Biochemical oscillations in delayed negative cyclic feedback: Existence and profiles

In this paper, we propose a theoretical framework to systematically analyze the existence and the profiles of chemical oscillations in gene regulatory networks with negative cyclic feedback. In particular, we analytically derive the existence conditions and the profiles of oscillations in terms of reaction kinetic parameters and reveal dimensionless quantities that essentially characterize the oscillatory dynamics. These discoveries then allow us to provide general biological insights that are useful for the design of synthetic gene circuits and wet-lab experiments. We point out that time delays due to splicing and transport play an important role for both of the existence and the profiles of oscillations. To this end, we first show that local instability leads to oscillations in cyclic gene regulatory networks, and we derive the existence conditions based on local instability analysis. Then, we analyze the period, phase and amplitude of oscillations using multivariable harmonic balance analysis. These results are demonstrated with two existing biochemical networks, the Pentilator and a self-repression network of a Hes protein.     

Absolute instability of Lur'e systems and its application to oscillation analysis of uncertain genetic networks

We derive instability criteria for Lur'e systems with sector‐bounded nonlinearities and uncertain external signals. First, we define absolute instability of an equilibrium and derive an absolute instability condition for a fixed equilibrium point in terms of a linear matrix inequality, which is analogous to the well‐known circle stability criterion. Then, the condition is extended to a parametric absolute instability condition, which is applicable to the instability test of a Lur'e system with an equilibrium point whose location is affected by uncertain nonlinearities and uncertain external signals. Finally, we show that the proposed analysis method is useful through the oscillation analysis of an uncertain genetic network model. Copyright © 2014 John Wiley & Sons, Ltd.     

限幅型非光滑吸振器模型的稳定性与周期运动研究

本文研究具有分段光滑特性的限幅型吸振器模型的稳定性与周期运动. 建立一类限幅型非光滑吸振器的动力学模型,探讨模型容许平衡点的存在性,通过Liénard-Chipart稳定性准则,分析容许平衡点的稳定性. 通过参数变换,将限幅型吸振器模型转化为具有两个切换流形的四维分段光滑系统. 通过计算系统首次积分,获得四维含参分段光滑动力系统在其未扰系统存在一族周期轨条件下的Melnikov函数. 探讨不同参数条件下系统周期轨的存在性及个数,并利用数值模拟方法给出其相图构型,验证理论结果的正确性. 研究结果表明不同的间隙参数影响系统周期轨个数及相对位置.     

Generating self-excited oscillations for underactuated mechanical systems via two-relay controller

A tool for the design of a periodic motion in an underactuated mechanical system via generating a self-excited oscillation of a desired amplitude and frequency by means of the variable structure control is proposed. First, an approximate approach based on the describing function method is given, which requires that the mechanical plant should be a linear low-pass filter–the hypothesis that usually holds when the oscillations are relatively fast. The method based on the locus of a perturbed relay systems provides an exact model of the oscillations when the plant is linear. Finally, the Poincaré map's design provides the value of the controller parameters ensuring the locally orbitally stable periodic motions for an arbitrary mechanical plant. The proposed approach is shown by the controller design and experiments on the Furuta pendulum.     

The Glansdorff–Prigogine stability criterion for biochemical reaction networks

Local stability analysis of steady states in mathematical models of biochemical reaction networks is an important tool for systems biology. The second variation of the Gibbs energy around a steady state is a positive definite function and a candidate for a Lyapunov function. A sufficient condition for the local stability is the local negative definiteness of the time derivative of this function. This is expressed by the Glansdorff–Prigogine stability criterion. Previously, the criterion was criticized to be overly conservative and difficult to check. Here, we derive an easily testable form of the criterion for models of biochemical networks. The criterion can be evaluated with incomplete knowledge of the parameters. For ideal mass-action kinetics, it depends only on the steady state fluxes. For reaction systems in ideal solutions, the Glansdorff–Prigogine criterion is overly conservative and we give a tighter criterion that depends on the same subset of the parameters as the Glansdorff–Prigogine criterion. Whenever these criteria are indefinite, there exist parameter values such that the steady state is unstable. By means of simple example systems we explore these aspects and discuss the possible uses of the criteria for systems biology.     

