Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $${\mathcal {A}}$$ A of real hyperplanes with multiplicities admitting the rational Baker–Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $${\mathcal {A}}$$ A -Hermite polynomials. These polynomials form a linear basis in the space of $${\mathcal {A}}$$ A -quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type $$A_N$$ A N this leads to a quasi-invariant version of the Lassalle–Nekrasov correspondence and its higher order analogues.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

Analytical Solutions for Stable and Unstable Period-1 Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

In this note, a closed-form solution of periodic motions in a periodically forced oscillator with quadratic nonlinearity is presented without any small parameters. The perturbation method is based on one harmonic term plus perturbation modification, and the traditional harmonic balance is to arbitrarily select harmonic terms with constant coefficients. If harmonic terms are not enough included in the approximate solution, such a solution is not an appropriate, analytical solution for periodic motions, and some analytical solutions cannot be caught.

Aerodynamic Analysis of Cup Anemometers Performance: The Stationary Harmonic Response

The effect of cup anemometer shape parameters, such as the cups’ shape, their size, and their center rotation radius, was experimentally analyzed. This analysis was based on both the calibration constants of the transfer function and the most important harmonic term of the rotor’s movement, which due to the cup anemometer design is the third one. This harmonic analysis represents a new approach to study cup anemometer performances. The results clearly showed a good correlation between the average rotational speed of the anemometer’s rotor and the mentioned third harmonic term of its movement.

The Rapid Series Method for Solving Differential Equation of the Odd Power Oscillator

The new rapid series method to solve the differential equation of the periodic vibration of the strongly odd power nonlinear oscillator has been put forward in this paper. By adding the exponentially decaying factor to each harmonic term of the Fourier series of the periodic solution, the high accurate solution can be obtained with a few harmonic terms. The number of truncated terms is determined by the requirement of accuracy. Comparing with other approximate methods, the calculation of rapid series method is very easy and the accurate degrees of solution can be control. By comparing the analytical approximate solutions obtained by this method with numerical solutions of the cubic and fifth power oscillators, it is proven that this method is valid.

Harmonics Identification with Artificial Neural Networks: Application to Active Power Filtering

This study proposes several high precision selective harmonics compensation schemes for an active power filter. Harmonic currents are identified and on-line tracked by novel Adaline-based architectures which work in different reference-frames resulting from specific currents or powers decompositions. Adalines are linear and adaptive neural networks which present an appropriate structure to fit and learn a weighted sum of terms. Sinusoidal signals with a frequency multiple of the fundamental frequency are synthesized and used as inputs. Therefore, the amplitude of each harmonic term can be extracted separately from the Adaline weights adjusted with a recursive LMS (Least Mean Squares) algorithm. A first method is based on the modified instantaneous powers, a second method optimizes the active currents, and a third method relies on estimated fundamental currents synchronized with the direct voltage components. By tracking the fluctuating harmonic terms, the Adalines learning process allows the compensation schemes to be well suited for on-line adaptive compensation. Digital implementations of the identification schemes are performed and their effectiveness is verified by experiments.