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     

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.     

Winkler地基上材料非线性矩形薄板主参数共振研究

研究Winkler地基上材料非线性矩形薄板受参数激励的参数共振动问题。按照弹性力学理论建立Winkler地基上材料非线性矩形薄板受参数激励的动力学方程。利用Galerkin方法将其转化为非线性振动方程。应用非线性振动的多尺度法求得系统满足主参数共振条件的一次近似解，并进行数值计算，分析定常解的稳定性。给出主参数共振系统参数平面的分岔集和幅频响应方程的分岔图。分析激励、调谐值、阻尼系数、非线性参数、几何参数对共振响应曲线的影响。     

Two-Fluid Hydrodynamics in Trapped Bose Gases and in Superfluid Helium

A review is given of recent theoretical work on the superfluid dynamics of trapped Bose gases at finite temperatures, where there is a significant fraction of non-condensate atoms. One can now reach large enough densities and collision cross-sections needed to probe the collective modes in the collision-dominated hydrodynamic region where the gas exhibits characteristic superfluid behavior involving the relative motions of the condensate and non-condensate components. The precise analogue of the Landau-Khalatnikov two-fluid hydrodynamic equations was recently derived from trapped Bose gases, starting from a generalized Gross-Pitaevskii equation for the condensate macroscopic wavefunction and a kinetic equation for the non-condensate atoms.     

A semianalytic meshless approach to the transient Fokker–Planck equation

In this paper, a semianalytic partition of unity finite element method (PUFEM) is presented to solve the transient Fokker–Planck equation (FPE) for high-dimensional nonlinear dynamical systems. Meshless spatial discretization of PUFEM with local p$p$-refinement (discussed in a previous paper) is employed to develop linear ordinary differential equations for the time varying coefficients of local shape functions. A similarity transformation to modal coordinates is shown to reveal numerous spurious modes in the eigenspace of the discretized FPE operator. Identification and elimination of these modes leads to an analytical solution of ODEs obtained from spatial discretization in terms of the remaining admissible modes, and a significant reduction in the size of the discretized transient problem. Initial equation error resulting from the set of admissible modes is shown to be an upper bound for all time, meaning that the reduced admissible set is sufficient for the FPE approximation for all time.     

压电夹层梁的分岔、混沌及主动控制

研究轴向激励作用下简支压电夹层梁的分岔、混沌振动及其主动控制。基于压电材料本构关系、von Karman型几何非线性应变位移关系,考虑耦合正、逆压电效应的比例微分控制策略,运用Hamilton原理建立了压电夹层梁的非线性横向运动偏微分方程并利用Galerkin方法对其进行离散化处理。通过采用数值模拟方法,研究了压电夹层梁的动态分岔。结果表明,通过比例控制增益和微分控制增益都可控制压电夹层梁的横向振动,阻止系统发生混沌运动,保持系统的稳定性。     

Bifurcation analyses and potential landscapes of a cortex–basal ganglia–thalamus model

The dynamics of cortical neuronal activity plays important roles in controlling body movement and is regulated by connection weights between neurons in a cortex–basal ganglia–thalamus (BGCT) loop. Beta‐band oscillation of cortical activity is closely associated with the movement disorder of Parkinson''s disease, which is caused by an imbalance in the connection weights of direct and indirect pathways in the BGCT loop. In this study, the authors investigate how the dynamics of cortical activity are modulated by connection weights of direct and indirect pathways in the BGCT loop under low dopamine levels through bifurcation analyses and potential landscapes. The results reveal that cortical activity displays rich dynamics under different connection weights, including one, two, or three stable steady states, one or two stable limit cycles, and the coexistence of one stable limit cycle with one stable steady state or two stable ones. For a low dopamine level, cortical activity exhibits oscillation for larger connection weights of direct and indirect pathways. The stability of these stable dynamics is explored by the potential landscapes.     

