Bounded perturbations of forced harmonic oscillators at resonance

Summary Let e be continuous and 2π-periodic, h continuous and bounded, and n>0 an integer. Sufficient conditions for the existence of 2π-periodic solutions of x″+n²x+h(x)= =e(t) are given. The proofs are based on a modification of Cesari's method and the Schauder fixed point theorem. Author is partially supported by N. S. F. under Grant 7447. Entrata in Redazione il 26 agosto 1968.     

Quasi-periodic solutions of forced isochronous oscillators at resonance

We deal with the existence of quasi-periodic solutions of forced isochronous oscillators with a repulsive singularity, the nonlinearity is a bounded perturbation. Using a variant of Moser's twist theorem of invariant curves, due to Ortega [R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999) 381-413], we show that there are many quasi-periodic solutions and the boundedness of all solutions.     

The stability of parametrically forced coupled oscillators in sum resonance

For a system of two damped parametrically forced oscillators in sum resonance the planar stability diagram of amplitude versus frequency of the forcing shows a discontinuity at damping zero. This is a well known phenomenon, for which we give a geometrical explanation. A linear stability analysis suffices. We show that a versal (i.e. a structurally stable) matrix unfolding for this problem needs four parameters, indicating that the stability diagram is actually four dimensional. The boundary of the stability region in parameter space is singular, this provides a geometric explanation of the discontinuity in the planar stability diagram.     

Analytical approximations to predict performance measures of manufacturing systems with general distributions,job failures and parallel processing

Parallel processing is prevalent in many manufacturing and service systems (i.e. some components may have to wait for other components before the assembly can begin). It is also common to observe manufacturing systems that deal with multiple products, resources shared between different products, and circulation due to random part failures. An example of such a system configuration is observed at a facility equipped to assemble and test web servers. The primary objective of this research was to develop analytical approximations to predict performance measures of a system with the above characteristics and evaluate its accuracy. Manufacturing systems with general distributions, multiple products, job circulation due to failures, resource sharing, and a fork and join system (to model parallel processing of some assembly operations) were studied using the parametric decomposition approach. The different work centers (or stations) in the manufacturing system is modeled as a network of queues and the parametric decomposition approach is applied to decompose the network of queues into individual queues to estimate the performance measure of the system. Existing analytical formulations were modified and appropriate correction terms were added to the approximations to bridge the gap in the error between the analytical approximation and the simulation models. Random instances were generated and the flow times from the approximations and simulation models were compared. The experimental study conducted indicates that the analytical approximations along with the correction terms can serve as a good estimate for the flow times of the manufacturing systems with the above characteristics.     

Exact solution of a model of three bound oscillators with Stark nonlinearity

A model of three bound boson oscillators with Stark nonlinearity is introduced and solved by the quantum inverse scattering method. For the trilinear oscillator, the eigenvalue problem is reduced to the spectral problem for the second-order homogeneous differential equation. Bibliography: 11 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 122–128. Translated by B. M. Bekker.     

Impulsive periodic and quasi-periodic orbits of coupled oscillators with essential stiffness nonlinearity

We study the impulsive responses of a grounded linear oscillator coupled to a light nonlinear attachment through an essentially nonlinear (nonlinearizable) stiffness. We analyze the periodic and quasi-periodic dynamics of the undamped system forced by a single impulse on the linear oscillator and being initially at rest, by considering separately low-, moderate- and high-energy impulsive motions. The motivation for studying the impulsive dynamics of this system centers on passive targeted energy transfer properties of the corresponding weakly damped one, that is, of the possibility of one-way, irreversible transfer of energy from the linear oscillator to the nonlinear attachment. A rather surprising aspect of this work is the complexity of the analysis required to study the impulsive dynamics of this system, due to its high degeneracy, as it undergoes a co-dimension three bifurcation.     

Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam

This paper employs the homotopy analysis method (HAM) to derive analytical approximate solutions for the nonlinear problem with high-order nonlinearity. Such a problem corresponds to the large-amplitude vibration of electrostatically actuated microbeams. The HAM is also optimized to accelerate the convergence of approximate solutions. To verify the accuracy of the present approach, illustrative examples are provided and compared with other analytical and exact solutions.     

Bifurcations of forced oscillators with fuzzy uncertainties by the generalized cell mapping method

In this paper, bifurcations in dynamical systems with fuzzy uncertainties are studied by means of the fuzzy generalized cell mapping (FGCM) method. A bifurcation parameter is modeled as a fuzzy set with a triangular membership function. We first study a boundary crisis resulting from a collision of a fuzzy chaotic attractor with a fuzzy saddle on the basin boundary. The fuzzy chaotic attractor together with its basin of attraction is eradicated as the fuzzy control parameter reaches a critical point. We also show that a saddle-node bifurcation is caused by the collision of a fuzzy period-one attractor with a fuzzy saddle on the basin boundary. The fuzzy attractor together with its basin of attraction suddenly disappears as the fuzzy parameter passes through a critical value.     

