Piecewise Linear Differential Equations Research Articles

Piecewise-linear and bilinear approaches to nonlinear differential equations approximation problem of computational structural mechanics

This paper presents a bilinear approach to nonlinear differential equations system approximation problem. Sometimes the nonlinear differential equations right-hand sides linearization is extremely difficult or even impossible. Then piecewise-linear approximation of nonlinear differential equations can be used. The bilinear differential equations allow to improve piecewise-linear differential equations behavior and reduce errors on the border of different linear differential equations systems areas. The matrices of bilinear differential equations system are estimated through nonlinear programming. The results of proposed approach application are presented.

Robust Unidirectional Airflow through Avian Lungs: New Insights from a Piecewise Linear Mathematical Model

Avian lungs are remarkably different from mammalian lungs in that air flows unidirectionally through rigid tubes in which gas exchange occurs. Experimental observations have been able to determine the pattern of gas flow in the respiratory system, but understanding how the flow pattern is generated and determining the factors contributing to the observed dynamics remains elusive. It has been hypothesized that the unidirectional flow is due to aerodynamic valving during inspiration and expiration, resulting from the anatomical structure and the fluid dynamics involved, however, theoretical studies to back up this hypothesis are lacking. We have constructed a novel mathematical model of the airflow in the avian respiratory system that can produce unidirectional flow which is robust to changes in model parameters, breathing frequency and breathing amplitude. The model consists of two piecewise linear ordinary differential equations with lumped parameters and discontinuous, flow-dependent resistances that mimic the experimental observations. Using dynamical systems techniques and numerical analysis, we show that unidirectional flow can be produced by either effective inspiratory or effective expiratory valving, but that both inspiratory and expiratory valving are required to produce the high efficiencies of flows observed in avian lungs. We further show that the efficacy of the inspiratory and expiratory valving depends on airsac compliances and airflow resistances that may not be located in the immediate area of the valving. Our model provides additional novel insights; for example, we show that physiologically realistic resistance values lead to efficiencies that are close to maximum, and that when the relative lumped compliances of the caudal and cranial airsacs vary, it affects the timing of the airflow across the gas exchange area. These and other insights obtained by our study significantly enhance our understanding of the operation of the avian respiratory system.

Solving Fractional Dynamical System with Freeplay by Combining Memory-Free Approach and Precise Integration Method

The Yuan-Agrawal (YA) memory-free approach is employed to study fractional dynamical systems with freeplay nonlinearities subjected to a harmonic excitation, by combining it with the precise integration method (PIM). By the YA method, the original equations are transformed into a set of first-order piecewise-linear ordinary differential equations (ODEs). These ODEs are further separated as three linear inhomogeneous subsystems, which are solved by PIM together with a predictor-corrector process. Numerical examples show that the results by the presented method agree well with the solutions obtained by the Runge-Kutta method and a modified fractional predictor-corrector algorithm. More importantly, the presented method has higher computational efficiency.

Periodic solutions of a generalized system of piecewise linear difference equations

In this article we consider the global behavior of the following system of piecewise linear difference equations: $x_{n+1}=|x_{n}|-y_{n}-b$ and $y_{n+1} = x_{n}+|y_{n}|$ where b is any positive real number and the initial condition $(x_{0}, y_{0})$ is an element of $R^{2}$ . By mathematical induction and direct computations we show that the solution to the system is eventually one of two particular prime period 3 solutions or the unique equilibrium solution.

Periodic orbits of a neuron model with periodic internal decay rate

In this paper we will study a non-autonomous piecewise linear difference equation which describes a discrete version of a single neuron model with a periodic internal decay rate. We will investigate the periodic behavior of solutions relative to the periodic internal decay rate. Furthermore, we will show that only periodic orbits of even periods can exist and show their stability character.

Coarse-grained optimization-driven design and piecewise linear modeling of synthetic genetic circuits

The aim of synthetic biology is to confer novel functions to cells by rationally interconnecting basic genetic parts into circuits. A key barrier in the design of synthetic genetic circuits is that only a qualitative description of the performance and interactions of the basic genetic parts is available in databases such as the Registry of Standard Biological Parts. Modeling approaches capable of harnessing this qualitative knowledge are thus timely. Here, we introduce an optimization-based framework, which makes use of the available qualitative information about basic biological parts to automatically identify the circuit elements and structures enabling a desired response to the presence/absence of input signals. Promoters and ribosome binding sites are categorized as high, medium or low efficiency and protein expressions in the circuit are described using piecewise linear differential equations. The desired function of the circuit is also mathematically described as the maximization/minimization of a constrained objective function. We employed this framework for the design of a toggle switch, a genetic decoder and a genetic half adder unit. The identified designs are consistent with previously constructed circuit configurations and in some cases point to completely new architectures. The identified non-intuitive circuit structures highlight the importance of accounting for ribosome binding site efficiencies and relative protein abundance levels in circuit design. Our results reaffirm the usefulness of the qualitative information for the coarse-grained genetic circuit design and simulation in the absence of detailed quantitative information.

