Search

Secure distributed estimation algorithms are designed to protect against a spectrum of attacks by exploring different attack models and implementing strategies to enhance the resilience of the algorithm. These models encompass diverse scenarios such as measurement sensor attacks and communication link attacks, which have been extensively investigated in existing literature. This paper, however, focuses on a specific type of attack: the multiplicative sensor attack model. To counter this, the paper introduces the Average diffusion least mean square (ADLMS) algorithm as a viable solution. Furthermore, the paper introduces the Average Likelihood Ratio Test (ALRT) detector, which provides a straightforward detection criterion. In the presence of communication link attacks, the paper considers the manipulation attack model and presents an ALRT adversary detector. The analysis extends to these ALRT detectors, encompassing the calculation of adversary detection probability and false alarm probability, both achieved in closed form. The paper also provides the mean convergence analysis of the proposed ADLMS algorithm. Simulation results reveal that the proposed algorithms exhibit enhanced performance compared to the DLMS algorithm, while the incremental complexity remains only marginally higher than that of the DLMS algorithm. [ABSTRACT FROM AUTHOR]

In the case of the numerical differentiation (ND), the numerical evidence shows that the round-off errors have a large random-like component when considered as a function of the size of the discretization parameter. If a derivative is evaluated on a computer many times with different but reasonable discretization parameters, the round-off errors have tendency to average out and one gets results where the related uncertainty is largely suppressed. Applying this approach to a situation where the round-off error dominates over the discretization error (i.e. the discretization parameter is chosen small), one can effectively increase the precision of the ND by several orders of magnitude in the absolute error. For a general differentiable (e.g. non-analytic) black-box function differentiated in the context of a fixed machine epsilon, the presented method is presumably the most precise way of performing the ND nowadays known. The method is of practical use and has a potential to be generalized to other numerical procedures. [ABSTRACT FROM AUTHOR]

This article presents some new criteria for the partial exponential stability of a slow–fast nonlinear system with a fast scalar variable using periodic averaging methods. Unlike classical averaging techniques, we construct an averaged system by averaging over this fast scalar variable instead of the time variable. We show that the partial exponential stability of the averaged system implies that of the original one. We then apply the obtained criteria to the study of remote synchronization of Kuramoto–Sakaguchi oscillators coupled by a star network with two peripheral nodes. We show that detuning the natural frequency of the central mediating oscillator increases the robustness of the remote synchronization against phase shifts. This article appears to be the first-known attempt to analytically study the phase-unlocked remote synchronization. [ABSTRACT FROM AUTHOR]

Vibrational control seeks to stabilize an unstable system by judiciously injecting a state-free high-frequency dither. This article presents some robustness properties of vibrational control with respect to additive disturbances in a nonlinear system. We assume that, without disturbances, the appropriately averaged system is regionally asymptotically stable. Using perturbation techniques, our first result shows that the stabilization realized through vibrational control is robust with respect to additive disturbances of sufficiently small amplitude. Indeed, the perturbed system of vibrational control has a robustness feature similar to input-to-state stability in a local region. In the case of periodic disturbances, our second result indicates that vibrational control naturally dampens disturbances of sufficiently high frequency, which allows for relatively high amplitude disturbances. A tight bound for the effect of such periodic disturbances on the ultimate deviation of states from the desired equilibrium is presented. Simulation results from a planar manipulator support the theoretic analysis. [ABSTRACT FROM AUTHOR]

We consider the notion of input-to-state multistability, which generalizes input-to-state stability to nonlinear systems evolving on Riemannian manifolds and possessing a finite number of compact, globally attractive, invariant sets, and in addition satisfies a specific condition of acyclicity. We prove that a parameterized family of dynamical systems whose solutions converge to those of a limiting system inherits such input-to-state multistability property from the limiting system in a semiglobal practical fashion. A similar result is also established for singular perturbation models whose boundary-layer subsystem is uniformly asymptotically stable and whose reduced subsystem is input-to-state multistable. Known results in the theory of perturbations, singular perturbations, averaging, and highly oscillatory control systems are here generalized to the multistable setting by replacing the classical asymptotic stability requirement of a single invariant set with attractivity and acyclicity of a decomposable invariant one. [ABSTRACT FROM AUTHOR]

