Nonlinear Negative Feedback Research Articles

Periodic Solutions in a Simple Delay Differential Equation

A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in the case with piecewise constant nonlinearities involved. The periodic dynamics are shown to persist under small perturbations of the equation, which make it smooth. The theoretical results are verified through extensive numerical simulations.

A SPICE Model for IGBTs and Power MOSFETs Focusing on EMI/EMC in High-Voltage Systems

We describe two models of Power Transistors (IGBT, MOSFET); both were successfully used for the analysis of electromagnetic interference (EMI) and electromagnetic compatibility (EMC) while modeling high-voltage systems (PFC, DC/DC, inverter, etc.). The first semi-mathematical–behavioral insulated-gate bipolar transistor (IGBT) model introduces nonlinear negative feedback generated in the semiconductor’s p+ and n+ layers, which are located near the metal contact of the IGBT emitter, to better describe the dynamic characteristics of the transistor. A simplified model of the metal–oxide-semiconductor field-effect transistor (MOSFET) in the IGBT is used to simplify this IGBT model. The second simpler behavioral model could be used to model both IGBTs and MOSFETs. Model parameters are obtained from datasheets and then adjusted using results from a single measurement test. Modeling results are compared with measured turn-on and turn-off waveforms for different types of IGBTs. To check the validation of the models, a brushless DC electric motor test setup with an inverter was created. Despite the simplicity of the presented models, a comparison of model predictions with hardware measurements revealed that the model accurately forecasted switch transients and aided EMI–EMC investigations.

Instability in large bounded domains—branched versus unbranched resonances

We study transitions from convective to absolute instability near a trivial state in large bounded domains for prototypical model problems in the presence of transport and negative nonlinear feedback. We identify two generic scenarios, depending on the nature of the linear mechanism for instability, which both lead to different, universal bifurcation diagrams. In the first, classical case of a linear branched resonance the transition is hard, that is, small changes in a control parameter lead to a finite-size state. In the second, novel case of an unbranched resonance, the transition is gradual. In both cases, the bifurcation diagram is determined by interaction of the leading edge of an invasion front with upstream boundary conditions. Technically, we analyze this interaction in a heteroclinic gluing bifurcation analysis that uses geometric desingularization of the trivial state.

Stability of Synchronous Slowly Oscillating Periodic Solutions for Systems of Delay Differential Equations with Coupled Nonlinearity

We study stability of so-called synchronous slowly oscillating periodic solutions (SOPSs) for a system of identical delay differential equations (DDEs) with linear decay and nonlinear delayed negative feedback that are coupled through their nonlinear term. Under a row sum condition on the coupling matrix, existence of a unique SOPS for the corresponding scalar DDE implies existence of a unique synchronous SOPS for the coupled DDEs. However, stability of the SOPS for the scalar DDE does not generally imply stability of the synchronous SOPS for the coupled DDEs. We obtain an explicit formula, depending only on the spectrum of the coupling matrix, the strength of the linear decay and the values of the nonlinear negative feedback function near plus/minus infinity, that determines the stability of the synchronous SOPS in the asymptotic regime where the nonlinear term is heavily weighted. We also treat the special cases of so-called weakly coupled systems, near uniformly coupled systems, and doubly nonnegative coupled systems, in the aforementioned asymptotic regime. Our approach is to estimate the characteristic (Floquet) multipliers for the synchronous SOPS. We first reduce the analysis of the multidimensional variational equation to the analysis of a family of scalar variational-type equations, and then establish limits for an associated family of monodromy-type operators. We illustrate our results with examples of systems of DDEs with mean-field coupling and systems of DDEs arranged in a ring.

