by the Painlev\'{e} - Laurent series solutions of a system of ODEs.
Restrictions on the number and form of quasi-monomial terms appearing in a
quasi-polynomial integral are obtained by an application of a theorem by
Yoshida (1983). The integrals are obtained by a proper balancing of the
coefficients in a quasi-polynomial function selected as initial ansatz for the
integral, so that all dependence on powers of the time is
eliminated. Both right and left Painlev\'{e} series are useful in the method.
Alternatively, the method can be used to show the non-existence of a
quasi-polynomial first integral. Examples from specific dynamical systems are
given.Comment: 16 pages, 0 figure
Partial Integrability of 3-d Bohmian Trajectories
In this paper we study the integrability of 3-d Bohmian trajectories of a
system of quantum harmonic oscillators. We show that the initial choice of
quantum numbers is responsible for the existence (or not) of an integral of
motion which confines the trajectories on certain invariant surfaces. We give a
few examples of orbits in cases where there is or there is not an integral and
make some comments on the impact of partial integrability in Bohmian Mechanics.
Finally, we make a connection between our present results for the integrability
in the 3-d case and analogous results found in the 2-d and 4-d cases.Comment: 18 pages, 3 figure
Origin of chaos in 3-d Bohmian trajectories
We study the 3-d Bohmian trajectories of a quantum system of three harmonic
oscillators. We focus on the mechanism responsible for the generation of
chaotic trajectories. We demonstrate the existence of a 3-d analogue of the
mechanism found in earlier studies of 2-d systems, based on moving 2-d `nodal
point - X-point complexes'. In the 3-d case, we observe a foliation of nodal
point - X-point complexes, forming a `3-d structure of nodal and X-points'.
Chaos is generated when the Bohmian trajectories are scattered at one or more
close encounters with such a structure.Comment: 7 pages, 8 figure
Hamiltonian formulation of the spin-orbit model with time-varying non-conservative forces
In a realistic scenario, the evolution of the rotational dynamics of a
celestial or artificial body is subject to dissipative effects. Time-varying
non-conservative forces can be due to, for example, a variation of the moments
of inertia or to tidal interactions. In this work, we consider a simplified
model describing the rotational dynamics, known as the spin-orbit problem,
where we assume that the orbital motion is provided by a fixed Keplerian
ellipse. We consider different examples in which a non-conservative force acts
on the model and we propose an analytical method, which reduces the system to a
Hamiltonian framework. In particular, we compute a time parametrisation in a
series form, which allows us to transform the original system into a
Hamiltonian one. We also provide applications of our method to study the
rotational motion of a body with time-varying moments of inertia, e.g. an
artificial satellite with flexible components, as well as subject to a tidal
torque depending linearly on the velocity.Comment: Accepted for publication in Communications in Nonlinear Science and
Numerical Simulatio
The theory of secondary resonances in the spin-orbit problem
We study the resonant dynamics in a simple one degree of freedom, time
dependent Hamiltonian model describing spin-orbit interactions. The equations
of motion admit periodic solutions associated with resonant motions, the most
important being the synchronous one in which most evolved satellites of the
Solar system, including the Moon, are observed. Such primary resonances can be
surrounded by a chain of smaller islands which one refers to as secondary
resonances. Here, we propose a novel canonical normalization procedure allowing
to obtain a higher order normal form, by which we obtain analytical results on
the stability of the primary resonances as well as on the bifurcation
thresholds of the secondary resonances. The procedure makes use of the
expansion in a parameter, called the detuning, measuring the shift from the
exact secondary resonance. Also, we implement the so-called `book-keeping'
method, i.e., the introduction of a suitable separation of the terms in orders
of smallness in the normal form construction, which deals simultaneously with
all the small parameters of the problem. Our analytical computation of the
bifurcation curves is in excellent agreement with the results obtained by a
numerical integration of the equations of motion, thus providing relevant
information on the parameter regions where satellites can be found in a stable
configuration.Comment: Accepted for publication in MNRA