Higher Order Perturbation Analysis Research Articles

Dynamic Instabilities and Pattern Formation in Diffusive Epidemic Spread

The COVID-19 pandemic has refocused research on mathematical modeling and analysis of epidemic dynamics. We analytically investigate a partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread, marked by strongly nonlinear infection forces representing the infection transmission mechanism. Employing higher-order perturbation analysis and computing the local Lyapunov exponent, we observe the emergence of dynamic instabilities induced by stochastic environmental forces driving the epidemic spread. Notably, the instabilities are uncovered using third-order perturbations whilst they are not observed under second-order perturbations. Moreover, the onset of instability is more likely with increasing noise strength of the stochastic environmental forces.

A higher-order perturbation analysis of the nonlinear Schrödinger equation

A well-known and thoroughly studied phenomenon in nonlinear wave propagation is that of modulation instability (MI). MI is usually approached as a perturbation to a pump, and its analysis is based on preserving only terms which are linear on the perturbation, discarding those of higher order. In this sense, the linear MI analysis is relevant to the understanding of the onset of many other nonlinear phenomena, such as supercontinuum generation, but it has limitations as it can only be applied to the propagation of the perturbation over short distances.In this work, we propose approximations to the propagation of a perturbation, consisting of additive white noise, that go beyond the linear modulation instability analysis, and show them to be in excellent agreement with numerical simulations and experimental measurements.

Nonlinear tuning of microresonators for dynamic range enhancement.

This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators. Using geometrically exact mechanical formulations, a nonlinear model is obtained that governs the transverse and longitudinal dynamics of multilayer microbeams, and also takes into account rotary inertia effects. The partial differential equations of motion are discretized, according to the Galerkin method, after being reformulated into a mixed form. A zeroth-order shift as well as a hardening effect are observed in the frequency response of the beam. These results are confirmed by a higher order perturbation analysis using the method of multiple scales. An inverse problem is then proposed for the continuation of the critical amplitude at which the transition to nonlinear response characteristics occurs. Path-following techniques are employed to explore the dependence on the system parameters, as well as on the geometry of bilayer microbeams, of the magnitude of the dynamic range in nano/microresonators.

A refined asymptotic perturbation method for nonlinear dynamical systems

In this paper, a refined asymptotic perturbation method for general nonlinear dynamical systems is proposed for the first time. This method can be considered as an alternative means for the traditional multiple scales method. Moreover, it is easier to be understood and used to carry out higher-order perturbation analysis. In addition, three examples including the Duffing equation, a system with quadratic and cubic nonlinearities to a subharmonic excitation, as well as the coupled van der Pol oscillator with parametrical excitations are investigated to illustrate the validity and usefulness of the proposed technique. The analytical and numerical results show good agreement.

Editorial

Editorial

Tidal mass loss from collisionless systems

We examine the problem tidally-induced mass loss from collisionless systems such as dark matter haloes. We develop a model for tidal mass loss, based upon the phase space distribution of particles, which accounts for how both tidal and Coriolis torques perturb the angular momentum of each particle in the system. This allows us to study how both the density profile and velocity anisotropy affect the degree of mass loss--we present basic results from such a study. Our model predicts that mass loss is a continuous process even in a static tidal field, a consequence of the fact that mass loss weakens the potential of the system making it easier for further mass loss to occur. We compare the predictions of our model with N-body simulations of idealized systems in order to check its validity. We find reasonable agreement with the N-body simulations except for in the case of very strong tidal fields, where our results suggest that a higher-order perturbation analysis may be required. The continuous tidally-induced mass loss predicted by our model can lead to substantial reduction in satellite mass in cases where the traditional treatment predicts no mass loss. As such, our results may have important consequences for the orbits and survival of low mass satellites in dark matter haloes.

The Response of an Oscillator Moving along a Parabola to an External Excitation

The behavior of a mass point moving along a parabola under theeffect of an external periodic excitation in resonance with the naturalfrequency of the oscillator is studied. The asymptotic perturbationmethod based on temporal rescaling and balancing of the harmonic termswith a simple iteration is used in order to determine the nonlinearmodulation equations for the amplitude and the phase of the oscillation.External force-response curves are shown and moreover jump phenomena arealso observed. In certain cases a second low frequency appears inaddition to the forcing frequency and then stable two-periodquasi-periodic motions are present with amplitudes depending on theinitial conditions. The value of the low frequency depends on theamplitude of the external excitation. A higher order perturbationanalysis is developed and the validity of the method is highlighted bycomparing the leading order and the higher order approximate analyticsolutions to numerical results.

Multiple-time higher-order perturbation analysis of the regularized long-wavelength equation.

By considering the long-wavelength limit of the regularized long wave (RLW) equation, we study its multiple-time higher-order evolution equations. As a first result, the equations of the Korteweg--de Vries hierarchy are shown to play a crucial role in providing a secularity-free perturbation theory in the specific case of a solitary-wave solution. Then, as a consequence, we show that the related perturbative series can be summed and gives exactly the solitary-wave solution of the RLW equation. Finally, some comments and considerations are made on the N-soliton solution, as well as on the limitations of applicability of the multiple-scale method in obtaining uniform perturbative series. \textcopyright{} 1996 The American Physical Society.

Numerical solution of a Rayleigh-Taylor instability problem

The numerical solution of a Rayleigh-Taylor instability problem where an inviscid liquid of finite depth is accelerated into a gas of semi-infinite extent is obtained by transforming the irregular flow domain into a rectangular domain by a coordinate transformation. The free surface equation is solved by a Crank-Nicolson procedure. The boundary condition at the free surface for the velocity potential ø which contains the time derivative of ø is also treated by an implicit scheme. The numerical results agree well with those obtained by higher order perturbation analysis.

A second- and higher-order perturbation analysis of two-body trajectories

An approach to the development of a second- and higher-order perturbation theory for two- body trajectories is presented. It is shown that the higher order analysis can be developed in a systematic manner in terms of series solutions to the two-body problem as functions of the time variable. Simple algebraic recursive formulas for the determination of the series coeffi- cients are derived and the radii of convergence for these series solutions are determined. Their accuracy and the rate of convergence are investigated in a number of numerical cases. ingly more precise mission objectives and with long flight times. It is the purpose of this paper to present an approach to the higher order perturbation analysis of two-body trajec- tories. A second-order theory, involving second partial derivatives of the six-dimensional state vector of a spacecraft at a given time with respect to the state vector at an initial epoch, is con- sidered in detail. It is shown that a second as well as higher order theory can be developed in a simple and systematic manner using power series solutions of the two-body problem as functions of the time variable, a technique suggested by the theory of Lie series.1 In this paper, the series representa- tions associated with the second-order theory are developed; they are uniformly valid for all two-body conies. It is shown that the coefficients of these series expansions can be generated by simple algebraic recursive formulas, and their radii of con- vergence can be determined analytically. The rate of convergence and truncation error associated with these series representations are investigated in a number of numerical cases. Results indicate that the convergence characteristics and accuracies can in general be estimated in terms of the eccentricity and the position vector at the initial epoch. Generally speaking, the results exhibit good con- vergence characteristics for elliptical orbits covering less than one orbital revolution. Results for hyperbolic orbits are less favorable.