This paper demonstrates how averaging over individual learning and forgetting curves gives rise to transformed averaged curves. In an earlier paper (Murre and Chessa, 2011), we already showed that averaging over exponential functions tends to give a power function. The present paper expands on the analyses with exponential functions. Also, it is shown that averaging over power functions tends to give a log power function. Moreover, a general proof is given how averaging over logarithmic functions retains that shape in a specific manner. The analyses assume that the learning rate has a specific statistical distribution, such as a beta, gamma, uniform, or half-normal distribution. Shifting these distributions to the right, so that there are no low learning rates (censoring), is analyzed as well and some general results are given. Finally, geometric averaging is analyzed, and its limits are discussed in remedying averaging artefacts. • Averaging over individual learning (or forgetting, etc.) curves changes their shape, where exponential functions may change into power functions and power functions into log power functions. • Averaging over logarithmic functions retains the shape. • For exponential and power functions, geometric averaging retains the shape. • Geometric averaging over logarithmic functions does not retain that shape. [ABSTRACT FROM AUTHOR]

We consider a simple pendulum whose suspension point undergoes fast vibrations in the plane of motion of the pendulum. The averaged over the fast vibrations system is a Hamiltonian system with one degree of freedom depending on two parameters. We give a complete description of bifurcations of phase portraits of this averaged system. [ABSTRACT FROM AUTHOR]

In this work, we present a modification for the classic and phasor extremum seeking control algorithms in order to improve the accuracy by removing or reducing the convergence error. The modulation signals were replaced by a sum of sinusoids in order to remove the equilibrium shift in the controlled variable of the averaged system. The convergence error is calculated as a function of the number of sinusoids used in the modulation signal. A simulation example is presented to illustrate the improvement. [ABSTRACT FROM PUBLISHER]

Oscillators with an antisymmetric constant restoring force are considered in this paper. Their general characteristics—the corresponding amplitude and period of motion, are discussed first. Then, the exact solutions for acceleration, velocity and motion of such conservative oscillators are presented. One of them includes the use of the square wave and triangular wave function, which are utilized to develop a procedure for finding approximations for the response of damped antisymmetric constant force oscillators. Comparisons with numerical solutions are provided to demonstrate their accuracy. [ABSTRACT FROM AUTHOR]

A general method to combine several estimators of the same quantity is investigated. In the spirit of model and forecast averaging, the final estimator is computed as a weighted average of the initial ones, where the weights are constrained to sum to one. In this framework, the optimal weights, minimizing the quadratic loss, are entirely determined by the mean squared error matrix of the vector of initial estimators. The averaging estimator is built using an estimation of this matrix, which can be computed from the same dataset. A non-asymptotic error bound on the averaging estimator is derived, leading to asymptotic optimality under mild conditions on the estimated mean squared error matrix. This method is illustrated on standard statistical problems in parametric and semi-parametric models where the averaging estimator outperforms the initial estimators in most cases. [ABSTRACT FROM AUTHOR]

We establish averaging type sufficient conditions for local asymptotic stability for nonlinear time-varying systems. Our main result is based on an extension of the averaging methodology employed in a recent paper by Tsinias and Stamati (published in the same journal), dealing with asymptotic stability for “slow” time-varying systems. [ABSTRACT FROM PUBLISHER]

This paper presents the theory and analysis of IF polar receiver (PRX) architecture. By using new quantization techniques in the polar domain, the proposed receiver can boost the signal to quantization noise ratio (SQNR) compared to a traditional rectangular (I/Q) receiver. The proposed PRX is composed of a magnitude and a phase quantizer. The magnitude quantizer is similar to the conventional rectangular quantizer in voltage domain. The phase quantizer employs a time-to-digital converter (TDC) for phase detection. Furthermore, an intuitive graphical method is used to analyze the quantization properties of the polar quantization. A 10 bit polar quantizer is designed and fabricated in 130 nm CMOS, and achieves 2- to 5-dB of SQNR improvement compared to rectangular quantizer for signal bandwidths as high as 20 MHz. [ABSTRACT FROM AUTHOR]

