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The three-dimensional description of rigid body kinematics is a key step in many studies in biomechanics. There are several options for describing rigid body orientation including Cardan angles, Euler angles, and quaternions; the utility of quaternions will be reviewed and elaborated.The orientation of a rigid body or a joint between rigid bodies can be described by a quaternion which consists of four variables compared with Cardan or Euler angles (which require three variables). A quaternion, q = (q0, q1, q2, q3), can be considered a rotation (Ω = 2 cos−1(q0)), about an axis defined by a unit direction vector $$ \left({q}_1/\sin \left(\frac{\Omega}{2}\right),{q}_2/\sin \left(\frac{\Omega}{2}\right),{q}_3/\sin \left(\frac{\Omega}{2}\right)\right) $$q1/sinΩ2q2/sinΩ2q3/sinΩ2. The quaternion, compared with Cardan and Euler angles, does not suffer from singularities or Codman’s paradox. Three-dimensional angular kinematics are defined on the surface of a unit hypersphere which means numerical procedures for orientation averaging and interpolation must take account of the shape of this surface rather than assuming that Euclidean geometry based procedures are appropriate. Numerical simulations demonstrate the utility of quaternions for averaging three-dimensional orientations. In addition the use of quaternions for the interpolation of three-dimensional orientations, and for determining three-dimensional orientation derivatives is reviewed.The unambiguous nature of defining rigid body orientation in three-dimensions using a quaternion, and its simple averaging and interpolation gives it great utility for the kinematic analysis of human movement.

International audience; In this paper, we develop a new strategy aimed at obtaining high-order asymp-totic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature and re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.

We apply Reeb’s theorem to prove the existence of periodic orbits in the rotating Hénon– Heiles system. To this end, a sort of detuned normal form is calculated that yields a reduced system with at most four non degenerate equilibrium points. Linear stability and bifurcations of equilibrium solutions mimic those for periodic solutions of the original system. We also determine heteroclinic connections that can account for transport phenomena. This work has been partly supported from the Spanish Ministry of Science and Innovation through the Projects MTM2014-59433-CO (Subprojects MTM2014-59433-C2-1-P and MTM2014-59433- C2-2-P), MTM2017-88137-CO (Subprojects MTM2017-88137-C2-1-P and MTM2017-88137-C2-2-P), and by University of La Rioja through Project REGI 2018751.

Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is obtained. Scaling arguments indicate that these estimates are optimal. In the linear limit, the proven regularity estimates are consistent with the optimal regularity of the linear case., 36 pages

We consider slow/fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter $H>{\frac{1}{2}}$. We show that unlike in the case $H={\frac{1}{2}}$, convergence to the averaged solution takes place in probability and the limiting process solves the ‘naively’ averaged equation. Our proof strongly relies on the recently obtained stochastic sewing lemma.

International audience; The subject matter of this paper concerns the Vlasov-Poisson equations in the framework of magnetic confinement. We study the behavior of the Vlasov-Poisson system with strong external magnetic field, when neglecting the curvature of the magnetic lines. The arguments rely on averaging techniques. We intend to determine second order approximations and to retrieve the usual electric cross field drift, the magnetic gradient drift, etc.

Passage through resonance of an unbalanced rotor mounted on a strongly damped oscillating system that is excited by an asynchronous motor of limited power is investigated using an averaging procedure for a partially strongly damped system. The approach is closely related to a singular perturbation technique. It is demonstrated that the system's dynamics can be reduced on the slow manifold to a single first-order differential equation predicting both the stationary and transient solutions of the original system. Furthermore, the technique provides insight into the system's dynamics, enabling an outline of two simple methods to avoid capture into resonance. The first method is based on an active compensation of the small averaged vibrational torque, and the second method is a passive mechanical device reducing vibration amplitudes. The approximate results obtained were compared with the direct numerical simulations of the full system and demonstrate very good accuracy everywhere except in the vicinity of the bifurcation point. In this vicinity, the asymptotically long time interval is not sufficient for predicting capture into resonance.

