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 $f_{\mu, \epsilon}$ defined on a $2d$-dimensional symplectic manifold $\mathcal M$ with exact symplectic form $\Omega$; we assume that $f_{\mu,\epsilon}$ satisfies $f_{\mu,\epsilon}^*\Omega=\lambda(\epsilon) \Omega$. We assume that the family depends on a $d$-dimensional parameter $\mu$ (called drift) and also on a small scalar parameter $\epsilon$. Furthermore, we assume that the conformal factor $\lambda$ depends on $\epsilon$, in such a way that for $\epsilon=0$ we have $\lambda(0)=1$ (the symplectic case). We study the domains of analyticity in $\epsilon$ near $\epsilon=0$ of perturbative expansions (Lindstedt series) of the parameterization of the quasi--periodic orbits of frequency $\omega$ (assumed to be Diophantine) and of the parameter $\mu$. 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 $\epsilon$ 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

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 $f_\mu$ of conformally symplectic maps which depend on a drift parameter $\mu$. We fix a Diophantine frequency of the torus and we assume to have a drift $\mu_0$ and an embedding of the torus $K_0$, which satisfy approximately the invariance equation $f_{\mu_0} \circ K_0 - K_0 \circ T_\omega$ (where $T_\omega$ denotes the shift by $\omega$). We also assume to have a splitting of the tangent space at the range of $K_0$ into three bundles. We assume that the bundles are approximately invariant under $D f_{\mu_0}$ and that the derivative satisfies some "rate conditions". Under suitable non-degeneracy conditions, we prove that there exists $\mu_\infty$, $K_\infty$ and splittings, close to the original ones, invariant under $f_{\mu_\infty}$. 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

An approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom

We construct an approximate renormalization transformation that combines Kolmogorov-Arnold-Moser (KAM)and renormalization-group techniques, to analyze instabilities in Hamiltonian systems with three degrees of freedom. This scheme is implemented both for isoenergetically nondegenerate and for degenerate Hamiltonians. For the spiral mean frequency vector, we find numerically that the iterations of the transformation on nondegenerate Hamiltonians tend to degenerate ones on the critical surface. As a consequence, isoenergetically degenerate and nondegenerate Hamiltonians belong to the same universality class, and thus the corresponding critical invariant tori have the same type of scaling properties. We numerically investigate the structure of the attracting set on the critical surface and find that it is a strange nonchaotic attractor. We compute exponents that characterize its universality class.Comment: 10 pages typeset using REVTeX, 7 PS figure