Hysteresis from dynamically pinned sliding states

We report a surprising hysteretic behavior in the dynamics of a simple one-dimensional nonlinear model inspired by the tribological problem of two sliding surfaces with a thin solid lubricant layer in between. In particular, we consider the frictional dynamics of a harmonic chain confined between two rigid incommensurate substrates which slide with a fixed relative velocity. This system was previously found, by explicit solution of the equations of motion, to possess plateaus in parameter space exhibiting a remarkable quantization of the chain center-of-mass velocity (dynamic pinning) solely determined by the interface incommensurability. Starting now from this quantized sliding state, in the underdamped regime of motion and in analogy to what ordinarily happens for static friction, the dynamics exhibits a large hysteresis under the action of an additional external driving force F_ext. A critical threshold value F_c of the adiabatically applied force F_ext is required in order to alter the robust dynamics of the plateau attractor. When the applied force is decreased and removed, the system can jump to intermediate sliding regimes (a sort of “dynamic” stick-slip motion) and eventually returns to the quantized sliding state at a much lower value of F_ext. Hysteretic behavior is also observed as a function of the external driving velocity.     

Dissipative oscillations in spatially restricted ecosystems due to long range migration

An ecosystem containing three interacting species is studied using both Mean Field approach and Kinetic Monte Carlo simulations on a lattice substrate. The so called 3rd order LLV model involves birth, death and reaction processes with 3rd order nonlinearities and feedbacks. At the mean field level this system exhibits conservative oscillations; the analytic form of the constant of motion is presented. The stochastic simulations show that the density oscillations disappear for sufficiently large lattices, while they are present locally, on small lattice windows. Introduction of mixing via long range migration in the two reacting species changes this picture. For small migration rates p, the behavior remains as with p = 0 and the system is divided into local asynchronous oscillators. As p increases the system passes through a phase transition and exhibits a weak disorder limit cycle through a supercritical Hopf-like bifurcation. The amplitude of the limit cycle depends on the rate p, on the range of migration r and on the system kinetic rates k₁, k₂ and k₃.     

The variable viscoelasticity oscillator

The complex dynamics of a variable viscoelasticity oscillator is studied using the novel concept of Variable‐Order (VO) Calculus. The damping force in the oscillator varies continuously between the elastic and viscous regimes depending on the position of the mass. The oscillator considered here is composed of a linear spring of stiffness k that inputs a restitutive force F_k = ‐k x, a VO damper of order q(x(t)) that generates a damping force F_q = ‐c_q ??^q(x(t)) x, and a mass m. A modified Runge‐Kutta method is used in conjunction with a trapezoidal numerical integration technique to yield a second‐order accurate method for the solution of the resulting VO Differential Equation (VODE). The VO oscillator is also modelled using a Constant Order (CO) formulation where a number of CO fractional order differentials are weighted to simulate the VO behavior. The CO formulation asymptotically approaches the VO results when a relatively large number of weights is used. For the viscoelastic range of 0 ≤ q ≤ 1, the dynamics of the oscillator is well approximated by the CO formulation when 5 or more fractional terms are included (e.g., 0, 1/4, 1/2, 3/4, and 1).     

Lifting the singular nature of a model for peeling of an adhesive tape

We investigate the dynamics of peeling of an adhesive tape subjected to a constant pull speed. Due to the constraint between the pull force, peel angle and the peel force, the equations of motion derived earlier fall into the category of differential-algebraic equations (DAE) requiring an appropriate algorithm for its numerical solution. By including the kinetic energy arising from the stretched part of the tape in the Lagrangian, we derive equations of motion that support stick-slip jumps as a natural consequence of the inherent dynamics itself, thus circumventing the need to use any special algorithm. In the low mass limit, these equations reproduce solutions obtained using a differential-algebraic algorithm introduced for the earlier singular equations. We find that mass has a strong influence on the dynamics of the model rendering periodic solutions to chaotic and vice versa. Apart from the rich dynamics, the model reproduces several qualitative features of the different waveforms of the peel force function as also the decreasing nature of force drop magnitudes.     

Nonlinear dynamics of the heartbeat: I. The AV junction: Passive conduit or active oscillator?