Non-linear dynamics of space-independent xenon oscillations

The space-independent xenon oscillation problem relevant to power nuclear reactors is studied. Xenon oscillations - both, without temperature feedback and in the presence of temperature feedback - have been analyzed semi-analytically and numerically. The effects of various parameters on the nature of bifurcation are studied. Bifurcation analysis shows that Hopf bifurcation occurs, and it is found that both sub- critical and super-critical Hopf bifurcation can occur in different regions of parameter space. Numerical experiments show that 'outsidey the unstable periodic solutions that exist for the sub-critical Hopf bifurcation case, there exist large amplitude, stable periodic solutions to which the initial conditions, outside the basin of attraction of the stable fixed point, evolve. Though the existence of sub-critical Hopf bifurcation indicates that large amplitude perturbations even in the stable region may lead to initially diverging oscillations, it is reassuring that the oscillation amplitude remains bounded     

Resonant macroscopic quantum tunneling in Josephson junctions

Many theoretical studies and some experiments have shown the possibility of several new secondary quantum effects in small Josephson junctions at low temperatures. We show that, within the well-established quantum picture of the junction, it is possible to experimentally observe sharp voltalge peaks at certain current values. Such peaks are related to a resonant macroscopic quantum tunneling between levels in neighboring wells of the proper washboard potential having close energies. The proposed experiment, with respect to experiments on Coulomb blockade or Block oscillations, requires a larger junction area, reducing the technological difficulties in making the samples.     

Contribution of time delays to p53 oscillation in DNA damage response

Although the oscillatory dynamics of the p53 network have been extensively studied, the understanding of the mechanism of delay‐induced oscillations is still limited. In this paper, a comprehensive mathematical model of p53 network is studied, which contains two delayed negative feedback loops. By studying the model with and without explicit delays, the results indicate that the time delay of Mdm2 protein synthesis can well control the pulse shape but cannot induce p53 oscillation alone, while the time delay required for Wip1 protein synthesis induces a Hopf bifurcation to drive p53 oscillation. In addition, the synergy of the two delays will cause the p53 network to oscillate in advance, indicating that p53 begins the repair process earlier in the damaged cell. Furthermore, the stability and bifurcation of the model are addressed, which may highlight the role of time delay in p53 oscillations.Inspec keywords: proteins, cellular biophysics, DNA, molecular biophysics, biomolecular effects of radiation, bifurcation, physiological models, cellular effects of radiation, oscillations, geneticsOther keywords: highlight, time delay, delayed negative feedback loops, murine double minute 2, Wip1 protein synthesis, explicit delays, Mdm2 protein synthesis, p53 network     

Spin Dynamics of a Trapped Spin-1 Bose Gas above the Bose-Einstein Transition Temperature

We study collective spin oscillations in a trapped spin-1 Bose gas above the Bose-Einstein transition temperature. Starting from the Heisenberg equation of motion, we derive a kinetic equation describing the dynamics of a thermal gas with the spin-1 internal degree of freedom. Applying the moment method to the kinetic equation, we study spin-wave collective modes with dipole symmetry. The dipole modes in the spin-1 system are classified into the three types of modes. These frequencies and damping rates are obtained as functions of the peak density. Moreover we find that the damping rate is characterized by three relaxation times associated with collisions.      

Heterogeneity and weak coupling may explain the synchronization characteristics of cells in the arterial wall

Vascular smooth muscle cells (SMCs) exhibit different types of calcium dynamics. Static vascular tone is associated with unsynchronized calcium waves and the developed force depends on the number of recruited cells. Global calcium transients synchronized among a large number of cells cause rhythmic development of force known as vasomotion. We present experimental data showing a considerable heterogeneity in cellular calcium dynamics in the vascular wall. In stimulated vessels, some SMCs remain quiescent, whereas others display waves of variable frequency. At the onset of vasomotion, all SMCs are enrolled into synchronized oscillation.Simulations of coupled SMCs show that the experimentally observed cellular recruitment, the presence of quiescent cells and the variation in oscillation frequency may arise if the cell population is phenotypically heterogeneous. In this case, quiescent cells can be entrained at the onset of vasomotion by the collective driving force from the synchronized oscillations in the membrane potential of the surrounding cells. Partial synchronization arises with an increase in the concentration of cyclic guanosine monophosphate, but in a heterogeneous cell population complete synchronization also requires a high-conductance pathway that provides strong coupling between the cells.     