Does exchange rate management affect the causality between exchange rates and oil prices? Evidence from oil-exporting countries

The bi-directional causality between exchange rates and oil prices of oil-exporting countries has been widely investigated since the oil crisis in the 1970s. The classical theoretical framework assumes that countries adopt free-floating exchange rate arrangements, and empirical studies seldom consider exchange rate restrictions when investigating oil exporters that always manage exchange rates. To address these limitations, this study designs a comparative analysis for three groups of oil-exporting economies that adopt different foreign exchange rate policies. The analysis aims to examine whether and how intervention behavior may distort the causal relationship between exchange rates and oil prices. We first show the non-linear causality relationships for countries with “free-floating exchange rate arrangements” in both long and short terms. Then, we establish how the managed floating policy can eliminate the impact of oil prices on exchange rates in bull markets but fail to prevent exchange rate depreciation in bear markets. In bear markets, a negative non-linear feedback from exchange rates to oil prices prevails in these countries. However, contrary to common wisdom, the strict “soft-pegged exchange rate arrangement” cannot remove all the linkages between exchange rates and oil prices. A small but significant response of exchange rates toward oil prices is detected for Kuwait and Saudi Arabia. A similar effect for Kazakhstan is only found in bear markets. In addition, this study re-investigates the relationship between exchange rate and the domestic currency price of oil. The conclusions are similar, except that the relationship is weakened in free-floating countries but strengthened in managed floating countries during the bear market periods.

Modeling & Analysis of Gene Expression as a Nonlinear Feedback Problem With State-Dependent Delay

This paper models the dynamics of gene expression as a nonlinear negative feedback system with a state-dependent delay. The delay accounts for the lag from the initiation of transcription until the appearance of the mature protein messenger RNA (mRNA). We do not consider this delay to be constant. Rather, we take this delay to be dependent on the instantaneous concentration of the protein. This gives rise to a nonlinear biological system with a state-dependent delay. We consider the degradation of mRNA and protein in the mathematical model. We give conditions for the asymptotic stability of the system. We also give examples and simulation results for which periodic (oscillatory) solutions/limit cycles are obtained.

Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model

Abstract. In this study, we construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. Compared to the 5DLM, the 7DLM includes two additional high wavenumber modes that are selected based on an analysis of the nonlinear temperature advection term, a Jacobian term (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos, as compared to an rc of 42.9 for the 5DLM as well as an rc of 24.74 for the 3DLM. The rc is determined by an analysis of ensemble Lyapunov exponents (eLEs) with a Prandtl number (σ) of 10. To examine the dependence of rc on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of the Rayleigh parameter (40 ≤ r ≤ 195) and the Prandtl number (5 ≤ σ ≤ 25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5 ≤ σ ≤ 25), the 7DLM requires a larger rc for the onset of chaos than the 5DLM. In addition to the negative nonlinear feedback illustrated and emulated by the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM reveals the hierarchical scale dependence of chaotic solutions. For chaotic solutions with r = 120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z1) and between the secondary and tertiary modes (i.e., Z1 and Z2) are 0.988 and 0.998, respectively. Here, Z, Z1, and Z2 represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher-dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.

System Implementation of Speed Restriction While Developing the Force Compensation Systems

The problems arising at operation of force compensating systems were considered. The necessity to apply additional speed restriction systems was proved. Two approaches to implement speed restriction systems were proposed due to application of a nonlinear negative feedback of speed and with use of the motor acceleration control. The ways of considered systems were proposed, the analysis of their operation in static and dynamic modes was made. Operation simulation of speed restriction systems was performed while operating under conditions of impact of various external efforts. Pilot studies’ results in relation to restriction systems operation were given. Advantages and disadvantages of proposed speed restriction systems were reviewed. The areas of the rational application of different systems of speed restriction were defined.

Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the streamfunction is referred to as a secondary streamfunction mode, while the two additional modes, which appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): "If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado." The implications of this and previous work, as well as future work, are also discussed.

A Compensation-Type Pyrometer

The operating principles and engineering solutions for constructing compensation-type pyrometers with nonlinear negative feedback and a built-in calibrator are considered.

