325 research outputs found
Transient times, resonances and drifts of attractors in dissipative rotational dynamics
In a dissipative system the time to reach an attractor is often influenced by
the peculiarities of the model and in particular by the strength of the
dissipation. In particular, as a dissipative model we consider the spin-orbit
problem providing the dynamics of a triaxial satellite orbiting around a
central planet and affected by tidal torques. The model is ruled by the
oblateness parameter of the satellite, the orbital eccentricity, the
dissipative parameter and the drift term. We devise a method which provides a
reliable indication on the transient time which is needed to reach an attractor
in the spin-orbit model; the method is based on an analytical result, precisely
a suitable normal form construction. This method provides also information
about the frequency of motion. A variant of such normal form used to
parametrize invariant attractors provides a specific formula for the drift
parameter, which in turn yields a constraint - which might be of interest in
astronomical problems - between the oblateness of the satellite and its orbital
eccentricity.Comment: 21 pages, 7 figures, colo
Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems
Many problems in Physics are described by dynamical systems that are
conformally symplectic (e.g., mechanical systems with a friction proportional
to the velocity, variational problems with a small discount or thermostated
systems). Conformally symplectic systems are characterized by the property that
they transform a symplectic form into a multiple of itself. The limit of small
dissipation, which is the object of the present study, is particularly
interesting.
We provide all details for maps, but we present also the modifications needed
to obtain a direct proof for the case of differential equations. We consider a
family of conformally symplectic maps defined on a
-dimensional symplectic manifold with exact symplectic form
; we assume that satisfies
. We assume that the family
depends on a -dimensional parameter (called drift) and also on a small
scalar parameter . Furthermore, we assume that the conformal factor
depends on , in such a way that for we have
(the symplectic case).
We study the domains of analyticity in near of
perturbative expansions (Lindstedt series) of the parameterization of the
quasi--periodic orbits of frequency (assumed to be Diophantine) and of
the parameter . Notice that this is a singular perturbation, since any
friction (no matter how small) reduces the set of quasi-periodic solutions in
the system. We prove that the Lindstedt series are analytic in a domain in the
complex plane, which is obtained by taking from a ball centered at
zero a sequence of smaller balls with center along smooth lines going through
the origin. The radii of the excluded balls decrease faster than any power of
the distance of the center to the origin
Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems
We consider a class of Hamiltonians with three degrees of freedom that can be
mapped into quasi-periodically driven pendulums. The purpose of this paper is
to determine the threshold of the break-up of invariant tori with a specific
frequency vector. We apply two techniques: the frequency map analysis and
renormalization-group methods. The renormalization transformation acting on a
Hamiltonian is a canonical change of coordinates which is a combination of a
partial elimination of the irrelevant modes of the Hamiltonian and a rescaling
of phase space around the considered torus. We give numerical evidence that the
critical coupling at which the renormalization transformation starts to diverge
is the same as the value given by the frequency map analysis for the break-up
of invariant tori. Furthermore, we obtain by these methods numerical values of
the threshold of the break-up of the last invariant torus.Comment: 18 pages, 4 figure
Promoting post-stroke recovery through focal or whole body vibration: criticisms and prospects from a narrative review
Objective: Several focal muscle vibration (fMV) and whole body vibration (WBV) protocols have been designed to promote brain reorganization processes in patients with stroke. However, whether fMV and WBV should be considered helpful tools to promote post-stroke recovery remains still largely unclear. Methods: We here achieve a comprehensive review of the application of fMV and WBV to promote brain reorganization processes in patients with stroke. By first discussing the putative physiological basis of fMV and WBV and then examining previous observations achieved in recent randomized controlled trials (RCT) in patients with stroke, we critically discuss possible strength and limitations of the currently available data. Results: We provide the first systematic assessment of fMV studies demonstrating some improvement in upper and lower limb functions, in patients with chronic stroke. We also confirm and expand previous considerations about the rather limited rationale for the application of current WBV protocols in patients with chronic stroke. Conclusion: Based on available information, we propose new recommendations for optimal stimulation parameters and strategies for recruitment of specific stroke populations that would more likely benefit from future fMV or WBV application, in terms of speed and amount of post-stroke functional recovery
Evolution and stability of Laplace-like resonances under tidal dissipation
