Deterministic Nonlinearity Research Articles

Multi-valued perturbations on stochastic evolution equations driven by fractional Brownian motions

We consider a stochastic evolution inclusion having deterministic multi-valued nonlinearity and fractional Brownian motion with nonlinear diffusion. We establish the nonemptiness and compactness of its solution set. After that, the upper semicontinuity with respect to random parameters and initial values of the corresponding solution map is proved. In particular, the results on nonemptiness and upper semicontinuity imply that the inclusion under consideration defines a multi-valued random dynamical system. Moreover, under an extra smooth assumption on the diffusion, it is demonstrated that the solution set has the topological structure of R δ -type.

An efficient data-driven multiscale stochastic reduced order modeling framework for complex systems

Suitable reduced order models (ROMs) are computationally efficient tools in characterizing key dynamical and statistical features of nature. In this paper, a systematic multiscale stochastic ROM framework is developed for complex systems with strong chaotic or turbulent behavior. The new ROMs fundamentally differ from the traditional Galerkin ROM (G-ROM) or those deterministic closure ROMs that aim to minimize path-wise errors and apply mainly to laminar systems. Here, the new ROM focuses on recovering the large-scale dynamics to the maximum extent. At the same time, it also exploits cheap but effective conditionally linear functions as the closure terms to capture the statistical features of the medium-scale variables and their feedback to the large scales. The new ROM mimics the model structure of many complex systems, allowing it to be more physically explainable. In addition, physics constraints of energy-conserving nonlinearities are incorporated into the new ROM. One unique feature of the resulting ROM is that it facilitates an efficient and accurate scheme for nonlinear data assimilation, the solution of which is provided by closed analytic formulae that do not require ensemble methods. Such an analytical solvable data assimilation solution significantly accelerates computational efficiency. It allows the new ROM to avoid many potential numerical and sampling issues in recovering the unobserved states from partial observations. The closure term calibration is efficient and systematic via explicit mathematical formulae. The new ROM framework is applied to complex nonlinear systems in which the intrinsic turbulent behavior is either triggered by external random forcing or deterministic nonlinearity. The out-of-sample numerical results show that the new ROM significantly outperforms the G-ROM in both scenarios regarding reproducing the dynamical and statistical features and recovering unobserved states via the efficient data assimilation scheme.

The Existence of Autonomous Chaos in EDM Process

The dynamical evolution of electrical discharge machining (EDM) has drawn immense research interest. Previous research on mechanism analysis has discussed the deterministic nonlinearity of gap states at pulse-on discharging duration, while describing the pulse-off deionization process separately as a stochastic evolutionary process. In this case, the precise model describing a complete machining process, as well as the optimum performance parameters of EDM, can hardly be determined. The main purpose of this paper is to clarify whether the EDM system can maintain consistency in dynamic characteristics within a discharge interval. A nonlinear self-maintained equivalent model is first established, and two threshold conditions are obtained by the Shilnikov theory. The theoretical results prove that the EDM system could lead to chaos without external excitation. The time series of the deionization process recorded in the EDM experiments are then analyzed to further validate this theoretical conclusion. Qualitative chaotic analyses verify that the autonomous EDM process has chaotic characteristics. Quantitative methods are used to estimate the chaotic feature of the autonomous EDM process. By comparing the quantitative results of the autonomous EDM process with the non-autonomous EDM process, a deduction is further made that the EDM system will evolve towards steady chaos under an autonomous state.

Fokker\u2013Planck dynamics of the El Ni\xf1o-Southern Oscillation

The asymmetric nature of the El Niño-Southern Oscillation (ENSO) is explored by using a probabilistic model (PROM) for ENSO. Based on a Fokker–Planck Equation (FPE), PROM describes the dynamics of a nonlinear stochastic ENSO recharge oscillator model for eastern equatorial Pacific temperature anomalies and equatorial Pacific basin-averaged thermocline depth changes. Eigen analyses of PROM provide new insights into the stationary and oscillatory solutions of the stochastic dynamical system. The first probabilistic eigenmode represents a stationary mode, which exhibits the asymmetric features of ENSO, in case deterministic nonlinearities or multiplicative noises are included. The second mode is linked to the oscillatory nature of ENSO and represents a cyclic asymmetric probability distribution, which emerges from the key dynamical processes. Other eigenmodes are associated with the temporal evolution of higher order statistical moments of the ENSO system. The model solutions demonstrate that the deterministic nonlinearity plays a stronger role in establishing the observed asymmetry of ENSO as compared to the multiplicative stochastic part.