Highlights: [•] We studied mixing in multiscale-flows. [•] We illustrated the method of partial averaging near resonances. [•] We computed the size of the localized domain of chaotic advection. [•] We estimated the rate of mixing. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we suggest averaging lateration estimates obtained using overlapped subgroups of distance measurements as opposed to obtaining a single lateration estimate from all of the measurements directly if a redundant number of measurements are available. Least squares based closed form equations are used in the lateration. In the case of Gaussian measurement noise the performances are similar in general and for some subgroup sizes marginal gains are attained. Averaging laterations method becomes especially beneficial if the lateration estimates are classified as useful or not in the presence of outlier measurements whose distributions are modeled by a mixture of Gaussians (MOG) pdf. A new modified trimmed mean robust averager helps to regain the performance loss caused by the outliers. If the measurement noise is Gaussian, large subgroup sizes are preferable. On the contrary, in robust averaging small subgroup sizes are more effective for eliminating measurements highly contaminated with MOG noise. The effect of high-variance noise was almost totally eliminated when robust averaging of estimates is applied to QR decomposition based location estimator. The performance of this estimator is just 1 cm worse in root mean square error compared to the Cramér–Rao lower bound (CRLB) on the variance both for Gaussian and MOG noise cases. Theoretical CRLBs in the case of MOG noise are derived both for time of arrival and time difference of arrival measurement data. [Copyright &y& Elsevier]

This paper investigates cooperative estimation of averaged moving target object poses in three dimensions for visual sensor networks. In particular, we consider the situation where multiple vision cameras see a common target object but the poses consistent with visual measurements differ from camera to camera due to a variety of uncertainties. Under the situation, we try to estimate an average of the contaminated poses not only for static but also for moving target objects by using only local negotiations. For this purpose, we present a cooperative estimation mechanism called networked visual motion observer. We then derive an upper bound of the ultimate error between the actual average and the estimates produced by the present estimation mechanism for both static and moving target objects. Finally the effectiveness of the networked visual motion observer is demonstrated through simulation. [ABSTRACT FROM AUTHOR]

Abstract: Clustering with fast algorithms large samples of high dimensional data is an important challenge in computational statistics. A new class of recursive stochastic gradient algorithms designed for the -medians loss criterion is proposed. By their recursive nature, these algorithms are very fast and are well adapted to deal with large samples of data that are allowed to arrive sequentially. It is proved that the stochastic gradient algorithm converges almost surely to the set of stationary points of the underlying loss criterion. A particular attention is paid to the averaged versions which are known to have better performances. A data-driven procedure that permits a fully automatic selection of the value of the descent step is also proposed. The performance of the averaged sequential estimator is compared on a simulation study, both in terms of computation speed and accuracy of the estimations, with more classical partitioning techniques such as -means, trimmed -means and PAM (partitioning around medoids). Finally, this new online clustering technique is illustrated on determining television audience profiles with a sample of more than 5000 individual television audiences measured every minute over a period of 24 hours. [Copyright &y& Elsevier]

Abstract: We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg–de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes. [Copyright &y& Elsevier]

We consider a path-following problem in which the goal is to ensure that the error between the system output and the geometric path is asymptotically less than a prespecified constant, while guaranteeing a forward motion along the path and boundedness of all states. Comparing with the results on this problem, we exploit averaging techniques to develop an alternative simpler solution for a class of nonlinear systems and for paths satisfying a certain geometric condition. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish sufficient conditions for local asymptotic stability and feedback stabilization for discrete-time systems with time depended dynamics. Our main results constitute generalizations of those developed by same authors in a recent paper, published in same journal, for the case of continuous-time systems. [ABSTRACT FROM AUTHOR]

This note presents an efficient distributed approach for computing a spatial average of parameters estimated by sensors in a wireless network. The most intuitive approach would rely on a two-step procedure. First, the nodes would estimate the local quantities, and second a distributed process would average these estimates over the network. Instead, the proposed algorithm combines both processes to foster the convergence while fulfilling the usual wireless sensor network requirements: simplicity, low memory/CPU usage, and asynchronicity. [ABSTRACT FROM AUTHOR]

Abstract: We use limit cycle oscillators to model bipolar II disorder, which is characterized by alternating hypomanic and depressive episodes and afflicts about 1% of the United States adult population. We consider two non-linear oscillator models of a single bipolar patient. In both frameworks, we begin with an untreated individual and examine the mathematical effects and resulting biological consequences of treatment. We also briefly consider the dynamics of interacting bipolar II individuals using weakly-coupled, weakly-damped harmonic oscillators. We discuss how the proposed models can be used as a framework for refined models that incorporate additional biological data. We conclude with a discussion of possible generalizations of our work, as there are several biologically-motivated extensions that can be readily incorporated into the series of models presented here. [Copyright &y& Elsevier]