AbstractThe Laplace resonance is a configuration that involves the commensurability between the mean motions of three small bodies revolving around a massive central one. This resonance was first observed in the case of the three inner Galilean satellites, Io, Europa, and Ganymede. In this work the Laplace resonance is generalised by considering a system of three satellites orbiting a planet that are involved in mean motion resonances. These Laplace-like resonances are classified in three categories: first-order (2:1&2:1, 3:2&3:2, 2:1&3:2), second-order (3:1&3:1) and mixed-order resonances (2:1&3:1). In order to study the dynamics of the system we implement a model that includes the gravitational interaction with the central body, the mutual gravitational interactions of the satellites, the effects due to the oblateness of the central body and the secular interaction of a fourth satellite and a distant star. Along with these contributions we include the tidal interaction between the central body and the innermost satellite. We study the survival of the Laplace-like resonances and the evolution of the orbital elements of the satellites under the tidal effects. Moreover, we study the possibility of capture into resonance of the fourth satellite
Short-term effects of focal muscle vibration on motor recovery after acute stroke: a pilot randomized sham-controlled study
Repetitive focal muscle vibration (rMV) is known to promote neural plasticity and long-lasting motor recovery in chronic stroke patients. Those structural and functional changes within the motor network underlying motor recovery occur in the very first hours after stroke. Nonetheless, to our knowledge, no rMV-based studies have been carried out in acute stroke patients so far, and the clinical benefit of rMV in this phase of stroke is yet to be determined. The aim of this randomized double-blind sham-controlled study is to investigate the short-term effect of rMV on motor recovery in acute stroke patients. Out of 22 acute stroke patients, 10 were treated with the rMV (vibration group–VG), while 12 underwent the sham treatment (control group–CG). Both treatments were carried out for 3 consecutive days, starting within 72 h of stroke onset; each daily session consisted of three 10-min treatments (for each treated limb), interspersed with a 1-min interval. rMV was delivered using a specific device (Cro®System, NEMOCO srl, Italy). The transducer was applied perpendicular to the target muscle's belly, near its distal tendon insertion, generating a 0.2–0.5 mm peak-to-peak sinusoidal displacement at a frequency of 100 Hz. All participants also underwent a daily standard rehabilitation program. The study protocol underwent local ethics committee approval (ClinicalTrial.gov NCT03697525) and written informed consent was obtained from all of the participants. With regard to the different pre-treatment clinical statuses, VG patients showed significant clinical improvement with respect to CG-treated patients among the NIHSS (p < 0.001), Fugl-Meyer (p = 0.001), and Motricity Index (p < 0.001) scores. In addition, when the upper and lower limb scales scores were compared between the two groups, VG patients were found to have a better clinical improvement at all the clinical end points. This study provides the first evidence that rMV is able to improve the motor outcome in a cohort of acute stroke patients, regardless of the pretreatment clinical status. Being a safe and well-tolerated intervention, which is easy to perform at the bedside, rMV may represent a valid complementary non-pharmacological therapy to promote motor recovery in acute stroke patients
Plasticity Induced in the Human Spinal Cord by Focal Muscle Vibration
The spinal cord spinal cord has in the past been considered a hardwired system which responds to inputs in a stereotyped way. A growing body of data have instead demonstrated its ability to retain information and modify its effector capabilities, showing activity-dependent plasticity. Whereas, plasticity in the spinal cord is well documented after different forms of physical exercise, whether exogenous stimulation can induce similar changes is still a matter of debate. This issue is both of scientific and clinical relevance, since at least one form of stimulation, i.e., focal muscle vibration (fMV), is currently used as a treatment for spasticity. The aim of the present study was to assess whether fMV can induce plasticity at the SC level when applied to different muscles of the upper limb. Changes in different electrophysiological measures, such as H-reflex testing homonymous and heteronymous pathways, reciprocal inhibition and somatosensory evoked potentials were used as outcomes. We found that fMV was able to induce long-term depression-like plasticity in specific spinal cord circuits depending on the muscle vibrated. These findings helped understand the basic mechanisms underlying the effects of fMV and might help to develop more advanced stimulation protocols
Whiskered KAM tori of conformally symplectic systems
We investigate the existence of whiskered tori in some dissipative systems,
called \sl conformally symplectic \rm systems, having the property that they
transform the symplectic form into a multiple of itself. We consider a family
of conformally symplectic maps which depend on a drift parameter .
We fix a Diophantine frequency of the torus and we assume to have a drift
and an embedding of the torus , which satisfy approximately the
invariance equation (where
denotes the shift by ). We also assume to have a splitting
of the tangent space at the range of into three bundles. We assume that
the bundles are approximately invariant under and that the
derivative satisfies some "rate conditions".
Under suitable non-degeneracy conditions, we prove that there exists
, and splittings, close to the original ones, invariant
under . The proof provides an efficient algorithm to construct
whiskered tori. Full details of the statements and proofs are given in
[CCdlL18].Comment: 15 pages, 1 figur
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