Homogenization method is used in the problem of layered array with a cylindrical cavity. Boundary conditions of two types are considered on the surfaces of the layers: an ideal contact and slipping without exfoliation. The only body force is the own weight of an array. Purpose of this article is to determine approximate values of elastic stresses in a given array. As a result, the array is modeled by a homogeneous transversely isotropic half-space.

In this paper, an analytical method is presented to study an initial-boundary value problem describing the transverse displacements of a vertically moving beam under boundary excitation. The length of the beam is linearly varying in time, i.e., the axial, vertical velocity of the beam is assumed to be constant. The bending stiffness of the beam is assumed to be small. This problem may be regarded as a model describing the lateral vibrations of an elevator cable excited at its boundaries by the wind-induced building sway. Slow variation of the cable length leads to a singular perturbation problem which is expressed in slowly changing, time-dependent coefficients in the governing differential equation. By providing an interior layer analysis, infinitely many resonance manifolds are detected. Further, the initial-boundary value problem is studied in detail using a three-timescales perturbation method. The constructed formal approximations of the solutions are in agreement with the numerical results.

The mathematical model of nonlinear oscillations of weightless string is analyzed. Coefficients of the mathematical model and initial conditions are periodical functions of the space variable. A multiscale perturbation technique and integrating along characteristics are used to construct asymptotic solution without secular members.

In the paper, an asymptotic analysis of G.F. Carrier’s mathematical model of string oscillation is presented. The model consists of a system of two nonlinear second order partial differential equations and periodic initial conditions. The longitudinal and transversal string oscillations are analyzed together when at the initial moment of time the system’s solutions have amplitudes proportional to a small parameter. The problem is reduced to a system of two weakly nonlinear wave equations. The resonant interaction of periodic waves is analyzed. An uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. Conditions of appearance of combinatoric resonances in the system have been established. The results of numerical experiments are presented.

The problem about the local dynamics of the logistic equation with rapidly oscillating time-periodic piecewise constant coefficient of delay was considered. It was shown that the averaged equation is a logistic equation with two delays. The criterion of equilibrium point stability was obtained. Dynamical properties of the original equation was considered provided that the critical case of equilibrium point stability problem was implemented. It was found that an increase of delay coefficient oscillation frequency may lead to an unlimited process of “birth” and “death” steady mode.

In earlier papers, it has been shown how formal series like those used nowadays to investigate the properties of numerical integrators may be used to construct high-order averaged systems or formal first integrals of Hamiltonian problems. With the new approach the averaged system (or the formal first integral) may be written down immediately in terms of (i) suitable basis functions and (ii) scalar coefficients that are computed via simple recursions. Here we show how the coefficients/basis functions approach may be used advantageously to derive exponentially small error bounds for averaged systems and approximate first integrals. A. Murua and J.M. Sanz-Serna have been supported by projects MTM2010-18246-C03-03 and MTM2010-18246-C03-01 respectively from Ministerio de Ciencia e Innovación. Publicado

A logistic equation with a delay feedback circuit and with periodic perturbation of parameters is considered. The problem parameters (a coefficient of the linear growth and a delay) are chosen close to the critical values at which a cycle is bifurcated from the equilibrium point. We assume that these values have a double-frequency relation to the time, the frequency of action being close to the doubled frequency of the natural vibration. Asymptotic analysis is performed under these assumptions and leads to a two-dimensional system of ordinary differential equations. The linear part of this system is periodic. If the parameter which defines the frequency detuning of the external action is large or small, we can apply standard asymptotic methods to the resulting system. Otherwise, numerical analysis is performed. Using the results of the numerical analysis, we clarify the main scenarios of phase transformations and find the area of chaotic oscillations. The main conclusion is that in case of parametric resonance the dynamics of the problem with double-frequency perturbation is more complicated than the dynamics of the problem with single-frequency perturbation.

In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

The mathematical model of nonlinear oscillations of a weightless string is analysed. The uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. A method for constructing special approximations of its solutions is proposed.