Dynamics in piecewise linear and continuous models of complex switching networks

Activities of genes are controlled in a combinatorial fashion by the concentrations of chemical called transcription factors. We model this type of network by piecewise linear differential equations formed by embedding a logical switching network in a differential equation. We generate continuous nonlinear equations by replacing the step function discontinuities in the piecewise linear equations by sigmoidal control functions. As the sigmoidal functions become steep, the continuous equations approach piecewise linear differential equations. We carry out numerical studies of the continuous and piecewise linear equations for a 4-dimensional example with particularly interesting and complex behavior, showing that the dynamics in the continuous equation approaches those in the piecewise linear equation as the sigmoids become steep.

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

We describe generic sliding modes of piecewise-linear systems of differential equations arising in the theory of gene regulatory networks with Boolean interactions. We do not make any a priori assumptions on regulatory functions in the network and try to understand what mathematical consequences this may have in regard to the limit dynamics of the system. Further, we provide a complete classification of such systems in terms of polynomial representations for the cases where the discontinuity set of the right-hand side of the system has a codimension 1 in the phase space. In particular, we prove that the multilinear representation of the underlying Boolean structure of a continuous-time gene regulatory network is only generic in the absence of sliding trajectories. Our results also explain why the Boolean structure of interactions is too coarse and usually gives rise to several non-equivalent models with smooth interactions.

Piecewise linear difference equations and convexity

This note presents a new method for analysing piecewise linear difference equations. The equations are considered in their natural phase space and interpreted via their associated semigroups and number theoretic graphs.

A Hybrid Model of Mammalian Cell Cycle Regulation

The timing of DNA synthesis, mitosis and cell division is regulated by a complex network of biochemical reactions that control the activities of a family of cyclin-dependent kinases. The temporal dynamics of this reaction network is typically modeled by nonlinear differential equations describing the rates of the component reactions. This approach provides exquisite details about molecular regulatory processes but is hampered by the need to estimate realistic values for the many kinetic constants that determine the reaction rates. It is difficult to estimate these kinetic constants from available experimental data. To avoid this problem, modelers often resort to ‘qualitative’ modeling strategies, such as Boolean switching networks, but these models describe only the coarsest features of cell cycle regulation. In this paper we describe a hybrid approach that combines the best features of continuous differential equations and discrete Boolean networks. Cyclin abundances are tracked by piecewise linear differential equations for cyclin synthesis and degradation. Cyclin synthesis is regulated by transcription factors whose activities are represented by discrete variables (0 or 1) and likewise for the activities of the ubiquitin-ligating enzyme complexes that govern cyclin degradation. The discrete variables change according to a predetermined sequence, with the times between transitions determined in part by cyclin accumulation and degradation and as well by exponentially distributed random variables. The model is evaluated in terms of flow cytometry measurements of cyclin proteins in asynchronous populations of human cell lines. The few kinetic constants in the model are easily estimated from the experimental data. Using this hybrid approach, modelers can quickly create quantitatively accurate, computational models of protein regulatory networks in cells.

Whether chaos can be achieved in piecewise linear boundary value problems

We show that chaos in piecewise linear partial differential equations with linear boundary conditions is realizable: even the simplest boundary value problems of this kind can possess chaotic solutions in an open region of the parameter space.

On the Global Character of the System of Piecewise Linear Difference Equations xn+1=|xn|−yn−1 and yn+1=xn−|yn|

We consider the system in the title where the initial condition We show that the system has exactly two prime period-5 solutions and a unique equilibrium point . We also show that every solution of the system is eventually one of the two prime period-5 solutions or else the unique equilibrium point.