Under physiologic conditions, the AV junction is traditionally regarded as a passive conduit for the conduction of impulses from the atria to the ventricles. An alternative view, namely that subsidiary pacemakers play an active role in normal electrophysiologic dynamics during sinus rhythm, has been suggested based on nonlinear models of cardiac oscillators. A central problem has been the development of a simple but explicit mathematical model for coupled nonlinear oscillators relevant both to stable and perturbed cardiac dynamics. We use equations describing an analog electrical circuit with an external d.c. voltage source (V₀) and two nonlinear oscillators with intrinsic frequencies in the ratio of 3:2, comparable to the SA node and AV junction rates. The oscillators are coupled by means of a resistor. 1:1 (SA:AV) phase-locking of the oscillators occurs over a critical range of V₀. Externally driving the SA oscillator at increasing rates results in 3:2 AV Wenckebach periodicity and a 2:1 AV block. These findings appear with no assumptions about conduction time or refractoriness. This dynamical model is consistent with the new interpretation that normal sinus rhythm may represent 1:1 coupling of two or more active nonlinear oscillators and also accounts for the appearance of an AV block with critical changes in a single parameter such as the pacing rate.     

Stock market and motion of a variable mass spring

We establish an analogy between the motion of spring whose mass increases linearly with time and volatile stock market dynamics within an economic model based on simple temporal demand and supply functions [E. Canessa, J. Phys. A 33 (2000) 3637]. The total system energy E_t is shown to be proportional to a decreasing time dependent spring constant k_t. This model allows to derive log-periodicity cos[log(t−t_c)] on commodity prices and oscillations (surplus and shortages) in the level of stocks. We also made an attempt to connect these results to the Tsallis statistics parameter q based on a possible force-entropy correlation [E. Canessa, Physica A 341(2004) 165] and find that the Tsallis second entropic term relates to the square of the demand (or supply) function.     

Existence and analytical predictions of periodic motions in a periodically forced,nonlinear friction oscillator

The grazing bifurcation, stick phenomena and periodic motions in a periodically forced, nonlinear friction oscillator are investigated. The nonlinear friction force is approximated by a piecewise linear, kinetic friction model with the static force. The total forces for the input and output flows to the separation boundary are introduced, and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces. The periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure. Illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction. The force responses are presented, which agreed very well with the force criteria. If the fully nonlinear friction force is modeled by several portions of piecewise linear functions, the periodically forced, nonlinear friction oscillator can be predicted more accurately. However, for the fully nonlinear friction force model, only the numerical investigation can be carried out.     

Quantum Nonlinear Oscillator with Two Degrees of Freedom in a Laser Field

The semiclassical dynamics of a quantum nonlinear oscillator with two degrees of freedom and anharmonicity of the fourth order in a periodic laser field is studied both analytically and numerically. In the absence of external excitation and dissipation, the equations of motion for the mean values of the coordinate and momentum operators of both degrees of freedom reduce to the equation of a onedimensional nonlinear pendulum. The general solution of this equation is written in terms of the Jacobian elliptic functions. As can be expected, the energy of the free oscillator is redistributed periodically between degrees of freedom. The periodic excitation of the nonlinear oscillator may substantially change its motion pattern. Using as an example an oscillator with two coupled vibrational degrees of freedom, it is numerically shown that the amount of laser photons absorbed depending on the parameter values and initial conditions may vary with time in a rather complex manner, including chaotic oscillations. A nonlinear oscillator is capable of manifesting bistable behavior with allowance for dissipation. The analytical condition for the origination of bistability is found. Examples of the bistable dependence of the number of quanta in the oscillator vibrational mode on the level of laser excitation are presented.     

Growth of attached actin filaments

In several studies of actin-based cellular motility, the barbed ends of actin filaments have been observed to be attached to moving obstacles. Filament growth in the presence of such filament-obstacle interactions is studied via Brownian dynamics simulations of a three-dimensional energy-based model. We find that with a binding energy greater than 24k _B T and a highly directional force field, a single actin filament is able to push a small obstacle for over a second at a speed of half of the free filament elongation rate. These results are consistent with experimental observations of plastic beads in cell extracts. Calculations of an external force acting on a single-filament-pushed obstacle show that for typical in vitro free-actin concentrations, a 3pN pulling force maximizes the obstacle speed, while a 4pN pushing force almost stops the obstacle. Extension of the model to treat beads propelled by many filaments suggests that most of the propulsive force could be generated by attached filaments.     

Estimation of Kauzmann temperature and isenthalpic temperature of metallic glasses

We propose expressions for the estimation of the isenthalpic temperature T ₀ (T ₀ = αT _m , α is a semi-empirical parameter and 0 ⩽ α < 1, T _m is the solidus temperature) and the Kauzmann temperature T _k (T _k = T _m exp(α−1)) for glass forming alloys. It is found that T _k estimated by T _k = T _m exp(α−1) is in agreement with that directly calculated from the heat capacity data, indicating that T _k = T _m exp(α − 1) can be used to estimate T _k of glass forming alloys. T ₀ estimated by T ₀ = αT _m , on the other hand, widely deviates from that of directly calculated from the heat capacity data. This suggests that the enthalpy difference of the under-cooled liquid and the crystal might be a nonlinear function of the temperature below T _k . Moreover, the Gibbs free energy difference ΔG is not sensitive to the deviation of α.     

