Bifurcation analysis of bistable and oscillatory dynamics in biological networks using the root‐locus method

Most of the biological systems including gene regulatory networks can be described well by ordinary differential equation models with rational non‐linearities. These models are derived either based on the reaction kinetics or by curve fitting to experimental data. This study demonstrates the applicability of the root‐locus‐based bifurcation analysis method for studying the complex dynamics of such models. The effectiveness of the bifurcation analysis in determining the exact parameter regions in each of which the system shows a certain dynamical behaviour, such as bistability, oscillation, and asymptotically equilibrium dynamics is shown by considering two mostly studied gene regulatory networks, namely Gardner''s genetic toggle switch and p53 gene network possessing two‐phase (mono‐stable/oscillation) dynamics.Inspec keywords: oscillations, curve fitting, differential equations, bifurcation, genetics, nonlinear dynamical systemsOther keywords: nonlinearities, reaction kinetics, root‐locus‐based bifurcation analysis method, complex dynamics, exact parameter regions, dynamical behaviour, equilibrium dynamics, studied gene regulatory networks, p53 gene network, bistable dynamics, oscillatory dynamics, biological networks, root‐locus method, biological systems, ordinary differential equation models     

具有时滞和阶段结构的生态-流行病模型的稳定性及Hopf分支

本文研究一类具有时滞和阶段结构的生态-流行病模型的稳定性及其Hopf分支．给出了边界平衡点和正平衡点存在的充分条件;通过分析特征方程,运用Hurwitz判定定理,讨论了边界平衡点和正平衡点的局部稳定性,并得到了正平衡点附近存在Hopf分支的充分条件;通过构造适当的Lyapunov泛函,运用LaSall不变集原理,讨论了边界平衡点和正平衡点的全局稳定性,从而得到了该生态模型永久持续生存与灭绝的充分条件．     

Nonlinear dynamics of tubes carrying a pulsatile flow

The nonlinear dynamics of a cantilever tube conveying a pulsatile flow and undergoing planar motions is investigated. The mean flow is near its critical value at which the downward vertical position of the tube gets unstable by flutter and executes limit-cycle oscillations. The pulsations in the flow are assumed to be small and harmonic with frequency nearly twice that of the limit cycle. To study the nonlinear dynamics, the method of averaging is utilized and the governing partial differential equation is reduced to a dynamic system on the plane. These two first-order differential equations depend on three parameters and govern the dynamics of the amplitude of motion of the tube. The planar system is studied for its qualitative behaviour using ideas from the local bifurcation theory and a local bifurcation set in the parameter plane is constructed. Using ideas from codimension-two unfolding of singularities, this bifurcation set is further refined. The resulting partial bifurcation set and the associated phase portraits in the various regions of the flowrate-detuning parameter plane show that the averaged equations have constant as well as periodic solutions. The stabifity type and the number of these solutions changes from one region to another. This explains how periodic or 'phase-locked' oscillations of the cantilever tube bifurcate into amplitude-modulated or almost-periodic motions.     

Prediction of vapour-liquid and vapour-liquid-liquid equilibria of nitrogen-hydrocarbon mixtures used in J-T refrigerators

Nitrogen-hydrocarbon mixtures are widely used as refrigerants in J-T refrigerators operating with mixtures, as well as in natural gas liquefiers. The Peng-Robinson equation of state has traditionally been used to simulate the above cryogenic process. Multi parameter Helmholtz energy equations are now preferred for determining the properties of natural gas. They have, however, been used only to predict vapour-liquid equilibria, and not vapour-liquid-liquid equilibria that can occur in mixtures used in cryogenic mixed refrigerant processes. In this paper the vapour-liquid equilibrium of binary mixtures of nitrogen-methane, nitrogen-ethane, nitrogen-propane, nitrogen-isobutane and three component mixtures of nitrogen-methane-ethane and nitrogen-methane-propane have been studied with the Peng-Robinson and the Helmholtz energy equations of state of NIST REFPROP and compared with experimental data available in the literature.     

