Reversing the stability of fixed points to generate oscillations in electrochemical systems: Simulations and experiments

In the present work, we report the induction of rhythms in electrochemical systems. Limit cycle oscillations are provoked by reversing the stability of a previously stable fixed point on which the system’s dynamics were originally settled. Superimposing an appropriate perturbation term, sign of the local eigenvalues (real part) of the fixed point was changed. This alters the stability of the fixed point without varying its location. This protocol was tested both in a numerical model simulating electrochemical corrosion as well as an experimental electrochemical cell. Both numerical results and experimental observations indicate that, employing suitable perturbations, it is indeed possible to generate oscillatory behavior in nonlinear systems whose autonomous dynamics exhibit steady state behavior.     

Using chaos to reduce oscillations: Experimental results

Chaotic dynamic systems are usually controlled in a way, which allows the replacement of chaotic behavior by the desired periodic motion. We give the example in which an originally regular (periodic) system is controlled in such a way as to make it chaotic. This approach based on the idea of dynamical absorber allows the significant reduction of the amplitude of the oscillations in the neighborhood of the resonance. We present experimental results, which confirm our previous numerical studies [D?browski A, Kapitaniak T. Using chaos to reduce oscillations. Nonlinear Phenomen Complex Syst 2001;4(2):206–11].     

Passive aerodynamics control of plasma instabilities

The possibility of using a smart-damping scheme to modify the dynamic responses of plasma oscillations governed by a two-fluid model is considered. The passive aerodynamics control strategy is used to address this issue. The control efficiency is found by analyzing the conditions satisfied by the control gain parameters for which, the amplitude of oscillations is reduced both in the harmonic and chaotic states. In the regular state, the analytical stability analysis uses for linear oscillations the Routh-Hurwitz criterion while the Whittaker method and Floquet theory are utilized for nonlinear harmonic oscillations. The stability boundaries in the control gain parameter space is derived. The agreement between the analytical and numerical results is good. In the chaotic states, numerical simulations are used to perform quenching of chaotic oscillations for an appropriate set of control parameters.     

Tubuloglomerular feedback in a dynamic nephron

A dynamic model for a short-looped mammalian nephron is developed to study tubuloglomerular feedback (TGF). Evolution equations for salt and urea concentrations and for fluid flux in the nephron are derived and coupled to a resistance network that serves as a schematic model of the glomerulus and associated structures. The evolution equations, which are semi-linear hyperbolic partial differential equations, are solved by the method of flux-corrected transport. The implementation and testing of this method is described and numerical results are presented. This investigation suggests that: (i) the concentrating nephron exhibits high gain, i.e., a small increase in single nephron glomerular filtration rate produces a large increase in the salt concentration of tubular fluid in the cortical thick ascending limb at the macula densa; (ii) the nephron, as a concentrating system, acts as a low-pass filter, i.e., high frequency pressure oscillations (1 Hz) of a prescribed amplitude at the proximal tubule produce relatively low amplitude oscillations in tubular concentrations, while low frequency oscillations (1/30 Hz) produce relatively high amplitude oscillations in tubular concentrations; and (iii) as a consequence of long time delay in TGF, some perturbations in afferent arteriolar blood pressure induce sustained periodic oscillations similar to those observed in recent experiments.     

A shock-capturing model based on flux-vector splitting method in boundary-fitted curvilinear coordinates

In this paper, a robust and accurate high-resolution finite-volume scheme is presented which employs flux-vector splitting (FVS) as the building block for solution of shallow water equations in boundary-fitted curvilinear coordinates. Eddy viscosity approach is used to accommodate shear stresses due to turbulence. Splitting of the convective terms is achieved via flux Jacobians whereas Liou–Steffen Splitting (LSS) technique, but in transformed coordinates, is used to split pressure terms. Limited flux gradients are also used to increase the computational accuracy of evaluation of interface fluxes and decrease the excessive numerical dissipation associated with FVS. This will completely remove spurious oscillations in high-gradient regions without introducing too much numerical dissipations. The method is tested for some classic simulations including hydraulic jump, 1D dam break and 2D dam break problems. The results show very satisfactory agreement with experimental data, analytical solutions and other numerical results.     

