Nonlinear Deterministic Component Research Articles

Energy scattering of hybrid FE-SEA model with nonlinear joints

Nonlinearity in structures causes energy scattering between different frequency ranges. This paper proposes a new hybrid FE-SEA formulation to analyse complex structures with nonlinear deterministic components. The formulation is derived in two stages. First the governing equation for deterministic components in time domain is transformed to frequency domain using the method of harmonic balance and the Galerkin method. Equivalent stiffness method is then applied in frequency domain to linearise the nonlinear terms. The second process is regarded to the derivation of the hybrid model. Based on a proposed nonlinear energy transfer model and the linearisation in the first stage, the linearised hybrid FE-SEA governing equation is derived, and the iterative procedure is given. For validation purpose, nonlinear Lagrange–Rayleigh–Ritz method plus Monte Carlo simulation is used as the benchmark model. Two case studies are applied for validation and investigation of the coupling loss factor (CLF). The coupling between subsystems at different frequencies can be reflected by the CLF. Good agreements between the benchmark model and proposed hybrid model can be observed.

Problems in Forecasting of the Decennial Solar Activity in Terms of TSI by the Method of Analogs

Reconstruction and forecasting the data series of total solar irradiation (TSI) is one of the most urgent scientific problems at the moment. In this paper, we review three reconstructions of TSI based on the data of cosmogenic 14C and 10Be isotopes and sunspots and tested their predictability by a modified analog method for a one-step prediction of decadal averages. The data of cosmogenic isotopes allow us to estimate changes of the open magnetic flux of the Sun, the magnitude of which is indirectly associated with TSI changes. We establish that the quality of the TSI forecast decreases if we consider a composite timeseries compiled from different data sources. In particular, the analog method does not work for reconstruction, various parts of which were recovered according to cosmogenic isotopes, the number of sunspots, and also according to modern satellite observations. However, at the same time, it operates for reconstructions recovered using cosmogenic isotopes for the whole series. Thus, the predictability of the series manifests the presence of a nonlinear deterministic component and is one of the criteria of the quality of reconstructions.

Robust and Powerful Tests for Nonlinear Deterministic Components

AbstractWe develop a test for the presence of nonlinear deterministic components in a univariate time series, approximated using a Fourier series expansion, designed to be asymptotically robust to the order of integration of the process and to any weak dependence present. We show that our proposed test has uniformly greater local asymptotic power than the existing tests of Harvey, Leybourne and Xiao (2010) when the shocks are I(1), identical local asymptotic power when the shocks are I(0), and also improved finite sample properties. We also consider the issue of determining the number of Fourier frequencies used to specify any nonlinear deterministic components.

A wide nonlinear analysis of reactive power time series related to electric arc furnaces

Prediction of electric arc furnace (EAF) reactive power is an appropriate solution to compensate for static VAr compensator delay and improve its performance in flicker reduction. A linear autoregressive moving average (ARMA) can only pull out the linear deterministic (LD) component of EAF reactive power time series. For the prediction to be made through both nonlinear deterministic (ND) and LD components, employing nonlinear models is necessary. However, before developing the nonlinear models for prediction, the necessity of the employing them should be verified by investigating the significance of the ND components in the process. This paper presents a novel approach for wide analysis of nonlinear behavior of EAFs reactive power time series related to eight ac EAFs installed in Mobarakeh steel industry, Isfahan, Iran to answer the question about the importance of their ND components. In the approach, a suitable linear auto regressive moving average (ARMA) model with order (4,4) is used for the time series to extract the residual time series. Then, a number of well established nonlinear analysis techniques such as time delay reconstruction, surrogate data, delay vector variance and recurrence plot methods are applied to the original and residual time series. To describe the nonlinear characteristics of the time series, some new indices are defined. They quantify the significance of the ND component in compare with LD component.

Testing for nonlinear deterministic components when the order of integration is unknown

We consider testing for the presence of nonlinearities in the deterministic component of a time series, approximating the potential nonlinear behaviour using a Fourier function expansion. In contrast to procedures that are currently available, we develop tests that are robust to the order of integration, in the sense that they are asymptotically correctly sized regardless of whether the stochastic component of the series is stationary or contains a unit root. The tests we propose take the form of Wald statistics based on cumulated series, together with a correction factor to line up the asymptotic critical values across the I(0) and I(1) environments. The local asymptotic power and finite sample properties of the tests are evaluated using various different correction factors. We envisage that the testing procedure we recommend should be very useful to applied researchers wishing to draw robust inference regarding the presence of nonlinear deterministic components in a series.