On the Global Character of the System of Piecewise Linear Difference Equations and

On the Global Character of the System of Piecewise Linear Difference Equations and

Fuzzy Logic in Genetic Regulatory Network Models

<p>Interactions between genes and the proteins they synthesize shape genetic regulatory networks (GRN). Several models have been proposed to describe these interactions, been the most commonly used those based on ordinary differential equations (ODEs). Some approximations using piecewise linear differential equations (PLDEs), have been proposed to simplify the model non linearities. However they not allways give good results. In this context, it has been developed a model capable of representing small GRN, combining characteristics from the ODE’s models and fuzzy inference systems (FIS). The FIS is trained through an artificial neural network, which forms an Adaptive Nertwork-based Fuzzy Inference System (ANFIS). This network allows to adapt the membership and output functions from the FIS according to the training data, thus, reducing the previous knowledge needed to model the specific phenomenon.<br /> In addition, Fuzzy Logic allows to express their rules through linguistic labels, which also allows to incorporate expert knowledge in a friendly way. The proposed model has been used to describe the Lac Operon in E. Coli and it has been compared with the models already mentioned. The outcome errors due to the training process of the ANFIS network are comparable with those of the models based on ODEs. Additionally, the fuzzy logic approach provides modeling flexibility and knowledge acquisition advantages.</p>

A note on some piecewise-linear difference equations with Mersenne-type periodic solutions

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2 k − 1) or 6(2 k − 1) for some k ≥ 1. The existence of solutions with such periods is also considered.

Modeling and simulation of nitrogen regulation in Corynebacterium glutamicum

Modeling the dynamic behaviour of biochemical systems at a molecular level aims at understanding and predicting the interactions of macromolecules inside the cell. Models of small subsystems based on differential equations not only prepare the way for the long-term goal of understanding a whole cell, but are inherently valuable due to their ability to predict the behaviour of the subsystem for varying external conditions or parameters. Nitrogen supply is essential for prokaryotes, thus the nitrogen uptake is an interesting target for model building. The goal is to provide new information about the interactions of the relevant proteins by performing various simulations. A model based on piecewise linear differential equations is formulated for the nitrogen uptake in Corynebacterium glutamicum. We theoretically derive a model for biochemical networks and introduce a general method for the parameter estimation which is also applicable in the case of very short time series. This approach is applied to a special system concerning the nitrogen uptake using Western blot experiments. The equations are set up for the main components of this system, the optimization problem for parameter estimation is formulated and solved, and simulations for the evaluation of the model as well as for predictions are carried out. We show that model building based on differential equations can also, when only a few measurements are performed, lead to a satisfactory model which provides valuable insights into the way it’s network components function. For example, we are able to make predictions about the maximal value of the time course as well as the steady-state level of the signal transduction protein GlnK in case of restricted activity of the proteases when considering the transition of nitrogen starvation to nitrogen excess or vice versa.

Origin of arrhythmias in a heart model

An investigation of the nonlinear dynamics of a heart model is presented. The model compartmentalizes the heart into one part that beats autonomously (the x oscillator), representing the pacemaker or SA node, and a second part that beats only if excited by a signal originating outside itself (the y oscillator), representing typical cardiac tissue. Both oscillators are modeled by piecewise linear differential equations representing relaxation oscillators in which the fast time portion of the cycle is modeled by a jump. The model assumes that the x oscillator drives the y oscillator with coupling constant α. As α decreases, the regular behavior of y oscillator deteriorates, and is found to go through a series of bifurcations. The irregular behavior is characterized as involving a large amplitude cycle followed by a number n of small amplitude cycles. We compute critical bifurcation values of the coupling constant, αn, using both numerical methods as well as perturbations.

Modeling gene regulatory networks with piecewise linear differential equations

Microarray chips generate large amounts of data about a cell’s state. In our work we want to analyze these data in order to describe the regulation processes within a cell. Therefore, we build a model which is capable of capturing the most relevant regulating interactions and present an approach how to calculate the parameters for the model from time-series data. This approach uses the discrete approximation method of least squares to solve a data fitting modeling problem. Furthermore, we analyze the features of our proposed system, i.e., which kinds of dynamical behaviors the system is able to show.

Frequency Coupling Matrix of a Voltage-Source Converter Derived From Piecewise Linear Differential Equations

When a power-electronic converter is introduced into a linear network, voltage and current harmonics of differing orders become coupled (through the modulation effect of the converter). The interharmonic coupling introduced by the modulation effect of a converter may be mathematically represented through a frequency coupling matrix (FCM). Given that the source of the coupling is a modulation process, researchers have, in the past, focused on deriving the FCM in the frequency domain - a process that requires the truncation of the harmonic representation of signals. This paper presents an alternate approach to evaluate the FCM based on a time-domain derivation. In contrast to frequency-domain-based methods, it is shown that the time-domain approach avoids truncation. Furthermore, the time-domain approach does not require system linearization about an operating point; thus, the FCM is not limited by small-signal assumptions.

Two classes of piecewise-linear difference equations with eventual periodicity three

In this paper, we will study two classes of difference equations which are piecewise-linear and of similar forms. We will show that all nontrivial solutions of one equation are eventually periodic with prime period three. We will show this result for one case of the second equation.