Nonlinear dynamics and stability of two streaming magnetic fluids

This paper concerns the linear and nonlinear instability of Kelvin–Helmholtz flows in magnetic fluids under external driving. The fluids are subjected to an oblique magnetic field. With the use of the method of multiple scaling, a generalized derivation of the amplitude equation is obtained in marginally unstable regions of parameter space. A Melnikov function is formulated for such an instability and it is shown that there exist transverse homoclinic orbits leading to chaos.     

Kolmogorov–Smirnov‐type testing for the partial homogeneity of Markov processes—with application to credit risk

In banking, the default behaviour of the counterpart is not only of interest for the pricing of transactions under credit risk but also for the assessment of a portfolio credit risk. We develop a test against the hypothesis that default intensities are chronologically constant within a group of similar counterparts, e.g. a rating class. The Kolmogorov–Smirnov‐type test builds up on the asymptotic normality of counting processes in event history analysis. The right censoring accommodates for Markov processes with more than one no‐absorbing state. A simulation study and two examples of rating systems demonstrate that partial homogeneity can be assumed, however occasionally, certain migrations must be modelled and estimated inhomogeneously. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonlinear dynamics of a machining system with two interdependent delays

The dynamics of turning by a tool head with two rows, each containing several cutters, is considered. A mathematical model of a process with two interdependent delays with the possibility of cutting discontinuity is analyzed. The domains of dynamic instability are derived, and the influence of technological parameters on system response is presented. The numeric analysis show that there exists specific conditions for given regimes in which one row of cutters produces an intermittent chip form while the other row produces continuous chips. It is demonstrated that the contribution of parametric excitation by shape roughness of an imperfect (unmachined) cylindrical workpiece surface is not substantial due to the special filtering properties of cutters that are uniformly distributed circumferentially along the tool head.     

Sensitivity analysis of regenerative queuing models

Queuing models described as regenerative processes driven by several sequences are under investigation. Occurrence times in these sequences are supposed to be asynchronous, in general. Sensitivity coefficients that are derivatives of some stationary performance measures with respect to definite parameters as well as derivatives of higher orders are derived for such queuing models. These sensitivity coefficients can be evaluated by simulation.     

Undiscounted approximations of discounted regenerative models

This paper approximates the discounted average cost in regenerative models by an undiscounted average cost. A financial holding cost is assessed on expenditures that are incurred earlier than the middle of each cycle. A financial variability cost that depends on the variability of the cycle time is also assessed. Explicit upper and lower bounds on the approximation are obtained in the deterministic case. Application to the EOQ model is made.     

Nonlinear character dependent models with constant time delay in population dynamics

A nonlinear character dependent model with constant time delay that extends the linear model of W. S. C. Gurney and R. M. Nisbet is introduced as a result of making the birth and death moduli dependent upon the total population. We reduce this model—an initial boundary value problem—to the solution of a pair of coupled nonlinear functional equations. Under suitable conditions, we establish the local existence, uniqueness, and continuation of solutions for all positive time. We also establish necessary and sufficient conditions for the local stability of time independent solutions in the form of a Lotka type characteristic equation similar to M. E. Gurtin and R. C. MacCamy's charateristic equation arising from their nonlinear extension of the age dependent MacKendrick-Von Foerster model.     

Global dynamics of two phytoplankton-zooplankton models with toxic substances effect

In this paper, we investigate phytoplankton-zooplankton models with toxic substances effect and two different kinds of predator functional responses. For Holling type II predator functional response, it is shown that the local stability of the positive equilibrium implies global stability if there exists a unique positive equilibrium. When there exist multiple positive equilibria, the local stability of the positive equilibrium with small phytoplankton population density implies that the model occurs bistable phenomenon. These results also hold for Holling type III predator functional response under certain conditions.     

Classical Skyrmions—static solutions and dynamics

Atomic nuclei can be modelled by Skyrmions with quantised spin and isospin. Skyrmions with a realistic value of the pion mass parameter should be quite compact structures, but few examples beyond baryon number 8 are known. The largest of these is the cubic Skyrmion with baryon number 32, which is a truncated piece of the Skyrme crystal. Here, it is argued that many Skyrmions are pieces of the Skyrme crystal. Particular attention is given to cubic crystal chunks and ways to reduce the baryon number by chopping corners off. It is also argued that quantised Skyrmion states can be approximated by classically spinning Skyrmions, and the orientations of spinning Skyrmions corresponding to polarised protons, neutrons and deuterons are identified. Nuclear collisions and the nuclear spin–orbit force are discussed in terms of classically spinning Skyrmions. Going beyond the rigid collective motions of Skyrmions, there are spatially modulated collective motions, oscillatory in time, which model vibrational excitations of nuclei and also giant resonance states. A speculative proposal for identifying quarks inside Skyrmions is briefly discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonlinear Transition Layers—The Second Painleve Transcendent