The purpose of this note is to provide sufficient conditions for asymptotic stability for time-varying systems, whose dynamics are unbounded with respect to time. An averaging approach is established that constitutes a generalization of the well known methodology adopted in the literature for systems whose dynamics are bounded with respect to time. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we carry out an asymptotic analysis of the elastic wave equations in random media in the parabolic white-noise regime. In this regime, the propagation distance is much larger than the initial beam width, which is itself much larger than the typical wavelength; moreover, the correlation length of the random medium is of the same order as the initial beam width, and the amplitude of the random fluctuations is small. In this distinguished limit we show that wave propagation is governed by a system of random paraxial wave equations. The equations for the shear waves and pressure waves have the form of Schrödinger equations driven by two correlated Brownian fields. The diffraction operators can be expressed in terms of transverse Laplacians. The covariance structure of the Brownian fields is determined by the two-point statistics of the density and Lamé parameters of the random medium. [Copyright &y& Elsevier]

An 8-bit 600 megasamples-per-second (MSPS) analog-to-digital converter (ADC) has been implemented in 0.18-nm CMOS to achieve a minimum signal-to-noise-and-distortion ratio (SNDR) of 40 dB and a spurious-free dynamic range (SFDR) of 45 dB with input-signal bandwidth up to 200 MHz. The ADC is also capable of sampling up to 1 gigasamples/s and maintaining 39-dB SNDR at an input-signal frequency of 55 MHz. Distributed track-and-hold (DT&H) is employed at the ADC front end to relieve the linearity burden on individual T&H subunit. Complementary resistor and capacitor averaging networks are employed before and after DT&H switches separately in order to alleviate offset- and switching-induced errors, respectively. The fabricated ADC occupies 0.5 mm² in chip area and consumes 207 mW from a 1.8-V supply. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the interaction effect of horizontal fast harmonic parametric excitation and time delay on self-excited vibration in van der Pol oscillator. We apply the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic of the oscillator. The method of averaging is then performed on the slow dynamic to obtain a slow flow which is analyzed for equilibria and periodic motion. This analysis provides analytical approximations of regions in parameter space where periodic self-excited vibrations can be eliminated. A numerical study is performed on the original oscillator and compared to analytical approximations. It was shown that in the delayed case, horizontal fast harmonic excitation can eliminate undesirable self-excited vibrations for moderate values of the excitation frequency. In contrast, the case without delay requires large excitation frequency to eliminate such motions. This work has application to regenerative behavior in high-speed milling. [Copyright &y& Elsevier]

An averaged mode-locking model is presented in which the nonlinear mode-coupling behavior in a wave-guide array, dual-core fiber, and/or fiber array is used to achieve stable and robust passive mode-locking. By using the discrete, nearest-neighbor spatial coupling of these nonlinear mode-coupling devices, low-intensity light can be transferred to the neighboring wave-guides and attenuated. In contrast, higher intensity light is self-focused in the launch wave-guide and remains largely unaffected. This nonlinear effect, which is described by linearly coupled nonlinear Schrödinger equations, leads to the temporal intensity discrimination required in the laser cavity for mode-locking. Computations of this pulse shaping mechanism show that stable and robust mode-locked soliton-like pulses can be produced in this master mode-locking model. [Copyright &y& Elsevier]

The distribution and interleaving (D&I) of signals is a common method for ripple attenuation in various engineering applications in such areas as control, communication, and power electronics. Similarities to this technique may also been found in nonengineering fields such as biology and medicine. This paper presents a mathematical exploration of distributed interleaved systems along with a simple frequency-domain model of interleaving. We are hoping that the insights provided by this mathematical framework and the newly proposed model for interleaved systems will lead to enhanced techniques for evaluating D&I processes, and facilitate the design of better systems. In particular, we hope this work results in new approaches to low-pass filtering that will exhibit fast dynamics and very efficient ripple attenuation (in theory, this can produce complete ripple removal in some cases) [ABSTRACT FROM PUBLISHER]

Abstract: Two aerodynamically excited pendula are considered as a simple example of two linearly coupled, self sustained mechanical oscillators, modelled by two coupled Van-der-Pol equations. The considered mechanical application admits of a systematic survey of synchronized regimes within the framework of standard nonlinear stability analysis. Using normal form theory and the prevailing direct averaging approach the occurring Hopf bifurcation with two distinct pairs of purely imaginary eigenvalues is studied in the non-resonant case and in the 1:1-resonance corresponding, respectively, to strong and weak coupling. In particular, for the resonant case a graphical approach permits a comprehensive interpretation relating the stable stationary solutions of the averaged system with synchronized regimes and allows an analytical computation of the oscillation amplitudes and the synchronous frequency. [Copyright &y& Elsevier]