Population dynamics of the modified theta model: macroscopic phase reduction and bifurcation analysis link microscopic neuronal interactions to macroscopic gamma oscillation

Gamma oscillations of the local field potential are organized by collective dynamics of numerous neurons and have many functional roles in cognition and/or attention. To mathematically and physiologically analyse relationships between individual inhibitory neurons and macroscopic oscillations, we derive a modification of the theta model, which possesses voltage-dependent dynamics with appropriate synaptic interactions. Bifurcation analysis of the corresponding Fokker–Planck equation (FPE) enables us to consider how synaptic interactions organize collective oscillations. We also develop the adjoint method (infinitesimal phase resetting curve) for simultaneous equations consisting of ordinary differential equations representing synaptic dynamics and a partial differential equation for determining the probability distribution of the membrane potential. This method provides a macroscopic phase response function (PRF), which gives insights into how it is modulated by external perturbation or internal changes of parameters. We investigate the effects of synaptic time constants and shunting inhibition on these gamma oscillations. The sensitivity of rising and decaying time constants is analysed in the oscillatory parameter regions; we find that these sensitivities are not largely dependent on rate of synaptic coupling but, rather, on current and noise intensity. Analyses of shunting inhibition reveal that it can affect both promotion and elimination of gamma oscillations. When the macroscopic oscillation is far from the bifurcation, shunting promotes the gamma oscillations and the PRF becomes flatter as the reversal potential of the synapse increases, indicating the insensitivity of gamma oscillations to perturbations. By contrast, when the macroscopic oscillation is near the bifurcation, shunting eliminates gamma oscillations and a stable firing state appears. More interestingly, under appropriate balance of parameters, two branches of bifurcation are found in our analysis of the FPE. In this case, shunting inhibition can effect both promotion and elimination of the gamma oscillation depending only on the reversal potential.     

Symmetry and bifurcations of planar configurations of the N-body and other problems

We describe a system of equations containing a real parameter β and an integer parameter N ≥2. Equilibria of these equations are in turn asymptotic shapes of systems of repelling particles for β= 0, central configurations with equal mass of the N-body problem for β= 1, and approximate solutions of a sphere-packing problem for β large. We introduce some asymmetric equilibria of these equations for N = 6 and 7, and identify and discuss the bifurcations that occur in this system for 5 ≤ N ≤ 8.     

A codimension-two resonant bifurcation from a heteroclinic cycle with complex eigenvalues

Robust heteroclinic cycles between equilibria lose stability either through local bifurcations of their equilibria or through global bifurcations. This paper considers a global loss of stability termed a 'resonant' bifurcation. This bifurcation is usually associated with the birth or death of a nearby periodic orbit, and generically occurs in either a supercritical or subcritical manner. For a specific robust heteroclinic cycle between equilibria with complex eigenvalues we examine the codimension-two point that separates the supercritical and subcritical. We investigate the bifurcation structure and show the existence of further bifurcations of periodic orbits.     

内平动齿轮副啮合综合刚度与系统的分岔特性

用机构反转法将内平动齿轮副转化为定轴齿轮副,然后用有限元方法分析了该内啮合定轴齿轮副的啮合综合刚度,并使用FFT变换得到其频谱特性,进而得到了内平动齿轮副的啮合综合刚度的频谱特性,在此基础上,考虑了齿侧间隙的非线性因素,进一步得到存在多齿接触的时变啮合刚度下内平动齿轮副的运动微分方程。然后,用经典的显式四阶Rouge-Kutta法对系统的各个参数进行了数值计算,得到系统的参数分岔图,并分析了各参数对系统动力学行为的影响。为内平动齿轮副的设计参数选择提供理论依据。     

Snakes and isolas in non-reversible conservative systems

Reversible variational partial differential equations such as the Swift–Hohenberg equation can admit localized stationary roll structures whose solution branches are bounded in parameter space but unbounded in function space, with the width of the roll plateaus increasing without bound along the branch: this scenario is commonly referred to as snaking. In this work, the structure of the bifurcation diagrams of localized rolls is investigated for variational but non-reversible systems, and conditions are derived that guarantee snaking or result in diagrams that either consist entirely of isolas.     

Perturbation method for phase-equilibrium calculations

In phase equilibrium problems, the thermodynamic model used often contains a small parameter. For example, for cubic equations of state the interaction coefficients occurring in van der Waals-type mixing rules are often numerically small. The small parameter (?) enters the model then via the formula $$a = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {x_i x_j \sqrt {a_i a_j } (1 - \varepsilon \theta _{ij} )} }$$ Other examples include mixtures with compounds whose characteristic parameters cover a narrow range, diluted solutions, small amounts of polydisperse material in a solvent, and so on. In this paper we develop a general scheme to obtain the solution of thermodynamic problems such as the prediction of phase equilibria, using an expansion in the small parameter ε. We also give a method to obtain a suitable zero-order (? = 0) system. The perturbation scheme may be helpful in the solution of difficult problems or as a tool in a sensitivity analysis. As an example we apply it to multicomponent mixtures, described through a two-parameter equation of state with small interaction coefficients. We show that for that case, if the number of components is large, it leads to computational savings.     