Elimination of overflow oscillations in digital filters employing saturation arithmetic

A criterion for the nonexistence of overflow oscillations in a class of digital filters employing saturation arithmetic is presented. The criterion is based on a novel characterization of the saturation nonlinearity (namely, the ‘reduced sector’ characterization) and, hence, is quite distinct from previously reported criteria.     

Forced resonant oscillations of nonlinear autonomous system in equilibrium neighborhood

Consideration was given to existence and stability of oscillations in the case of resonance where in the neighborhood of equilibrium the autonomous nonlinear system is subjected to periodic perturbations. For each of the probable cases where the equilibrium is surrounded or not by a family of periodic oscillations, established were the sufficient conditions solving the problem. The amplitudes of the resonant oscillations were estimated in terms of the small parameter.     

时滞和扩散驱动下信息物理融合系统恶意病毒传播动力学

本文提出了信息物理融合系统(CPS)中具有反应扩散效应的时滞恶意病毒传播模型, 研究了恶意病毒传播 的空间格局动态演化机制, 为恶意病毒在信息物理融合系统中预测和控制提供了战略指导. 给出了模型的基本再 生产数, 并分析了空间中无病毒平衡点和地方病毒平衡点的存在性. 在无时滞条件下建立了扩散引发的图灵不稳 定条件; 在有时滞条件下得到了时滞依赖的稳定性条件和Hopf分岔判据. 最后, 通过数值仿真验证了理论分析的正 确性.     

Acoustic modes of self-excited oscillations in struts

The acoustic modes of self-excited oscillations are investigated numerically for two-dimensional generic struts. In profiles with short chord lengths, hydrodynamic instability is responsible for wake bifurcation and for generation of narrow band tones. However, it is found, another mechanism is the major factor for periodic oscillations in long profiles. Two points of wall contour develops an acoustic feedback between each other. These points are the inflection point and the trailing edge point. Consequently, limits of two different modes of periodic oscillations are outlined.Moreover, it is determined that acoustic radiation has dipole character which agrees well with the theory of sound radiation for acoustically compact surfaces.     

A distributed predictive control approach for periodic flow-based networks: application to drinking water systems

This paper proposes a distributed model predictive control approach designed to work in a cooperative manner for controlling flow-based networks showing periodic behaviours. Under this distributed approach, local controllers cooperate in order to enhance the performance of the whole flow network avoiding the use of a coordination layer. Alternatively, controllers use both the monolithic model of the network and the given global cost function to optimise the control inputs of the local controllers but taking into account the effect of their decisions over the remainder subsystems conforming the entire network. In this sense, a global (all-to-all) communication strategy is considered. Although the Pareto optimality cannot be reached due to the existence of non-sparse coupling constraints, the asymptotic convergence to a Nash equilibrium is guaranteed. The resultant strategy is tested and its effectiveness is shown when applied to a large-scale complex flow-based network: the Barcelona drinking water supply system.     

Nonlinearizable single-input control systems do not admit stationary symmetries

The main result of the paper states that almost any analytic single-input control system, which is truly nonlinear, that is not feedback linearizable, with controllable linearization at an equilibrium point, does not admit any symmetry preserving that point. By almost any system, we mean that we exclude a small class of odd systems, that admit just one nontrivial symmetry conjugated to minus identity. The obtained results are based on a recent classification of nonlinear single-input systems under formal feedback. We also describe symmetries of feedback linearizable systems.     

The Andronov-Hopf bifurcation with weakly oscillating parameters

Consideration is given to the problem of local bifurcations in neighborhoods of stationary states of dynamical systems with parameters evolving according to the periodic law. Scenarios of the bifurcation behavior of the system are studied and criteria for its stability are presented. It is shown that in the natural formulation, the Andronov-Hopf bifurcation of the dynamical system is transformed to a bifurcation of quasi-periodic oscillations. Asymptotic formulae are defined for occurring oscillations as well as recommendations for construction of solutions.     