Application of the differential transformation method to the free vibrations of strongly non-linear oscillators

This paper adopts the differential transformation method to obtain the free vibration behavior of an oscillator with fifth-order non-linearities. The principle of differential transformation is briefly introduced, and is then applied in the derivation of a set of difference equations for the free vibration oscillator problem. The solutions are subsequently solved by a process of inverse transformation. The time responses of the oscillator are presented under different parameter conditions, and the current results are then compared with those derived from the established Runge–Kutta method in order to verify the accuracy of the proposed method. It is shown that there is excellent agreement between the two sets of results. This finding confirms that the proposed differential transformation method is a powerful and efficient tool for solving non-linear problems.     

On the sensitivity of strongly nonlinear autonomous oscillators and oscillatory waves to small perturbations

Original asymptotic solutions are determined for two autonomousdifferential equations. The application of initial conditionsfor the energy, wave number and phase shift proves to be lesscomplicated than in previous work. For the damped simple pendulum,explicit solutions demonstrate the dependence on the initialconditions. For strongly nonlinear wave packets of the Klein–Gordonequation, asymptotic solutions are compared. In both cases,the phase shift is shown to be highly sensitive to small perturbationsin the initial conditions.     

Asymptotic stability of solitons of the gKdV equations with general nonlinearity

We consider the generalized Korteweg-de Vries equation (gKdV)

Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators

A new approach, namely the global residue harmonic balance method, was advanced to determine the accurate analytical approximate periodic solution of a class of strongly nonlinear oscillators. A class of nonlinear jerk equation containing velocity-cubed and velocity times displacements-squared was taken as a typical example. Unlike other harmonic balance methods, all the former residual errors are introduced in the present approximation to improve the accuracy. Comparison of the result obtained using this approach with the exact one and simplicity and efficiency of the proposed procedure. The method can be easily extended to other strongly nonlinear oscillators.     

Analytical approximations to predict performance measures of markovian type manufacturing systems with job failures and parallel processing

Manufacturing or service systems with multiple product classes, job circulation due to random failures, resources shared between product classes, and some portions of the manufacturing or assembly carried in series and the rest in parallel are commonly observed in real-life. The web server assembly is one such manufacturing system which exhibits the above characteristics. Predicting the performance measures of these manufacturing systems is not an easy task. The primary objective of this research was to propose analytical approximations to predict the flow times of the manufacturing systems, with the above characteristics, and evaluate its accuracy. The manufacturing system is represented as a network of queues. The parametric decomposition approach is used to develop analytical approximations for a system with arrival and service rates from a Markovian distribution. The results from the analytical approximations are compared to simulation models. In order to bridge the gap in error, correction terms were developed through regression modeling. The experimental study conducted indicates that the analytical approximations along with the correction terms can serve as a good estimate for the flow times of the manufacturing systems with the above characteristics.     

Analytical approximations for the periodic motion of the Duffing system with delayed feedback

In this study, the homotopy analysis method is developed to give periodic solutions of delayed differential equations that describe time-delayed position feedback on the Duffing system. With this technique, some approximate analytical solutions of high accuracy for some possible solutions are captured, which agree well with the numerical solutions in the whole time domain. Two examples of dynamic systems are considered, focusing on the periodic motions near a Hopf bifurcation of an equilibrium point. It is found that the current technique leads to higher accurate prediction on the local dynamics of time-delayed systems near a Hopf bifurcation than the energy analysis method or the traditional method of multiple scales.     

Number and stability of nonlinear parametric oscillations in a system of weakly coupled oscillators with pendulum-type nonlinearity

The range of parameters for which the nonlinear system under study has three stable cycles with certain symmetry properties is determined. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 369–376, March, 1999.     

Finite-dimensional linear approximations of solutions to general irregular nonlinear operator equations and equations with quadratic operators

A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.     

Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity

This paper deals with the initial boundary value problem for strongly damped semilinear wave equations with logarithmic nonlinearity $u_{tt}-\Delta u-\Delta u_t={\phi }_p\left(u\right)log\left|u\right|$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$. We discuss the existence, uniqueness and polynomial or exponential energy decay estimates of global weak solutions under some appropriate conditions. Moreover, we derive the finite time blow up results of weak solutions, and give the lower and upper bounds for blow-up time by the combination of the concavity method, perturbation energy method and differential–integral inequality technique.