Nonlinear analysis of solar wind parameters

We have analyzed time series of solar wind speed, density, temperature and interplanetary magnetic field observed by ‘Advanced Composition Explorer’ (ACE) at the Earth orbits during solar maximum and minimum using nonlinear time series tools. Evidence for deterministic nonlinearity is present in all solar wind parameters. The dimension of the attractor obtained from each solar wind parameters varies from 2.5 to 4.5. Our result suggests that this variation is due to a different amount of noise in the solar wind parameters. We have also found that the dynamics of solar wind is almost identical during solar maximum and minimum at 1 AU. Our result supports the interaction of neighboring individual flow during its evolution from Sun and 1 AU.

A Machine Learning-Based Detection Technique for Optical Fiber Nonlinearity Mitigation

We investigate the performance of a machine learning classification technique, called the Parzen window, to mitigate the fiber nonlinearity in the context of dispersion managed and dispersion unmanaged systems. The technique is applied for detection at the receiver side, and deals with the non-Gaussian nonlinear effects by designing improved decision boundaries. We also propose a two-stage mitigation technique using digital back propagation and Parzen window for dispersion unmanaged systems. In this case, digital back propagation compensates for the deterministic nonlinearity and the Parzen window deals with the stochastic nonlinear signal-noise interactions, which are not taken into account by digital back propagation. A performance improvement up to 0:4 dB in terms of Q factor is observed.

Strongly Coupled Single-Quantum-Dot–Cavity System Integrated on a CMOS-Processed Silicon Photonic Chip

Quantum photonic integrated circuit (QPIC) is a promising tool for constructing integrated devices for quantum technology applications. In the optical regime, silicon photonics empowered by complementary-metal-oxide-semiconductor (CMOS) technology provides optical components useful for realizing large-scale QPICs. Optical nonlinearity at the single-photon level is required for QPIC to facilitate photon-photon interaction. However, to date, realization of optical elements with deterministic( i.e., not probabilistic) single-photon nonlinearity by using silicon-based components is challenging, despite the enhancement of the functionality of QPICs based on silicon photonics. In this study, we realize for the first time a strongly coupled InAs/GaAs quantum dot-cavity quantum electrodynamics (QED) system on a CMOS-processed silicon photonic chip. The heterogeneous integration of the GaAs cavity on the silicon chip is performed by transfer printing. The cavity QED system on the CMOS photonic chip realized in this work is a promising candidate for on-chip single-photon nonlinear element, which constitutes the fundamental component for future applications based on QPIC, such as, coherent manipulation and nondestructive measurement of qubit states via a cavity, and efficient single-photon filter and router.

Nonlinearity in data with gaps: Application to ecological and meteorological datasets

Datagaps are ubiquitous in real world observational data. Quantifying nonlinearity in data having gaps can be challenging. Reported research points out that interpolation can affect nonlinear quantifiers adversely, artificially introducing signatures of nonlinearity where none exist. In this paper we attempt to quantify the effect that datagaps have on the multifractal spectrum ($f(\alpha)$), in the absence of interpolation. We identify tolerable gap ranges, where the measures defining the $f(\alpha)$ curve do not show considerable deviation from the evenly sampled case. We apply this to the multifractal spectra of multiple data-sets with missing data from the SMEAR database. The datasets we consider include ecological datasets from SMEAR I, namely CO$_2$ exchange variation, photosynthetically active radiation levels and soil moisture variation time series, and meteorological datasets from SMEAR II, namely dew point variation and air temperature variation. We could establish multifractality due to deterministic nonlinearity in two of these datasets, even in the presence of gaps.

Comparison of DSP-based nonlinear equalizers for intra-channel nonlinearity compensation in coherent optical OFDM.

A novel versatile digital signal processing (DSP)-based equalizer using support vector machine regression (SVR) is proposed for 16-quadrature amplitude modulated (16-QAM) coherent optical orthogonal frequency-division multiplexing (CO-OFDM) and experimentally compared to traditional DSP-based deterministic fiber-induced nonlinearity equalizers (NLEs), namely the full-field digital back-propagation (DBP) and the inverse Volterra series transfer function-based NLE (V-NLE). For a 40Gb/s 16-QAM CO-OFDM at 2000km, SVR-NLE extends the optimum launched optical power (LOP) by 4dB compared to V-NLE by means of reduction of fiber nonlinearity. In comparison to full-field DBP at a LOP of 6dBm, SVR-NLE outperforms by ∼1 dB in Q-factor. In addition, SVR-NLE is the most computational efficient DSP-NLE.