Numerical investigation of impinging gas jets onto deformable liquid layers

Impinging jets over liquid surfaces are a common practice in the metallurgy and chemical industries. This paper presents a numerical study of the fluid dynamics involved in this kind of processes. URANS simulations are performed using the volume of fluid (VOF) method to deal with the multiphase physics. This unsteady approach with the appropriate computational domain allows resolution of the big eddies responsible for the low frequency phenomena. The solver we used is based on the finite volume method and turbulence is modelled with the realisable k-? model. Two different configurations belonging to the dimpling and splashing modes are under consideration. The results are compared with PIV and LeDaR experimental data previously obtained by the authors. Attention is focused on the surroundings of the impingement, where the interaction between jet and liquid film is much stronger. Finally, frequency analysis is carried out to study the flapping motion of the jet and cavity oscillations.     

Fourth-Order Symmetric DIRK Methods for Periodic Stiff Problems

New symmetric DIRK methods specially adapted to the numerical integration of first-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion conditions for symmetric DIRK methods as well as symmetric stability functions with real poles and maximal dispersion order. Two new fourth-order symmetric methods with four and five stages are obtained. One of the methods is fourth-order dispersive whereas the other method is symplectic and sixth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with the symplectic DIRK method derived by Sanz-Serna and Abia (SIAM J. Numer. Anal. 28 (1991) 1081–1096).     

An Exact Riemann Solver for a Fluidized Bed Model

We study a 2 x 2 hyperbolic system of conservation laws withsource term arising in a fluidized bed model. The system issolved numerically and results are presented to demonstratethe occurrence of ‘slugging’ in the full model equations.The numerical procedure is based on operator splitting and Godunov'smethod, for which we derive the exact solution of the Riemannproblem. A second-order improvement due to Davis (1988) mayproduce small oscillations near shocks and these can be reducedif the underlying flux limiter of the Davis method is replacedby the minmod limiter.     

Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order

New SDIRKN methods specially adapted to the numerical integration of second-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion and P-stability conditions for SDIRKN methods which are assumed to be zero dissipative. Two four-stage symplectic and P-stable methods with algebraic order 4 and high order of dispersion are obtained. One of the methods is symmetric and sixth-order dispersive whereas the other method is nonsymmetric and eighth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with other methods derived by Sharp et al. [IMA J. Numer. Anal. 10 (1990) 489–504].     

Stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type

This paper is concerned with the numerical properties of θ-methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments. Using two θ-methods, namely the one-leg θ-method and the linear θ-method, the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained, and the conditions of oscillations for the θ-methods are also obtained. It is proved that oscillations of the analytic solution are preserved by the θ-methods. Furthermore, the relationships between stability and oscillations are revealed. Some numerical experiments are presented to illustrate our results.     

Spiral waves on static and moving spherical domains

The FitzHugh–Nagumo-type model on static and periodically oscillating surface of the sphere is studied. The numerical investigation of the model is performed in both cases and detailed numerical results are presented for the two-arm spiral wave and its rotation on both manifolds. On the static surface, meandering waves are obtained and it is shown that these waves are stable. On the periodically oscillating surface, the initial excitation gives rise to an irregular (chaotic) meandering rotation, depending on the frequency and the amplitude of the oscillations.     

Numerical Stability and Oscillations of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments of Advanced Type

For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.     

Structure and properties of four-kink multisolitons of the sine-Gordon equation

The dynamics of nonlinear waves of the sine-Gordon equation with a spatially modulated periodic potential are studied using analytical and numerical methods. The structure and properties of four-kink multisolitons excited on two identical attracting impurities are determined. For small-amplitude oscillations, an analytical spectrum of the oscillations is obtained, which is in qualitatively agreement with the numerical results.     

单物种人口模型指数隐式Euler方法的振动性

本文研究了用以描述单物种人口模型的延迟Logistic方程的数值振动性.对方程应用隐式Euler方法进行求解,针对离散格式定义了指数隐式Euler方法,证明了该方法的收敛阶为1.根据线性振动性理论获得了数值解振动的充分条件.进而还对非振动数值解的性质作了讨论.最后用数值算例对理论结果进行了验证.     