We consider a simple model for the fluctuating hydrodynamics of a flexible polymer in a dilute solution, demonstrating geometric ergodicity for a pair of particles that interact with each other through a nonlinear spring potential while being advected by a stochastic Stokes fluid velocity field. This is a generalization of previous models which have used linear spring forces as well as white-in-time fluid velocity fields.We follow previous work combining control theoretic arguments, Lyapunov functions, and hypo-elliptic diffusion theory to prove exponential convergence via a Harris chain argument. In addition we allow the possibility of excluding certain “bad” sets in phase space in which the assumptions are violated but from which the system leaves with a controllable probability. This allows for the treatment of singular drifts, such as those derived from the Lennard-Jones potential, which is a novel feature of this work.

In this note we present a unified approach, based on pde methods, for the study of averaging principles for (small) stochastic perturbations of Hamiltonian flows in two space dimensions. Such problems were introduced by Freidlin and Wentzell and have been the subject of extensive study in the last few years using probabilistic arguments. When the Hamiltonian flow has critical points, it exhibits complicated behavior near the critical points under a small stochastic perturbation. Asymptotically the slow (averaged) motion takes place on a graph. The issues are to identify both the equations on the sides and the boundary conditions at the vertices of the graph. Our approach is very general and applies also to degenerate anisotropic elliptic operators which could not be considered using the previous methodology.

An averaging result is presented for local uniform asymptotic stability of nonlinear differential equations without requiring a fast time-varying vectorfield. The nonlinearity plays a crucial role: close to the origin, the trajectories vary slowly compared to the time dependence of the vectorfield. The result generalises averaging results which prove stability properties for systems having a homogeneous vectorfield with positive order. The result is illustrated with several examples.

The mathematical model of nonlinear oscillations of weightless string is analyzed. To find an asymptotic solution of the problem, uniformly valid in a long interval of time, an averaged system of integral differential equations has been constructed. A method for constructing special approximations of its solutions is proposed.

We show how B-series may be used to derive in a systematic way the analytical expressions of the high-order stroboscopic averaged equations that approximate the slow dynamics of highly oscillatory systems. For first-order systems we give explicitly the form of the averaged systems with $\mathcal{O}(\epsilon^{j})$errors, j=1,2,3 (2π e denotes the period of the fast oscillations). For second-order systems with large $\mathcal{O}(\epsilon^{-1})$forces, we give the explicit form of the averaged systems with $\mathcal{O}(\epsilon^{j})$errors, j=1,2. A variant of the Fermi–Pasta–Ulam model and the inverted Kapitsa pendulum are used as illustrations. For the former it is shown that our approach establishes the adiabatic invariance of the oscillatory energy. Finally we use B-series to analyze multiscale numerical integrators that implement the method of averaging. We construct integrators that are able to approximate not only the simplest, lowest-order averaged equation but also its high-order counterparts.

International audience; The subject matter of this paper concerns the asymptotic analysis of mathematical models for strongly magnetized plasmas. We concentrate on the finite Larmor radius regime with non uniform magnetic field in three dimensions. We determine the limit model and establish convergence results for any initial conditions, not necessarily well prepared. This study relies on a two-scale approach, based on the mean ergodic theorem, which allows us to separate between the fast and slow dynamics. The method adapts to many models. In particular it is possible to incorporate collision operators and to compute the effective diffusion matrices of the limit models. The average advection field and average diffusion matrix field appear as the long time limit for some parabolic problems, which allows us to obtain good approximations of the limit models, in the case of non uniform magnetic fields (when exact formulae are not available).

In this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrödinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang splitting method applied to the cubic Schrödinger equation is adapted to a wider class of nonlinearities. In a second step, the dependence of the global error on the decisive parameter 0 > ε > > 1 0 > \varepsilon >\!\!> 1 , defining the length of the period, is examined. The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor ε \varepsilon , provided that the time stepsize is chosen as an integer fraction of the period. This improved error behavior over a time interval of fixed length, which is independent of the period, is due to an averaging effect. The extension of the convergence result to higher-order splitting methods and numerical illustrations complement the investigations.

For several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e. the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: (a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; (b) a Taylor expansion of the fibre constitutive function in the structure tensor; (c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument. Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e. different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation.

We study a two-time-scale system of jump-diffusion stochastic differential equations. The main goal is to study the convergence rate of the slow components to the effective dynamics. The convergence established here is in the strong sense, i. e., uniformly in time. For the ergodicity assumptions, we use the existence of a Lyapunov function to control the return times. This assumption is weaker than the one- sided Lipschitz condition, frequently used for deriving rates.