Nonlinearity in high-frequency stock returns: Evidence from the Athens Stock Exchange

This study investigates empirically the presence of nonlinearities in the Athens Composite Share Price Index high-frequency returns. A preliminary analysis indicates that volatility exhibits a periodic intraday inverse J-shaped pattern, associated with the opening and closing of the market. Periodicity is then removed employing a Flexible Fourier Form. Subsequently, an ARMA–FIGARCH model over several frequencies yields that return volatility is long memory and self-similar. Nonlinear analysis with the use of the embedding dimension suggests that the filtered return process does not exhibit deterministic or higher-order stochastic nonlinearity. Rather, it is reminiscent of a random process. We conclude that the ACSPI data are nonlinear; however, nonlinearity is attributed to persistent ARCH effects.

Static Response of Geometrically Nonlinear Laminated Composite Plates Having Uncertain Material Properties

The effect of uncertainty in material properties on the transverse bending of laminated composite plate is investigated. The transverse shear and large rotations have been included in the system equation in the framework of higher order shear deformation theory. The analysis uses Green–Lagrange nonlinear strain displacement equations to model geometric nonlinearity. The stochastic finite element analysis is performed using a direct iteration approach to handle deterministic geometric nonlinearity and perturbation approach to handle the randomness in the material properties. Mean and variance of the transverse deflection have been obtained by employing a C0 isoparametric nonlinear finite element model.

Buckling and Post-Buckling Response of Laminated Composite Plate with Random System Properties

This article addresses the buckling and post-buckling stochastic analysis of a laminated composite plate having random system properties. The transverse shear effects have been included in the system equation in the framework of higher order shear deformation theory. The analysis uses Green Lagrange nonlinear strain displacement equations to model geometric nonlinearity. Direct iteration approach is used to handle deterministic geometric nonlinearity and perturbation approach is presented to handle the randomness in the system properties. Mean and variance of the random buckling and post-buckling strength have been obtained by employing a C 0 isoparametric nonlinear finite element model. Comparisons with the published results show the accuracy of the proposed probabilistic procedure. Results covering various features are presented for a laminated composite plate with different boundary conditions.

Geometrically Nonlinear Free Vibration of Laminated Composite Plate Embedded With Piezoelectric Layers Having Uncertain Material Properties

In this paper, the nonlinear free vibration stochastic characteristic of a smart laminated composite plate having random system properties is presented. The transverse shear effects have been included in the system equation in the frame work of higher order shear deformation theory. The analysis uses the Green-Lagrange nonlinear strain displacement relationship to model geometric nonlinearity. The direct iteration approach is used to handle deterministic geometric nonlinearity, and the perturbation approach is employed to handle the randomness in the system properties. Mean and variance of the random natural frequencies have been obtained by employing a C0 isoparametric nonlinear finite element model. Comparisons with the published results show the accuracy of the proposed procedure. A few results covering various features have been presented for a laminated composite plate with different boundary conditions.

Stochastic Parameterization Schemes for Use in Realistic Climate Models

Abstract Stochastic parameterizations of fast-evolving, subgrid-scale processes are increasingly being used in a range of models from conceptual models to general circulation models. However, stochastic terms are generally included in an ad hoc fashion. In this study, a systematic method—“Hasselmann’s method”—of stochastic parameterization is developed through the direct application of rigorously justified limit theorems that predict the effective slow dynamics in systems with coupled slow and fast variables. The multiple Hasselmann models form a hierarchy of models ordered by the time scales over which they are expected to provide good approximations to the slowly evolving variables. Adaptable, efficient algorithms for integrating these reduced models are developed that require minimal changes to the unreduced model. Hasselmann’s method is tested on an O(10 000)-dimensional (planetary and synoptic scale) quasigeostrophic model of atmospheric low-frequency variability. Low-dimensional deterministic and stochastic models in the planetary-scale modes alone are derived, which accurately generate the statistics of the corresponding modes of the unreduced model, including the statistical signatures of jet regime behavior. It is shown that deterministic nonlinearity through slow forcing averaged with respect to the fast modes distribution dominates over multiplicative noise in generating the regime behavior.