We investigate a large class of weakly nonlinear second-order ordinary differential equations with slowly varying coefficients. We show that the standard two-timing perturbation solution is not valid during the transition from oscillatory to exponentially decaying behavior. In all cases this difficulty is remedied by a nonlinear transition layer, whose leading-order character is described by one special nonlinear differential equation known as the second Painlevé transcendent (in essence a nonlinear Airy equation). The method of matched asymptotic expansions yields the desired connection formula. The second Painlevé transcendent also provides two other types of transitions: (1) between weakly nonlinear solutions (either oscillatory or exponentially decaying) and special fully nonlinear solutions, and (2) between two of these special nonlinear solutions. These special solutions are of three: different kinds: (a) slowly varying stable equilibrium solutions, (b) “exploding” solutions, and (c) solutions depending on both the fast and slow scales (which emerge from the unstable zero equilibrium solution).     

Nonlinear dynamics of unicycles in leader–follower formation

In this paper, a dynamical systems analysis is presented for characterizing the motion of a group of unicycles in leader–follower formation. The equilibrium formations are characterized along with their local stability analysis. It is demonstrated that with the variation in control gain, the collective dynamics might undergo Andronov–Hopf and Fold–Hopf bifurcations. The vigor of quasi-periodicity in the regime of Andronov–Hopf bifurcation and heteroclinic bursts between quasi-periodic and chaotic behavior in the regime of Fold–Hopf bifurcation increases with the number of unicycles. Numerical simulations also suggest the occurrence of global bifurcations involving the destruction of heteroclinic orbit.     

Comparison of two models for managing reliability growth during product design

Relying on reliability growth testing to improve system designis neither usually effective nor efficient. Instead it is importantto design in reliability. This requires models to estimate reliabilitygrowth in the design that can be used to assess whether goalreliability will be achieved within the target timescale forthe design process. Many models have been developed for analysisof reliability growth on test, but there has been much lessattention given to reliability growth in design. This paperdescribes and compares two models: one motivated by the practicalengineering process; the other by extending the reasoning ofstatistical reliability growth modelling. Both models are referencedin the recently revised edition of international standard IEC61164. However, there has been no reported evaluation of theirproperties. Therefore, this paper explores the commonalitiesand differences between these models through an assessment oftheir logic and their application to an industrial example.Recommendations are given for the use of reliability growthmodels to aid management of the design process and to informproduct development.     

Cyclic and unstable chaotic dynamics in models of two populations of sturgeon fish

This paper considers nonlinear effects in the dynamics of biological models. We describe two dynamic systems elaborated for simulating populations of Russian sturgeon and stellate sturgeon and based on formalization of the relationship between the spawning stock and recruitment according to the analysis of observational data. For the numerical study of differential equations with a structurally changing right-hand side, we use the method of representing models based on maps of states with conditional transitions. For dynamic systems, the presence of qualitatively different modes of the behavior of trajectories is revealed: stable periodic oscillations (sturgeon model) and unstable chaotic oscillations (stellate model) realized in a limited time interval due to a chaotic subset not being an attractor, which is present in the phase space.     

Nonlinear dynamics of endothelial cells

We propose a mathematical model that governs endothelial cell pattern formation on a biogel surface. The model accounts for diffusion and chemotactic motion of the cells, diffusion of the growth factor and effective biochemical reactions. The model admits a basic steady state that corresponds to a spatially uniform distribution of both the cells and the growth factor. We perform a weakly nonlinear stability analysis of the basic state in order to determine whether spatially nonuniform steady patterns can appear in the system when the basic state becomes unstable. The main results can be summarized as follows. No steady patterns can bifurcate from the basic state if the rate of decay of the growth factor is small. Increasing the rate of decay of the growth factor allows one to observe steady patterns, provided that diffusion of the growth factor is sufficiently slow. Specifically, the work focuses on the occurrence of hexagons and stripes. Most often hexagons are observed. In order for stripes to occur, the chemotactic sensitivity of the endothelial cells and/or their biochemical activity have to be reduced.     