Nonlinear Feedback in a Five-Dimensional Lorenz Model

Abstract In this study, based on the number of modes, the original three-dimensional Lorenz model (3DLM) is generalized with two additional modes [five-dimensional Lorenz model (5DLM)] to examine their role in the predictability of the numerical solutions and to understand the underlying processes that increase the solution stability. As a result of the simplicity of the 5DLM with respect to existing generalized Lorenz models (LMs), the author is able to obtain the analytical solutions of its critical points and identify the role of the major nonlinear term in the solution’s stability, which have previously not been documented in the literature. The nonlinear Jacobian terms of the governing equations are analyzed to highlight the importance of selecting new modes for extending the nonlinear feedback loop of the 3DLM and thus effectively increasing the degree of nonlinearity (i.e., the nonlinear mode–mode interactions) in the 5DLM. It is then shown that numerical solutions in the 5DLM require a larger normalized Rayleigh number r for the onset of chaos and are more predictable than those in the 3DLM when r is between 25 and 40 and the Prandtl number σ is 10. The improved predictability is attributable to the negative nonlinear feedback enabled by the new modes. The role of the (negative) nonlinear feedback is further verified using a revised 3DLM with a parameterized nonlinear eddy dissipative term. The finding of the increased stability in the 5DLM and revised 3DLM with respect to the 3DLM is confirmed with the linear stability analysis and the analysis of the Lyapunov exponents using different values of r and σ. To further understand the impact of an additional heating term, results from the 5DLM and a higher-dimensional LM [e.g., the six-dimensional LM (6DLM)] are analyzed and compared.

Cluster synchronisation of non‐linearly coupled Lur'e networks with identical and non‐identical nodes and an asymmetrical coupling matrix

In this study, the cluster synchronisation problem of Lur’e networks is focused with non-linear coupling. There are both identical and non-identical nodes in the dynamical system and the coupling matrix is asymmetrical. By using the linear and non-linear negative feedback control schemes, the Lyapunov stability theorem and linear matrix inequality, sufficient conditions are obtained that guarantee the realisation of the cluster synchronisation pattern for all initial values. Numerical simulation results are also given to support the validity of the main results.

Additive noise quenches delay-induced oscillations

Noise has significant impact on nonlinear phenomena. Here we demonstrate that, in opposition to previous assumptions, additive noise interferes with the linear stability of scalar nonlinear systems when these are subject to time delay. We show this by performing a recently designed time-dependent delayed center manifold (DCM) reduction around a Hopf bifurcation in a model of nonlinear negative feedback. Using this, we show that noise intensity must be considered as a bifurcation parameter and thus shifts the threshold at which delay-induced rhythmic solutions emerge.

Exploring New Models for Population Prediction in Detecting Demographic Phase Change for Sparse Census Data

The logistic model has some limitations when applied to the sparse census data sets, typically available for developing countries. In such a situation, the relative growth rates (RGR) exhibit some unusual trends which are different from the common decreasing trend of logistic law. To explain such irregular trends we extend the logistic law by incorporating nonlinear positive and negative feedback terms. We performed RGR modelling as a function of time, as the size covariate model is not analytically solvable and the underlying model is better identifiable in the former case. It can also detect the demographic phase change point of developing country.

Follow the Money: The Monetary Roots of Bubbles and Crashes

We propose a reduced form model for the Minskian dynamics of liquidity and of asset prices in terms of the so-called financial accelerator mechanism. In a nutshell, credit creation is driven by the market value of the financial assets employed as collateral in the bank loans. This leads to a self-reinforcing feedback between financial prices and liquidity that we model by coupled non-linear stochastic processes. We show that the resulting dynamics are characterized by a transient super-exponential growth qualifying a bubble regime. Unchecked, this would lead to a finite time singularity (FTS). The underlying singularity expresses the unsustainable dynamics of the corresponding economy and announces a regime change, such as a crash. We propose to describe the dynamics of the crisis by the same coupled non-linear stochastic process with inverted signs, i.e., non-linear negative feedback of value and money on their growth rates. Casting the financial accelerator dynamics into a simple macroeconomic model, we show that the cycle of booms and bursts of financial assets and liquidity determines economic recessions in the form of increasing aggregate default rates and decreasing GDP. Finally, by exploiting the implications of the proposed model on the dynamics of financial asset returns, we introduce a generalized GARCH process, called FTS-GARCH, that can provide an early warning identification of bubbles. Estimating the FTS-GARCH on well-known historical bubble episodes suggest the possibility to diagnose in real-time the presence of bubbles in financial time series.