The mathematical model of two strings nonlinear oscillations is presented. To find uniformly valid in a long time interval asymptotic solution of the problem an averaging scheme is constructed.

We consider a non-autonomous system of ordinary differential equations. Assume that the time dependence is periodic with a very high frequency 1 / ɛ , where ɛ is a small parameter and differentiability with respect to the parameter is lost when ɛ equals zero. We derive from Arenstorf's implicit function theorem a set of conditions to show the existence of periodic solutions. These conditions look formally like the standard analytic continuation method, namely, checking that a certain minor does not vanish. We apply this result to show the existence of a new class of periodic orbits of very large radii in the three-dimensional elliptic restricted three-body problem for arbitrary values of the masses of the primaries.

Averaging schemes for functional differential inclusions in Banach spaces with slow and fast time variables are studied. Under mild suppositions on the regularity, the periodic case and the case of non-existence of an average are investigated. The accuracy of the averaging technique is considered as well. In particular, for periodic systems, the usual linear approximation is achieved. Under stronger regularity conditions, approximation orders for Krylov–Bogoliubov–Mitropolskii type right-hand sides are derived.

Proton binding plays a critical role in protein structure and function. We report pKa calculations for three aspartates in two proteins, using a linear response approach, as well as a "standard" Poisson-Boltzmann approach. Averaging over conformations from the two endpoints of the proton-binding reaction, the protein's atomic degrees of freedom are explicitly modeled. Treating macroscopically the protein's electronic polarizability and the solvent, a meaningful model is obtained, without adjustable parameters. It reproduces qualitatively the electrostatic potentials, proton-binding free energies, Marcus reorganization free energies, and pKa shifts from explicit solvent molecular dynamics simulations, and the pKa shifts from experiment. For thioredoxin Asp-26, which has a large pKa upshift, we correctly capture the balance between unfavorable carboxylate desolvation and favorable interactions with a nearby lysine similarly for RNase A Asp-14, which has a large pKa downshift. For the unshifted thioredoxin Asp-20, desolvation by the protein cavity is overestimated by 2.9 pKa units several effects could explain this. "Standard" Poisson-Boltzmann methods sidestep this problem by using a large, ad hoc protein dielectric but protein charge-charge interactions are then incorrectly downscaled, giving an unbalanced description of the reaction and a large error for the shifted pKa values of Asp-26 and Asp-14. © 2005 by the Biophysical Society. 88 6 3888 3904 Cited By :56

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.

The exponential stability of singularly perturbed time-varying systems is investigated. It turns out that, under natural conditions, exponential stability of an averaged system is equivalent to exponential stability of the perturbed system for small perturbation parameters. Explicit estimates for both, the approximation of single trajectories and the order of the exponential decay, are obtained. The method of proof does not require smoothness of the averaged system.

We consider a seasonally forced SIS epidemic model where the population is spatially divided into two patches. We consider that periodicity occurs in the contact rates by switching between two levels. The epidemic dynamics are described by a switched system. We prove the existence of an invariant domain D containing at least one periodic solution. By considering small migrations, we rewrite the SIS model as a slow-fast dynamical system and show that it has a harmonic periodic solution which lies in a small tubular neighborhood of a curve Γ m . We deduce from this study the persistence or not of the disease in each patch.

We establish new regularity estimates, in terms of Sobolev spaces, of the solution f to a kinetic equation. The right-hand side can contain partial derivatives in time, space and velocity, as in classical averaging, and f is assumed to have a certain amount of regularity in velocity. The result is that f is also regular in time and space, and this is related to a commutator identity introduced by Hormander for hypoelliptic operators. In contrast with averaging, the number of derivatives does not depend on the L p space considered. Three type of proofs are provided: one relies on the Fourier transform, another one uses Hormander's commutators, and the last uses a characteristics commutator. Regularity of averages in velocity are deduced. We apply our method to the linear Fokker–Planck operator and recover the known optimal regularity, by direct estimates using Hormander's commutator.