Circuit techniques using resistor strings (R-strings) and resistor rings (R-rings) for phase averaging and interpolation are described. Phase averaging can reduce phase errors, and phase interpolation can increase the number of available phases. In addition to the waveform shape, the averaging and the interpolation performances of the R-strings and R-rings are determined by the clock frequency normalized by a RC time constant of the circuits. To attain better phase accuracy, a smaller RC time constant is required, but at the expense of larger power dissipation. To demonstrate the resistor ring's capability of phase averaging and interpolation, a 125-MHz 8-bit digital-to-phase converter (DPC) was designed and fabricated using a standard 0.35-μm SPQM CMOS technology. Measurement results show that the DPC attains 8-bit resolution using the proposed phase averaging and interpolation technique. [ABSTRACT FROM AUTHOR]

Abstract: Correlation is a fundamental technique to detect signals buried in noise. However, due to noise, the fluctuations of the output signal may be significant and the detection of the desired signal can then fail. It is known that a simple one-bit correlator receiver can give nearly identical performance to the full precision correlator in determining time delays in medium signal-to-noise ratio situations, when averaging several successive correlation signals. An oversampled and averaged version of the correlator, with input quantization to a low number of bits, is studied here using simulation. It is demonstrated that, by oversampling the received signal at a multiple of the correlation code clock rate and averaging over several successive samples, performance similar to that of the full precision correlator may be attained to in low SNR situations. [Copyright &y& Elsevier]

We propose two modeling approaches of additive-increase/multiplicative-decrease (AIMD) congestion control mechanisms. The first separates the increase and decrease parts whereas the second describes the rate evolution of the congestion window as a continuous process governed by a differential equation. We relate these approaches and show that the second (fluid-flow) model results from deterministic time averaging of the first (discontinuous) model. A generalized class of nonlinear protocols, which includes Floyd's HighSpeed TCP, is then proposed and analyzed. Our findings are validated by simulation. [ABSTRACT FROM AUTHOR]

This paper presents a new method to characterize the behavior of an oscillator's slow-varying modes, e.g., amplitude and phase. Key to the method is the projection of the original set of circuit equations onto the slow manifold defined by the oscillator's core system. The core system corresponds to the terms of the circuit equations that generate the fast-varying oscillations. The corresponding slow manifold is made up by the core system's steady-state solutions. By means of perturbation and averaging techniques, we then extract the equations that govern the oscillator's motion over the slow manifold. These equations correspond to the (slow-varying) dynamics of, for instance, the oscillator's amplitude, phase or common-mode level. In many cases, they capture that part of the oscillator's behavior that is of greatest interest. Moreover, since the method here presented explicitly separates the analysis of an oscillator's fast- and slow-varying behavior, it can be used to improve simulation efficiency or for behavioral model extraction. [ABSTRACT FROM AUTHOR]

The averaging technique is used in flash analog-to- digital converters to reduce nonlinearities resulting from random from random offset voltages of the pre-amplifiers, which stand before the comparators. The main contribution of this paper is to provide further insight into this technique, through exact closed-form expressions obtained for the output voltage, gun, integral nonlinearity and differential nonlinearity in averaged pre-amplifiers. These theoretical results are compared with HSPICE simulations, and a very good agreement is found. Finally an automatic design procedure is described, which is based on the expressions derived, and a design example is given. [ABSTRACT FROM AUTHOR]

In a previous paper [Chaos Solitons Fract. 14(2) (2002) 173], the authors investigated the dynamics of the equation:d2x/dt2+(δ+∊cost)x+∊Ax3+Bx2dx/dt+Cxdx/dt2+Ddx/dt+(δ+∊cost)x+∊Ax3+Bx2dx/dt+Cxdx/dt2+Ddx/dt+Cxdx/dt2+Ddx/dt2+Ddx/dt3=0We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the resulting slow flow and some of the bifurcations near this point were looked at. In this work we present additional results concerning the bifurcations around this point using analytic techniques and AUTO. An analytic approximation for a heteroclinic bifurcation curve is obtained. Additional results on the bifurcations of periodic orbits in the slow flow are also presented. [Copyright &y& Elsevier]