Nonlinear dynamics of high-dimensional models of in-plane and out-of-plane vibration in an axially moving viscoelastic beam

Nonlinear dynamics of high-dimensional models of an axially moving viscoelastic beam with in-plane and out-of-plane vibration with combined parametric and forcing excitations are investigated by the incremental harmonic balance (IHB) method in this paper. Governing equations of transverse in-plane and out-of-plane and longitudinal vibration are obtained basing on the Hamilton's principle. The Galerkin method is used to separate time variable and spatial variable to obtain a set of multi-order differential equations. The IHB method with the fast Fourier transform (FFT) is used to solve periodic response of high-dimensional models of the beam for which convergent mode is reached. Stability of the steady-state periodic solutions is analyzed using the multivariable Floquet theory. Particular attention is paid to in-plane and out-of-plane vibration on convergent mode of the beam with combined parametric and forcing excitations. Multiple solutions are observed, and jump phenomena between in-plane and out-of-plane vibration with different transverse cross sections are discovered.     

Nonlinear models of suspension bridges

Nonlinear variational equations describing one type of suspension bridges are proposed and studied. The variational equations describe the behaviour of road bed, main cables and cable stays. The road bed is described by two functions connected with vertical and horizontal deformation of any cross section. The main cable is considered to be perfectly flexible and inextensible. The cable stays only resist tensile forces. The variational equations are derived from the principle of minimum potential energy. The existence of solution is based on the Brouwer Fixed Point Theorem. The local uniqueness and continuous dependence on the data represented by gravitational forces acting on the road bed are studied. The local results are based on the Implicit Function Theorem for Banach spaces. A certain stability criterion for suspension bridges is formulated and this criterion indicates how to influence the stability of suspension bridges.     

New approach in dynamics of regenerative chatter research of turning

In this paper, regenerative chatter phenomena in a turning process is discussed from impulsive dynamical point of view. By introducing the instantaneous pulse when vibration occurs and the vibratory condition set, we optimize the models and present a certain kind of second-order impulsive differential systems, which is a specific discontinuous dynamical system. Then we search for the general results of the nonoccurrence of chatter phenomena by discussing the number of the vibration pulse times, utilizing the method of flow theory in discontinuous systems and transversal property at the boundary. Our results give a convenient way to estimate the available parameters to keep the turning process stable.     

Identification of chaos in a regenerative cutting process by the 0-1 test

We examine the regenerative cutting process by using a single degree of freedom non-smooth model with a friction component and a time delay term. Instead of the standard Lyapunov exponent calculations, we propose a statistical 0-1 test for chaos. This approach reveals the nature of the cutting process signaling regular or chaotic dynamics. We are able to show that regular or chaotic motion occur in the investigated model depending on the delay time. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chaotic dynamics of a simple oscillator — a pictorial introduction

That seemingly simple mechanical oscillators show already many general phenomena including local and global bifurcations as well as chaotic behavior makes their study so important in the theory of nonlinear dynamics. A considerable body of knowledge has been developed in recent years for analyzing and evaluating this variety of behavior. By studying a Duffing oscillator some well approved techniques are demonstrated and discussed. The presentation is highly illustrated to improve understanding.     

Nonlinear time-domain models of human controllers

Data from a compensatory tracking task are analyzed by using time-domain models. The linear time-domain results are transformed and compared with frequency-domain results. The nonlinear time-domain model of the same data reduced the remnant or residual power by only a small amount. The need for testing models on independent data is discussed. A novel, but attractive, method of generating functions for an efficient functional expansion of time-domain models is offered.Notation c Pilot output (control deflection), inches - E Error matrix - e Error, radians - F[·] Fourier transform - h Time interval, seconds - h _i Sample of impulse response of pilot - h _p Impulse response of pilot, inches/radian or inches/degree - i Input (external disturbance function), radians - M Maximum value ofm, M=T _M/ - m Index for the argument ofh _p - N Maximum value ofn - n Index for time - o Linear output of pilot model (control deflection), inches - r Remnant signal of pilot model (control deflection), inches - S Matrix - s Laplace variable - T _M Maximum memory time of the pilot model, seconds - t Time, seconds - Y _c Transfer function of controlled element - Y _c j) Controlled-element transfer function, radians/inch - Y _p j) Pilot-describing function, inches/radian - Argument ofh _p, seconds - Incremental value of, seconds - Frequency, radians/second - ^ Estimate - Absolute value - Phase angle