Computational design of synthetic regulatory networks from a genetic library to characterize the designability of dynamical behaviors

The engineering of synthetic gene networks has mostly relied on the assembly of few characterized regulatory elements using rational design principles. It is of outmost importance to analyze the scalability and limits of such a design workflow. To analyze the design capabilities of libraries of regulatory elements, we have developed the first automated design approach that combines such elements to search the genotype space associated to a given phenotypic behavior. Herein, we calculated the designability of dynamical functions obtained from circuits assembled with a given genetic library. By designing circuits working as amplitude filters, pulse counters and oscillators, we could infer new mechanisms for such behaviors. We also highlighted the hierarchical design and the optimization of the interface between devices. We dissected the functional diversity of a constrained library and we found that even such libraries can provide a rich variety of behaviors. We also found that intrinsic noise slightly reduces the designability of digital circuits, but it increases the designability of oscillators. Finally, we analyzed the robust design as a strategy to counteract the evolvability and noise in gene expression of the engineered circuits within a cellular background, obtaining mechanisms for robustness through non-linear negative feedback loops.

A new oscillatory mechanism for the electro-oxidation of iodide involving two phase transitions and a disproportional reaction

Oscillations have been first observed during iodide oxidation in alkaline solution on a static platinum electrode, where phase transitions at the interface of electrode/solution are essential. The film formation of solid iodine acts as a negative nonlinear feedback slowing down the iodide oxidation due to its poor conductivity, and oxygen evolution as a positive nonlinear feedback by destroying the iodine film mechanically and by promoting its dissolution through convection mainly via a disproportional reaction. Based on the results of electrochemical experiments and in situ Raman spectroscopy, a tentative mechanism is given concerning the interfacial phase transitions and the disproportional reaction.

On the possibility of using a semiconductor crystal as a saturable absorber and a negative feedback element

The possibility of using a semiconductor crystal simultaneously as an element of nonlinear negative feedback and as a saturable absorber was studied. Numerical computer simulation of a laser model was performed. The calculations were carried out for a GaAs crystal. Analysis of the results showed that, at a corresponding concentration of defect levels and crystal thickness, it is possible to realize both nonlinear negative feedback and saturable absorption in the same crystal for mode locking.

Complex Metabolic Oscillations in Plants Forced by Harmonic Irradiance

Plants exposed to harmonically modulated irradiance, ≈1 + cos( ωt), exhibit a complex periodic pattern of chlorophyll fluorescence emission that can be deconvoluted into a steady-state component, a component that is modulated with the frequency of the irradiance ( ω), and into at least two upper harmonic components (2 ω and 3 ω). A model is proposed that accounts for the upper harmonics in fluorescence emission by nonlinear negative feedback regulation of photosynthesis. In contrast to simpler linear models, the model predicts that the steady-state fluorescence component will depend on the frequency of light modulation, and that amplitudes of all fluorescence components will exhibit resonance peak(s) when the irradiance frequency is tuned to an internal frequency of a regulatory component. The experiments confirmed that the upper harmonic components appear and exhibit distinct resonant peaks. The frequency of autonomous oscillations observed earlier upon an abrupt increase in CO 2 concentration corresponds to the sharpest of the resonant peaks of the forced oscillations. We propose that the underlying principles are general for a wide spectrum of negative-feedback regulatory mechanisms. The analysis by forced harmonic oscillations will enable us to examine internal dynamics of regulatory processes that have not been accessible to noninvasive fluorescence monitoring to date.

All-optical inverted operational amplifier derivedfrom negative nonlinear feedback systems

An all-optical inverted operational amplifier that has the characteristic of being both an optical inverter and an amplifier designed based on the negative nonlinear absorption effect has been realised by using an erbium-doped fibre amplifier with a feedback ring resonator. A NOR-type logic gate operation has been demonstrated using two optical signals of 1.53 µm.