Neuromuscular resetting and entrainment

In a recent work (O?uztöreli and Stein, 1979) some of the theoretical characteristics of the interacting resetting and entrainment oscillations in an experimentally based model of the mammalian neuromuscular system has been investigated. The present work is devoted to the numerical solution of the nonlinear integrodifferential difference equation which describes the complex nature of these oscillations.     

Experimental and theoretical investigation of creep groan of brakes through minimal models

Creep groan of brake systems is a low frequency vibration phenomenon occurring at low speeds which can make passengers feel uncomfortable. This phenomenon is caused by the stick-slip-effect resulting in limit cycle oscillations with frequencies lower than 200 Hz. For the experimental investigation of this problem, an idealized brake test rig is designed concentrating on the investigation of the frictional contact by realizing low damping and small disturbances in the system. Different sensors are utilized in the test rig. Limit cycles and bifurcation effects can be observed in the experimental results. With respect to modeling, a one degree-of-freedom (DOF) model using Coulomb's friction law and a two DOF model using the bristle friction law are considered. In a comparative study of experimental and simulation results, the parameters of both friction laws can be identified from the dynamic experimental results, such as the static and dynamic friction coefficients, contact stiffness and Stribeck velocity. Experimental and theoretical results show a very good concordance. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulation of Supersonic Axisymmetric Gas Jets

The oscillations of free supersonic jets are simulated using the axisymmetric Euler equation model. A third order accurate through difference scheme with artificial viscosity is used. Jet flows from a sonic nozzle with nozzle-to-ambient pressure ratio between 1.2 and 2 are investigated. The numerical results are compared with physical experiments.     

Nonadiabatic couplings and chaos in a dynamical potential well model

The mechanism of nonadiabatic couplings between quantum states of a potential well model with finite heights and a dynamical width coordinate is investigated in detail. The system is described in a mixed quantum-classical approach in which the oscillations of the classical width coordinate induce transitions between the quantum states of a particle trapped inside the well. The dynamics of the system is considered in detail for transitions between two quantum states and resulting coupled Bloch-oscillator equations. Poincaré sections showing a mixed phase space with chaotic and regular behaviour are found by a numerical investigation. In particular, chaos results for high energies of the well width oscillations when the mixing between the adiabatic reference states is strong. The inclusion of relaxation is considered and shown that in this case the regimes of chaotic and regular dynamics are not separated as in the relaxation free case. In particular, for some initial conditions chaos can become a transient phenomena placed in a time window between regular oscillations of the system.     

Bifurcation analysis of the mathematical model of the NO+CO/Pt(100) reaction

The article presents a detailed bifurcation analysis of steady and periodic states for a new mathematical model of the NO+CO/Pt(100) reaction. Various bifurcation diagrams are constructed in the planes of partial reagent pressures and surface temperature. Regions with oscillations and multiple steady solutions are investigated. Isolated branches of steady and periodic states are identified. An “explosive” bifurcation of the periodic solution leading to chaotic alternation of small-and large-amplitude oscillations is detected and analyzed for the first time. A good quantitative fit is demonstrated between modeling results and experimental data. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 52–78.     

Exponential asymptotics of woodpile chain nanoptera using numerical analytic continuation

Traveling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These oscillations have been studied using exponential asymptotic methods, which typically require an explicit form for the leading-order behavior. For many nonlinear systems, such as granular woodpile chains, it is not possible to calculate the leading-order solution explicitly. We show that accurate asymptotic approximations can be obtained using numerical approximation in place of the exact leading-order behavior. We calculate the oscillation behavior for Toda woodpile chains, and compare the results to exponential asymptotics based on previous methods from the literature: long-wave approximation and tanh-fitting. We then use numerical analytic continuation methods based on Padé approximants and the adaptive Antoulas–Anderson (AAA) method. These methods are shown to produce accurate predictions of the amplitude of the oscillations and the mass ratios for which the oscillations vanish. Exponential asymptotics using an AAA approximation for the leading-order behavior is then applied to study granular woodpile chains, including chains with Hertzian interactions—this method is able to calculate behavior that could not be accurately approximated in previous studies.