Local and global controllability for nonlinear systems

A technique to study local and global controllability properties for a wide class of nonlinear systems is presented. In addition to an extensive study of systems on ² we propose — as an application of our method — a new criterion for global controllability of systems with polynomial drift term, defined on ⁿ. In the case of linear systems, the criterion reduces to the classical Kalman condition.A study of local controllability at the equilibrium point of the drift term of systems defined on ² enables us not only to rederive a local controllability result derived by Hermes [4,5], but also to extend this result when no assumptions are made on the boundedness of the controls.     

Existence of periodic orbits of stable saturated systems

The saturation of linear controllers produces the undesirable existence of equilibrium points or periodic orbits of the closed-loop system. This typical nonlinear behavior has been observed in real systems or by means of simulation of certain examples. However, there are only a few studies in which the properties of saturated systems have been examined rigorously and, a proof of the existence of periodic orbits created by the saturation of the controller is lacking. In this paper we choose an example of an open-loop stable linear control system with an stabilizing saturated linear feedback to prove rigorously the existence of a periodic orbit.     

稳定性切换点法在时滞系统的鲁棒稳定性中的应用

讨论了一类参数与时滞相关的时滞系统的鲁棒稳定性.在"稳定性切换几何判据法"的基础上提出了"稳定性切换点法",使用该方法可得到相应方程零解稳定的参数变化区域.针对向日葵方程这一实际例子,利用文中所提出的方法并结合Maple软件作图可以容易地得到稳定性区域和不稳定性区域以及两区域的分界线、Hopf分岔点等;进一步通过对时滞...     

Control over chaotic price dynamics in a price competition model

The dynamic model of price competition in which processes of strategic interaction between companies on an imperfect competition market are described with the game-theoretic approach and methods of nonlinear dynamics. The pricing dynamics for the companies is modeled with difference equations (mappings). We study the stability of the fixed point of the price mapping. Results of our numerical modeling have shown the existence of periodic and chaotic solutions in the price competition model. We present intra-company adaptation mechanisms based on changing the prices in a way proportional to the rate of change in the companies’ profits; this lets us reduce the prices to a local Nash equilibrium and stabilize the chaotic dynamics of the market.     

具免疫时滞的HBV传染病模型的稳定性和Hopf分支

考虑了溶解性和非溶解性机制下的一类具有免疫时滞的HBV感染动力学模型。分析了无感染平衡点及感染无免疫平衡点的全局稳定性,讨论了感染免疫平衡点的局部稳定性和Hopf分支的存在条件。数值模拟结果表明：当易感细胞生成率的取值使得基本再生数满足平衡存在条件且低于某一临界值时,时滞对平衡点的稳定性没有影响;当大于该临界值时,随着时滞增大,平衡点不稳定,出现一系列Hopf分支,最终表现为周期波动模式。     

非高斯噪声下多循环平稳特征频谱检测

研究非高斯噪声下基于循环平稳特征的主用户检测问题。结合主用户信号的多循环平稳特征,通过求和准则和最大准则对多个循环频率的谱相关函数进行整合,提高检测结果的可信度。在等权求和准则的基础上,利用不同循环频率处不同程度的循环平稳特征,提出一种基于谱相关函数的加权求和准则。仿真结果表明,多循环频率合作可以提高检测效率,并且加权求和准则的检测性能优于等权求和 准则。     

Analysis of the energy‐based swing‐up control of the Acrobot

This paper addresses the energy‐based swing‐up control problem for the Acrobot, a two‐link underactuated robot with a single actuator at the joint of the two links. In line with the energy‐based control approach, this paper presents a necessary and sufficient condition for non‐existence of any singular point in the derived control law, and provides a complete analysis of the convergence of the energy and the motion of the Acrobot. Specifically, for any initial state of the Acrobot, this paper shows clearly how to choose control parameters such that the Acrobot will eventually either be swung up to any arbitrarily small neighbourhood of the upright equilibrium point, or remain in a set containing a finite number of equilibrium points. Moreover, this paper shows that these equilibrium points are unstable. Furthermore, imposing a stronger condition on a control parameter yields that the equilibrium set contains only the downward equilibrium point, which is shown to be hyperbolic and unstable. This proves that the Acrobot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero. Copyright © 2007 John Wiley & Sons, Ltd.