Analysis of partly wrinkled membranes by the method of dynamic relaxation

A version of the method of dynamic relaxation is developed to analyze equilibrium configurations of partly wrinkled membranes. In this method equilibria are regarded as long time limits of a damped dynamical problem. The membrane theory considered is based on the concept of a relaxed strain energy function that automatically incorporates the effects of wrinkling. For neo-Hookean materials, existence theorems of nonlinear elasticity are used to show that the relaxed potential energy possesses minimizers in a certain function space. Moreover, solutions of the equilibrium equations furnish global minima of the energy, for certain classes of boundary data. Such deformations are automatically stable according to the minimum-energy criterion. This result motivates the search for solutions of the equilibrium equations, although the existence theory does not guarantee that energy minimizers possess the degree of regularity required by these equations. Several examples of two-and three-dimensional deformations are presented.     

自同步振动系统的稳定性与分岔

用偏心转子相位差角微分方程代替双激振器反向回转式自同步振动系统的运动方程，讨论了系统的平衡点稳定性和分岔特性。首先，根据振动系统的运动方程推导出关于偏心转子相位差角的微分方程。然后，基于该方程建立了同步运动的必要性条件，应用Lyapunov稳定性理论，讨论系统的平衡点稳定性及其分岔特性，最后，结合仿真，考察了系统质量、刚度参数和激振器参数的影响规律。     

Influence of aerodynamics on the critical parameters of thermal ignition

A numerical analysis of the critical conditions of thermal ignition for a two-dimensional cylindrical reactor with flow field occurrence was carried out. The critical conditions of thermal ignition were defined as bifurcation points of the limit type. The dependence of the bifurcation points on aerodynamics was analysed by a sequential variation of the Peclet number. The flow field in the reactor was generated previously by a numerical solution of the Navier-Stokes equations. The nonlinear parabolic equation of heat balance and linear eigenvalue boundary problems occurring in the proposed non-stationary method for calculating the bifurcation points was solved numerically using the finite difference technique. It was shown that the critical points of thermal ignition form a surface bounded by the line of ‘limit of criticality’ in the parameter space. For large values of the Peclet number, the dependence of the critical parameter of ignition on Peclet number becomes linear.     

DRBEM solution of exterior nonlinear wave problem using FDM and LSM time integrations

The nonlinear wave equation is solved numerically in an exterior region. For the discretization of the space derivatives dual reciprocity boundary element method (DRBEM) is applied using the fundamental solution of Laplace equation. The time derivative and the nonlinearity are treated as the nonhomogenity. The boundary integrals coming from the far boundary are eliminated using rational and exponential interpolation functions which have decay properties far away from the region of interest. The resulting system of ordinary differential equations in time are solved using finite difference method (FDM) with a relaxation parameter and least squares method (LSM). The proposed methods are examined with numerical test problems in which the behaviours of solutions are known. Although it gives almost the same accuracy with the DRBEM+FDM procedure, DRBEM+LSM solution procedure is preferred, since it is a direct method without the need of a parameter.     

直线运动柔性梁非线性动力学--主参数共振与内共振联合激励

针对基础直线运动柔性梁，基于Kane方程建立了相应的非线性动力学方程。采用多尺度法并结合笛卡尔坐标变换，导出了系统受前两阶模态间3：1内共振及第二阶模态主参激共振时的非线性调制方程组．数值求解了该方程组的定常解及相应的稳定性问题。研究表明，系统的平凡、单模态、双模态稳态解共存，超临界及亚临界叉形分岔只发生在单模态状态下，相反，鞍结分岔及Hopf分岔只在双模态状态下产生，一些稳定的极限环随参数变化经一系列倍周期分岔后导致运动的突然跳跃。     

Buckling of a compressible elastic ring

Summary We study stability of a circular ring subjected to uniform hydrostatic pressure. The constitutive equations of a ring are taken in a generalized form, allowing the compression of the ring axis. We show that the null space of the linearized differential equations has dimension three. On the basis of a single bifurcation equation the bifurcation diagram is obtained. It is shown that bifurcation could be both super- and subcritical, depending on the values of parameters.     

Relative equilibria of tethered satellite systems and their stability for very stiff tethers

We study the existence and stability of the relative equilibria of systems of two satellites joined by a tether. Since tethers used in practice are very stiff we consider a stiff tether as a perturbation of an inextensible tether. We show that the equations for relative equilibria and the stability conditions are continuous as stiffness approaches infinity and limit on the equations and conditions relevant to an inextensible tether. We obtain a numerical bifurcation diagram for a class of relative equilibria in the case of an inextensible tether.