Assessment of the dynamical properties in EDM process—detecting deterministic nonlinearity of EDM process

Time series of gap state were often used as feedback signal in electrical discharge machining (EDM) adaptive control systems. However, models precisely describing the EDM process have never been built because of the once believed stochastic nature of the EDM process. In this case, the power of adaptive controls in EDM had not been fully brought into play. Before building a feasible model, it is prerequisite to determine whether an efficient stable EDM process is nonlinear or linear, deterministic or stochastic. The main purpose of this paper is to investigate the deterministic nonlinearity of the process. A discriminating method was first provided to judge states in the gap at sampling intervals from voltage and current. Gap state was then statistically quantified from a train of discriminated states at sampling intervals within a specified period of time. Based on a time series of gap state data, we took use of surrogate data method to detect the nonlinearity of the process. From the results of two kinds of tests, it can be concluded that the deterministic nonlinearity of the process reflected by gap states is intrinsic.

Detecting spatiotemporal nonlinear dynamics in resting state of human brain based on fMRI datasets

In this work, a nonlinear dynamics method, coupled map lattices, was applied to functional magnetic resonance imaging (fMRI) datasets to examine the spatiotemporal properties of resting state blood oxygen level-dependent (BOLD) fluctuations. Spatiotemporal Lyapunov Exponent (SPLE) was calculated to study the deterministic nonlinearity in resting state human brain of nine subjects based on fMRI datasets. The results show that there is nonlinearity and determinism in resting state human brain. Furthermore, the results demonstrate that there is a spatiotemporal chaos phenomenon in resting state brain, and suggest that fluctuations of fMRI data in resting state brain cannot be fully attributed to nuclear magnetic resonance noise. At the same time, the spatiotemporal chaos phenomenon suggests that the correlation between voxels varies with time and there is a dynamic functional connection or network in resting state human brain.

Controlling chaos based on an adaptive nonlinear compensator mechanism

The control problems of chaotic systems are investigated in the presence of parametric uncertainty and persistent external disturbances based on nonlinear control theory. By using a designed nonlinear compensator mechanism, the system deterministic nonlinearity, parametric uncertainty and disturbance effect can be compensated effectively. The renowned chaotic Lorenz system subjected to parametric variations and external disturbances is studied as an illustrative example. From the Lyapunov stability theory, sufficient conditions for choosing control parameters to guarantee chaos control are derived. Several experiments are carried out, including parameter change experiments, set-point change experiments and disturbance experiments. Simulation results indicate that the chaotic motion can be regulated not only to steady states but also to any desired periodic orbits with great immunity to parametric variations and external disturbances.

Tissue specificity of nonlinear dynamics in baseline fMRI

In this work, recent advances in the field of nonlinear dynamics (NLD) were applied to fMRI data to examine the spatio-temporal properties of BOLD resting state fluctuations. Five human subjects were imaged during resting state (visual fixation) at 3T using single-shot gradient echo planar imaging (EPI). Respiration and cardiac signals were concurrently recorded for retrospectively removing fluctuations due to these physiologic activities. Patterns of singularity in the complex plane (PSC) and Lempel-Ziv complexity (LZ) were used to study the deterministic nonlinearity in resting state fMRI data. The results show that there is greater nonlinearity (higher PSC) and determinism (lower LZ) in gray matter compared to white matter and CSF. In addition, the removal of respiratory and cardiac pulsations decreases the nonlinearity and determinism but does not alter the relative difference between gray matter and white matter. Therefore, our results demonstrate that determinism and nonlinearity in the fMRI data are tissue-specific, suggesting that they reflect native physiologic and metabolic fluctuations and are not a result of physiologic artifacts due to respiration and cardiac pulsation.

Consistency of Nonlinear System Response to Complex Drive Signals

The consistency of a nonlinear system's response to a repeated complex waveform drive signal is an important consideration in classical and quantum systems as diverse as lasers, neuronal networks, and manufacturing plants. We show from a consideration of different characteristic waveforms that there is typically an optimal drive amplitude for the most consistent response; internal noise sources dominate for small amplitude driving while deterministic system nonlinearity reduces consistency for large amplitudes. We test this general concept and its measurement experimentally and numerically on the specific example of a laser system.

Determinism in Financial Time Series

The attractive possibility that financial indices may be chaotic has been the subject of much study. In this paper we address two specific questions: Masked by stochasticity, do financial data exhibit deterministic nonlinearity?, and If so, so what?. We examine daily returns from three financial indicators: the Dow Jones Industrial Average, the London gold fixings, and the USD-JPY exchange rate. For each data set we apply surrogate data methods and nonlinearity tests to quantify determinism over a wide range of time scales (from 100 to 20,000 days). We find that all three time series are distinct from linear noise or conditional heteroskedastic models and that there therefore exists detectable deterministic nonlinearity that can potentially be exploited for prediction.