A Van der Pol-Mathieu equation for the dynamics of dust grain charge in dusty plasmas

The chaotic profile of dust grain dynamics associated with dust-acoustic oscillations in a dusty plasma is considered. The collective behaviour of the dust plasma component is described via a multi-fluid model, comprising Boltzmann distributed electrons and ions, as well as an equation of continuity possessing a source term for the dust grains, the dust momentum and Poisson's equations. A Van der Pol–Mathieu-type nonlinear ordinary differential equation for the dust grain density dynamics is derived. The dynamical system is cast into an autonomous form by employing an averaging method. Critical stability boundaries for a particular trivial solution of the governing equation with varying parameters are specified. The equation is analysed to determine the resonance region, and finally numerically solved by using a fourth-order Runge–Kutta method. The presence of chaotic limit cycles is pointed out.

Twentieth Century Modern Architecture and the Countryside: Ludwig Mies van der Rohe's design for a country gold clubhouse for the Krefeld Golf Club Association

This paper investigates relationships between modernity and monumentality in the architecture of Ludwig Mies van der Rohe. In his Modern Architecture, the critic and historian Kenneth Frampton separated Mies’ work into two historical periods, 1921-1933 and 1933-1967; the first he entitled ‘Mies van der Rohe and the significance of fact,’ the second ‘Mies van der Rohe and the monumentalisation of technique.’ The two historical periods correspond to two different geopolitical phases of Mies’ career, the first in Weimar Germany the second in the United States. By looking at a number of designs and texts made by Mies in the 1930’s and 1940’s, this essay questions the validity of separating Mies’ architecture into such clear-cut categories, where each one can enjoy a seeming independence from the other. The fulcrum for the discussion is Mies’ design of 1930 for a country golf clubhouse for the industrial town of Krefeld in north-western Germany. Our attention to the golf clubhouse design was prompted by the recent installation (2013), in which a 1-1 model of the design, made primarily from plywood, was erected in a field close the the site of Mies' original proposal.

Complex-order forced van der Pol oscillator

In this paper we consider a complex-order forced van der Pol oscillator. The complex derivative Dα1jβ, with α, β ∈ ℝ+, is a generalization of the concept of an integer derivative, where α = 1, β = 0. The Fourier transforms of the periodic solutions of the complex-order forced van der Pol oscillator are computed for various values of parameters such as frequency ω and amplitude b of the external forcing, the damping μ, and parameters α and β. Moreover, we consider two cases: (i) b = 1, μ = {1.0, 5.0, 10.0}, and ω = {0.5, 2.46, 5.0, 20.0}; (ii) ω = 20.0, μ = {1.0, 5.0, 10.0}, and b = {1.0, 5.0, 10.0}. We verified that most of the signal energy is concentrated in the fundamental harmonic ω0. We also observed that the fundamental frequency of the oscillations ω0 varies with α and μ. For the range of tested values, the numerical fitting led to logarithmic approximations for system (7) in the two cases (i) and (ii). In conclusion, we verify that by varying the parameter values α and β of the complex-order derivative in expression (7), we accomplished a very effective way of perturbing the dynamical behavior of the forced van der Pol oscillator, which is no longer limited to parameters b and ω.

Complex order van der Pol oscillator

In this paper a complex-order van der Pol oscillator is considered. The complex derivative Dα±ȷβ , with α,β∈R + is a generalization of the concept of integer derivative, where α=1, β=0. By applying the concept of complex derivative, we obtain a high-dimensional parameter space. Amplitude and period values of the periodic solutions of the two versions of the complex-order van der Pol oscillator are studied for variation of these parameters. Fourier transforms of the periodic solutions of the two oscillators are also analyzed.