Modelling Autoregressive Processes with a Shifting Mean

In this paper we introduce an autoregressive model with a deterministically shifting intercept. This implies that the model has a shifting mean and is thus nonstationary but stationary around a nonlinear deterministic component. The shifting intercept is defined as a linear combination of logistic transition functions with time as the transition variables. The number of transition functions is determined by selecting the appropriate functions from a possibly large set of alternatives using a sequence of specification tests. This selection procedure is a modification of a similar technique developed for neural network modelling by White (2006). A Monte Carlo experiment is conducted to show how the proposed modelling procedure and some of its variants work in practice. The paper contains two applications in which the results are compared with what is obtained by assuming that the time series used as examples may contain structural breaks instead of smooth transitions and selecting the number of breaks following the technique of Bai and Perron (1998).

Nonlinear dynamics of cycle-to-cycle combustion variations in a lean-burn natural gas engine

Temporal dynamics of the combustion process in a lean-burn natural gas engine was studied by the analysis of time series of consecutive experimental in-cylinder pressure data in this work. Methods borrowed to the nonlinear dynamical system theory were applied to analyze the in-cylinder pressure time series under operating conditions with different equivalence ratio. Phase spaces were reconstructed from the in-cylinder pressure time series and Poincaré section calculated from each phase space. Poincaré sections show that the in-cylinder combustion process involves chaotic behavior. Furthermore, return maps plotted from time series of indicated mean effective pressure show that both nonlinear deterministic components and stochastic components are involved in the dynamics of cycle-to-cycle combustion variations in the lean burn natural gas engine. There is a transition from stochastic behavior to noisy nonlinear determinism as equivalence ratio decreases from near stoichiometric to very lean conditions.

Detecting determinism and nonlinearity in river-flow time series

The temporal evolution of river-flow dynamics is investigated using methods from nonlinear time series analysis. After a discussion on the possible sources of nonlinear deterministic components, different methods for the detection of nonlinear and deterministic components in river flow time series are illustrated and the application to three river-flow time series from basins of different dimensions and characteristics is presented. All the results of different nonlinear techniques show noticeable evidence of determinism and nonlinearity at high discharge values, while at low discharge values, and especially during recessions, the series seem to be the result of a mostly linear dynamics. The complexity of the dynamics and the degree of nonlinearity are also discussed in relation to the basin dimensions and the kind of hydroclimatic regime.

Geometrical characteristics of magnetospheric energetic ion time series: evidence for low dimensional chaos

Abstract. In the first part of the paper we study the geometrical characteristics of the magnetospheric ions’ time series in the reconstructed phase space by using the SVD extended chaotic analysis, and we test the strong null hypothesis supposing that the ions’ time series is caused by a linear stochastic process perturbed by a static nonlinear distortion. The SVD reconstructed spectrum of the ions’ signal reveals a strong component of high dimensional, external coloured noise, as well as an internal low dimensional nonlinear deterministic component. Also, the stochastic Lorenz system produced by coloured noise perturbation of the deterministic Lorenz system was used as an archetype model in comparison with the dynamics of the magnetrospheric ions.Key words. Magnetospheric physics (energetic particles) – Radio science (nonlinear phenomena)

Coherent Resonant Millennial-Scale Climate Oscillations Triggered by Massive Meltwater Pulses

The role of mean and stochastic freshwater forcing on the generation of millennial-scale climate variability in the North Atlantic is studied using a low-order coupled atmosphere–ocean–sea ice model. It is shown that millennial-scale oscillations can be excited stochastically, when the North Atlantic Ocean is fresh enough. This finding is used in order to interpret the aftermath of massive iceberg surges (Heinrich events) in the glacial North Atlantic, which are characterized by an excitation of Dansgaard–Oeschger events. Based on model results, it is hypothesized that Heinrich events trigger Dansgaard–Oeschger cycles and that furthermore the occurrence of Heinrich events is dependent on the accumulated climatic effect of a series of Dansgaard–Oeschger events. This scenario leads to a coupled ocean–ice sheet oscillation that shares many similarities with the Bond cycle. Further sensitivity experiments reveal that the timescale of the oscillations can be decomposed into stochastic, linear, and nonlinear deterministic components. A schematic bifurcation diagram is used to compare theoretical results with paleoclimatic data.

Effects of noise on some dynamical models in ecology.

We investigate effects of random perturbations on the dynamics of one-dimensional maps (single species difference equations) and of finite dimensional flows (differential equations for n species). In particular, we study the effects of noise on the invariant measure, on the "correlation" dimension of the attractor, and on the possibility of detecting the nonlinear deterministic component by applying reconstruction techniques to the time series of population abundances. We conclude that adding noise to maps with a stable fixed-point obscures the underlying determinism. This turns out not to be the case for systems exhibiting complex periodic or chaotic motion, whose essential properties are more robust. In some cases, adding noise reveals deterministic structure which otherwise could not be observed. Simulations suggest that similar results hold for flows whose attractor is almost two-dimensional.