• Mathematical model of governing differential equations is presented. • Radial, along-track and cross-track slowly varying parameters are developed. • Averaging method is applied to the nonlinear equation of relative motion. • The averaged model is useful for rendezvous and formation flying control approaches. The relative orbital motion problem of a deputy spacecraft with respect to a chief spacecraft is, generally, described using a set of differential equations governing the motion of the spacecraft relative to each other instead of describing their motion, separately, relative to the Earth. In this paper, new time-varying, time periodic cubic approximation model of spacecraft relative motion is developed from the original nonlinear relative equations of motion with the chief in elliptical orbit. Afterwards, averaging method is applied to the cubic model to obtain asymptotic approximations and periodic solutions. The formulation of the averaging solutions using averaging method affords us the opportunity to have a better insight of the relative motion dynamics. From the numerical simulations, the averaging model is in close agreement with the nonlinear model. This can be attributed to the inclusion of cubic nonlinear terms. The model is amenable to the long-term prediction of the behavior of the relative motion and useful for spacecraft formation flying analysis. [ABSTRACT FROM AUTHOR]

• Kaula's recurrences for long-term geopotential motion are discussed. • They show better performance over other recursions in the literature. • They are useful in exploring analytically the long-term dynamics. • They help in disclosing the required complexity of the dynamical model. The long-term dynamics of perturbed Keplerian motion is usually analyzed in simplified models as part of the preliminary design of artificial satellites missions. It is commonly approached by averaging procedures that deal with literal expressions in expanded form. However, there are cases in which the correct description of the dynamics may require full, contrary to simplified, potential models, as is, for instance, the case of low-altitude, high-inclination lunar orbits. In these cases, dealing with literal expressions is yet possible with the help of modern symbolic algebra systems, for which memory handling is no longer an issue. Still, the efficient evaluation of the averaged expressions related to a high fidelity potential is often jeopardized for the expanded character of the output of the automatic algebraic process, which unavoidably provides huge expressions that commonly comprise tens of thousands of literal terms. Rearrangement of the output to generate an efficient numerical code may solve the problem, but automatization of this kind of post-processing is a non trivial task due to the ad-hoc heuristic simplification procedures involved in the optimization process. However, in those cases in which the coupling of different perturbations is not of relevance for the analysis, the averaging procedure may preserve the main features of the structure of the potential model, thus avoiding the need of the typical blind computer-based brut force perturbation approach. Indeed, we show how standard recursions in the literature may be used to efficiently replace the brut force approach, in this way avoiding the need of further simplification to improve performance evaluation. In particular, Kaula's seminal recursion formulas for the gravity potential reveal clearly superior to the use of both expanded expressions and other recursions more recently proposed in the literature. Thus, after making a general assessment on the computational efficiency of the different approaches to compute the zonal gravitational potential in mean elements, the need of using high degrees of the gravitational potential for mission design purposes is illustrated for the case of a low lunar orbit. [ABSTRACT FROM AUTHOR]

• Transport in Hamiltonian systems with slow stochastic parameters is the result of two different mechanisms, which we characterize. • We provide an asymptotically exact stochastic equation for long-term transport of action variables when the Hamiltonian is integrable. • When the Hamiltonian is chaotic, we show -using averaging- that long-term transport is equivalent to a simple Markov process. Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are not order one perturbations of a given Hamiltonian. This class of problems is very important for many applications, for instance in celestial mechanics. As an example, we study a class of one-dimensional Hamiltonians that depend explicitly on time and on stochastic external parameters. The variations of the external parameters are responsible for a distortion of the phase space structures: chaotic, weakly chaotic and regular sets change with time. We show that theoretical predictions of transport rates can be made in the limit where the variations of the stochastic parameters are very slow compared to the Hamiltonian dynamics. Exact asymptotic results can be obtained in the one-dimensional case where the Hamiltonian dynamics is integrable for fixed values of the parameters. For the more interesting chaotic Hamiltonian dynamics case, we show that two mechanisms contribute to the transport. For some range of the parameter variations, one mechanism -called "transport by migration with the mixing regions" - is dominant. We are then able to model transport in phase space by a Markov model, the local diffusion model, and to give reasonably good transport estimates. [ABSTRACT FROM AUTHOR]