3,728 research outputs found
Intrinsic localized modes in parametrically driven arrays of nonlinear resonators
We study intrinsic localized modes (ILMs), or solitons, in arrays of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). The analysis is performed using an amplitude equation in the form of a nonlinear Schrödinger equation with a term corresponding to nonlinear damping (also known as a forced complex Ginzburg-Landau equation), which is derived directly from the underlying equations of motion of the coupled resonators, using the method of multiple scales. We investigate the creation, stability, and interaction of ILMs, show that they can form bound states, and that under certain conditions one ILM can split into two. Our findings are confirmed by simulations of the underlying equations of motion of the resonators, suggesting possible experimental tests of the theory
Galerkin approximations for the optimal control of nonlinear delay differential equations
Optimal control problems of nonlinear delay differential equations (DDEs) are
considered for which we propose a general Galerkin approximation scheme built
from Koornwinder polynomials. Error estimates for the resulting
Galerkin-Koornwinder approximations to the optimal control and the value
function, are derived for a broad class of cost functionals and nonlinear DDEs.
The approach is illustrated on a delayed logistic equation set not far away
from its Hopf bifurcation point in the parameter space. In this case, we show
that low-dimensional controls for a standard quadratic cost functional can be
efficiently computed from Galerkin-Koornwinder approximations to reduce at a
nearly optimal cost the oscillation amplitude displayed by the DDE's solution.
Optimal controls computed from the Pontryagin's maximum principle (PMP) and the
Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE
systems, are shown to provide numerical solutions in good agreement. It is
finally argued that the value function computed from the corresponding reduced
HJB equation provides a good approximation of that obtained from the full HJB
equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042
Reduced-order modeling and dynamics of nonlinear acoustic waves in a combustion chamber
For understanding the fundamental properties of unsteady motions in combustion chambers, and for applications of active feedback control, reduced-order models occupy a uniquely important position. A framework exists for transforming the representation of general behavior by a set of infinite-dimensional partial differential equations to a finite set of nonlinear second-order ordinary
differential equations in time. The procedure rests on an expansion of the pressure and velocity fields in modal or basis functions, followed by spatial averaging to give the set of second-order equations in time. Nonlinear gasdynamics
is accounted for explicitly, but all other contributing processes require modeling. Reduced-order models of the global behavior of the chamber dynamics, most importantly of the pressure, are obtained simply by truncating the
modal expansion to the desired number of terms. Central to the procedures is a criterion for deciding how many modes must be retained to give accurate results. Addressing that problem is the principal purpose of this paper. Our
analysis shows that, in case of longitudinal modes, a first mode instability problem requires a minimum of four modes in the modal truncation whereas, for a second mode instability, one needs to retain at least the first eight modes. A second important problem concerns the conditions under which a linearly stable system becomes unstable to sufficiently large disturbances. Previous work has given a partial answer, suggesting that nonlinear gasdynamics alone cannot produce pulsed or 'triggered' true nonlinear instabilities; that suggestion is now theoretically established. Also, a certain form of the nonlinear energy
addition by combustion processes is known to lead to stable limit cycles in a linearly stable system. A second form of nonlinear combustion dynamics with a new velocity coupling function that naturally displays a threshold character
is shown here also to produce triggered limit cycle behavior
Fractional dynamics of systems with long-range interaction
We consider one-dimensional chain of coupled linear and nonlinear oscillators
with long-range power wise interaction defined by a term proportional to
1/|n-m|^{\alpha+1}. Continuous medium equation for this system can be obtained
in the so-called infrared limit when the wave number tends to zero. We
construct a transform operator that maps the system of large number of ordinary
differential equations of motion of the particles into a partial differential
equation with the Riesz fractional derivative of order \alpha, when 0<\alpha<2.
Few models of coupled oscillators are considered and their synchronized states
and localized structures are discussed in details. Particularly, we discuss
some solutions of time-dependent fractional Ginzburg-Landau (or nonlinear
Schrodinger) equation.Comment: arXiv admin note: substantial overlap with arXiv:nlin/051201
A note on the dependence of solutions on functional parameters for nonlinear sturm-liouville problems
We deal with the existence and the continuous dependence of solutions on functional
parameters for boundary valued problems containing the Sturm-Liouville equation.
We apply these result to prove the existence of at least one solution for a certain class of
optimal control problems
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
Some dynamics of acoustic oscillations with nonlinear combustion and noise
The results given in this paper constitute a continuation of progress with nonlinear analysis of coherent oscillations in combustion chambers. We are currently focusing attention on two general problems of nonlinear behavior important to practical applications: the conditions under which a linearly unstable system will execute stable periodic limit cycles; and the conditions under which a linearly stable system is unstable to a sufficiently large disturbance. The first of these is often called 'soft' excitation, or supercritical bifurcation; the second is called 'hard' excitation, 'triggering,' or subcritical bifurcation and is the focus of this paper. Previous works extending over more than a decade have established beyond serious doubt (although no formal proof exists) that nonlinear gasdynamics alone does not contain subcritical bifurcations. The present work has shown that nonlinear combustion alone also does not contain subcritical bifurcations, but the combination of nonlinear gasdynamics and combustion does. Some examples are given for simple models of nonlinear combustion of a solid propellant but the broad conclusion just mentioned is valid for any combustion system.
Although flows in combustors contain considerable noise, arising from several kinds of sources, there is sound basis for treating organized oscillations as distinct motions. That has been an essential assumption incorporated in virtually all treatments of combustion instabilities. However, certain characteristics of the organized or deterministic motions seem to have the nature of stochastic processes. For example, the amplitudes in limit cycles always exhibit a random character and even the occurrence of instabilities seems occasionally to possess some statistical features. Analysis of nonlinear coherent motions in the presence of stochastic sources is therefore an important part of the theory. We report here a few results of power spectral densities of acoustic amplitudes in the presence of a subcritical bifurcation associated with nonlinear combustion and gasdynamics
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