Chaos, coexistence of attractors and fractal basin boundaries of attraction in a model system coupling activation and inhibition in parallel

The dynamical behaviour of a theoretical model featuring activation and inhibition coupled in parallel is studied in two different parameter regions. The coexist- ence of attractors between various complex oscillations and between period-2 oscillation and chaos has been found. Coexisting period-1 and other periodic attractors are also observed. A range of bifurcation behaviours is found, including periud duubling cascades to chaos and the existence of two period-1 regimes produced from sub- and super-critical Hopf bifurcations respectively. For the coexisting attractors, the fractal basin boundaries of attraction are observed. Finally, we discuss the implications of the model for enzymatic reactions     

Chaos in a railway bogie

Summary Railway bogies may perform sustained lateral oscillations — the hunting motion —when the speed reaches a certain value. We examine the hunting motion in the complex Cooperrider bogie and find that the wheel flanges have a strong influence on the behaviour. When the bogie has flangeless wheels we have found a symmetry breaking bifurcation, by which we mean, that there is a transition from a symmetric periodic oscillation to an asymmetric periodic oscillation. The periodic motions are examined using the residual map. When the bogie has flanged wheels (the flange is represented as a dead band spring), we do instead find chaotic behaviour confirmed by a positive Liapunov number. The dynamical equations for the bogie model are strongly nonlinear, and we use computer methods to examine the dynamical behaviour.With 10 Figures     

On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media

Macroscopic balance equations for components, momentum and energy are established for a multiphase flow with diffusion, chemical reactions, heat transfer and exchanges of components between phases in a porous medium. These equations are established separately for each fluid phase, for the solid part of the medium, and for interfaces, by starting from the corresponding equations valid at the pore level and taking their mean value around each point. Then macroscopic entropy balance equations are derived. The entropy source density shows clearly the generalized fluxes and forces which appear in the problem, and suggests how to choose phenomenological equations. A simple example illustrating the method is given in the last paragraph, for a single phase flow with heat transfer in a porous medium. One obtains a generalized form of Darcy's equation. Rigorous conditions along the interfaces and contact lines in a multiphasic medium are given in Appendix.     

Interference of the Bloch oscillations in the Josephson junctions

A quantum motion of a particle with dissipation in a titled periodic potential, adequate to the dynamics of the Josephson junctions, is considered. A concept of band motion of a particle, leading to the Bloch oscillations, holds at finite scales of time only, determined by dissipation. At large times there occurs an interference of the Bloch oscillations, resulting in vanishing of the resonance feature on the I–V characteristics under the action of the ac current with the frequency near _B. There exist two relaxation times: ₁ is a width of a separate Stark and ₂ is the time of the mutual damping of the Bloch oscillation.     

Stiff and soft active control of friction by vibrations and their energy efficiency

The present paper builds upon the results of two recent theoretical studies on the influence of friction by normal and sideways oscillations. The findings are in part rewritten to a more compact and dimensionless form so as to present the results for both oscillation modes side by side in a consistent manner. Thereby, it is shown that for the considered system the macroscopic coefficient of friction is only a function of a dimensionless sliding velocity and a dimensionless oscillation amplitude. Furthermore, the energy efficiency is characterized for both modes for the first time by comparing the total energy needed for a sliding motion which includes the superimposed oscillations with the energy needed for the same sliding motion without the additional oscillations. It is shown that this ratio is also only a function of the two dimensionless system parameters. We consider a simple one-spring model in a displacement-controlled setting. Any system-dynamical feedback is neglected. The lower end of the spring either slides, sticks or jumps on a rigid plane. In the case of normal oscillations, the macroscopic coefficient of friction can be reduced only when the contact point undergoes a stick-slip motion (“stiff control of friction”) whereas with sideways oscillations the macroscopic coefficient of friction can be reduced also when the contact point is continuously sliding (“soft control of friction”). It is found that the motion with superimposed sideways oscillations requires more energy for any combination of system parameters, than the corresponding motion without the oscillations. For the case of normal oscillations however, there are combinations of system parameters for which the motion with the superimposed oscillations requires less, the same, or more energy than for the reference case without the oscillations.