High Dimensional Harmonic Balance Method Research Articles

Prediction of Period-1 oscillations of the state-dependent delayed compound TCP model with PIE queue management policy via HDHBM

State-dependent time-delayed systems are more realistic delayed systems in nature and human society. Their dynamic problems are hot issues and difficult questions due to the fact that the classical RFDE theory is not suitable for them any more. And its corresponding rigorous theory is to be established. The study on the existence and stability of their periodic solutions is an important topic and a difficult question. Compound TCP is a widely used transport control protocol, and proportional integral enhanced (PIE) queue policy is an effective and simple queue policy in the Internet. Thus it is significant to study the dynamics and control of the compound TCP-PIE system. In this paper, Period-1 oscillations of the state-dependent delayed TCP models with PIE queue management policy are predicted by the high-dimensional harmonic balance method (HDHBM). Compared with the traditional harmonic balance method, HDHBM avoids the complex integration of the Fourier coefficients of the nonlinear right hand of the systems by solving the time domain variables of the Fourier coefficients at equally spaced sub-time over the period of one cycle of motion. Thus it transforms the problem into a relatively simple nonlinear algebraic equations, which greatly reduces the computation. By HDHBM, the approximate expressions of the Period-1 solutions for this state-dependent delayed compound TCP-PIE system are obtained. We compare their amplitudes and phase portraits with those of the DDE-BIFTOOL. They agree well with each other. It suggests the validity, effectiveness and simplicity of the employed method. It also turns out that many parameters of the state-dependent delayed TCP model with PIE queue management policy play a very important role in its dynamic behaviours. It follows that PIE parameter μ and update time T may play the most important roles in the Compound TCP-PIE congestion control model and packet loss probability p(t˜) may dominate the dynamics and performances of the network. As a result, we should control the parameters in a proper range to achieve our desired network performances.

A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity

A time domain collocation method for the study of the motion of a two dimensional aeroelastic airfoil with a cubic structural nonlinearity is presented. This method first transforms the governing ordinary differential equations into a system of nonlinear algebraic equations (NAEs), which are then solved by a Jacobian-inverse-free NAE solver. Using the aeroelastic airfoil as a prototypical system, the time domain collocation method is shown here to be mathematically equivalent to the well known high dimensional harmonic balance method. Based on the fact that the high dimensional harmonic balance method is essentially a collocation method in disguise, we clearly explain the aliasing phenomenon of the high dimensional harmonic balance method. On the other hand, the conventional harmonic balance method is also applied. Previous studies show that the harmonic balance method does not produce aliasing in the framework of solving the Duffing equation. However, we demonstrate that a mathematical type of aliasing occurs in the harmonic balance method for the present self-excited nonlinear dynamical system. Besides, a parameter marching procedure is used to sufficiently eliminate the effects of aliasing pertaining to the time domain collocation method. Moreover, the accuracy of the time domain collocation method is compared with the harmonic balance method.

A time domain collocation method for obtaining the third superharmonic solutions to the Duffing oscillator

In this study, a simple time domain collocation method (TDC) is applied to investigate the third superharmonic solutions of the Duffing oscillator. Upon using the proposed scheme, the multivaluedness, jump phenomenon, and transitional region of the third superharmonic response are explored. The amplitude frequency response curves for various values of damping, nonlinearity, and external force are obtained and compared. In addition, instead of collocating at N points so that the resulting nonlinear algebraic system is well determined, we extend the time domain collocation method to a new version by collocating at M>N points. The resulting over determined system is solved by the least square method. The extended time domain collocation method can significantly relieve the nonphysical solution phenomenon, which may be severe in the time domain collocation method, and its equivalent high dimensional harmonic balance method. Finally, numerical examples confirm the simplicity, efficiency, and accuracy of the proposed scheme.

A Simple Collocation Scheme for Obtaining the Periodic Solutions of the Duffing Equation, and its Equivalence to the High Dimensional Harmonic Balance Method: Subharmonic Oscillations

A Simple Collocation Scheme for Obtaining the Periodic Solutions of the Duffing Equation, and its Equivalence to the High Dimensional Harmonic Balance Method: Subharmonic Oscillations

High dimensional harmonic balance dealiasing techniques for a Duffing oscillator

A dealiasing study of the high dimensional harmonic balance method is conducted using a Duffing oscillator as a prototypical nonlinear dynamical system. Previous study has shown that aliasing can occur when the high dimensional harmonic balance method is used to solve systems that contain nonlinearities. We demonstrate that frequency filtering techniques, such as the Fourier smoothing method, can sufficiently reduce or eliminate the effects of aliasing. The drawback is that Fourier filtering requires redundant coordinate transformations between the time and frequency domains, resulting in unnecessary computational expense. As an alternative, temporal filters are constructed based upon compact finite difference schemes with spectral-like resolution. It is shown that the temporal filtering schemes successfully replicate the performance of the Fourier smoothing method with improved computational economy.

A novel harmonic balance analysis for the Van Der Pol oscillator

This study focuses on a novel harmonic balance formulation, the high-dimensional harmonic balance method. To investigate a non-linearity in the damping term, the system chosen for study is the Van der Pol's oscillator. Both unforced and forced oscillators are analyzed. The results from the analysis are compared with those obtained from the classical harmonic balance and the time marching (Runge–Kutta) methods.