A new perturbative approach to the classical anharmonic oscillator

The periodic motion of the classical anharmonic oscillator characterized by the potentialV(x)=1/2x ²+λ/2k x ^2k is considered. The period is first determined to all orders inλ in a perturbative series. Making use of this, the solution of the nonlinear equation of motion is then expressed in the form of a Fourier series. The Fourier coefficients are obtained by solving simple algebraic relations. Secular terms are inherently absent in this perturbative scheme. Explicit solution is presented for generalk up to the second order, from which the Duffing and the sextic oscillator results follow as special cases.     

Instability regimes and self-excited vibrations in deep drilling systems

This paper analyzes the stability of the discrete model proposed by Richard et al. (2004 [1], 2007 [2]) to study the self-excited axial and torsional vibrations of deep drilling systems. This model, which relies on a rate-independent bit/rock interaction law, reduces to a coupled system of state-dependent delay differential equations governing the axial and angular perturbations to the stationary motion of the bit. A linear stability analysis indicates that, although the steady-state motion of the bit is always unstable, the nature of the instability depends on the nominal angular velocity Ω₀ of the drillstring imposed at the rig. On the one hand, if Ω₀ is larger than a critical velocity Ω_c, the angular dynamics is responsible for the instability. However, on the timescale of the resonance period of the drillstring viewed as a torsional pendulum, the system behaves like a marginally stable one, provided that exogenous perturbations are of limited magnitude. The instability then only appears on a much larger timescale, in the form of slowly growing oscillations that ultimately lead to an undesired drilling regime such as bit-bouncing or stick-slip vibrations. On the other hand, if Ω₀ is smaller than Ω_c, the instability manifests itself on the timescale of the bit motion due to a dominating unstable axial dynamics; perturbations to the steady-state motion then rapidly degenerate into stick-slip limit cycles or bit-bouncing. For typical deep drilling field conditions, the critical angular velocity Ω_c is virtually independent of the axial force acting on the bit and of the bit bluntness. It can be approximated by a power law monomial, a function of known parameters of the drilling system and of the intrinsic specific energy (a quantity characterizing the energy required to drill a particular rock). This approximation holds on account that the dissipation in the drilling structure is negligible with respect to that taking place through the bit/rock interaction, as is typically the case. These findings are further illustrated on an example of deep drilling and shown to match the trends observed in the field.     

A dynamical transition of a chain of charged particles in a 2D substrate potential

The transport of a chain of charged particles with a transverse degree of freedom is investigated in a 2D asymmetric potential. Here, the energy of the periodic driving force is converted into motion in the vertical direction. The analysis exhibits a transition from stick-slip motion to periodic oscillation. The chain velocity can be controlled to an optimized value by adjusting system parameters, such as the amplitude and frequency of the periodic force. The existence of a resonance platform indicates resonance between the motion of the chain and the periodic force as coupling strength increases adiabatically. The atomic configuration and the transverse degree of freedom also play key roles in the control.     

High frequency forcing on nonlinear systems

High-frequency signals are pervasive in many science and engineering fields.In this work,the effect of high-frequency driving on general nonlinear systems is investigated,and an effective equation for slow motion is derived by extending the inertial approximation for the direct separation of fast and slow motions.Based on this theory,a high-frequency force can induce various phase transitions of a system by changing its amplitude and frequency.Numerical simulations on several nonlinear oscillator systems show a very good agreement with the theoretic results.These findings may shed light on our understanding of the dynamics of nonlinear systems subject to a periodic force.     

Surface morphology characterization of pentacene thin film and its substrate with under-layers by power spectral density using fast Fourier transform algorithms

Surface morphology of pentacene thin films and their substrates with under-layers is characterized by using atomic force microscopy (AFM). The power values of power spectral density (PSD) for the AFM digital data were determined by the fast Fourier transform (FFT) algorithms instead of the root-mean-square (rms) and peak-to-valley value. The PSD plots of pentacene films on glass substrate are successfully approximated by the k-correlation model. The pentacene film growth is interpreted the intermediation of the bulk and surface diffusion by parameter C of k-correlation model. The PSD plots of pentacene film on Au under-layer is approximated by using the linear continuum model (LCM) instead of the combination model of the k-correlation model and Gaussian function. The PSD plots of SiO₂ layer on Au under-layer as a gate insulator on a gate electrode of organic thin film transistors (OTFTs) have three power values of PSD. It is interpreted that the specific three PSD power values are caused by the planarization of the smooth SiO₂ layer to rough Au under-layer.     

Effect of tangential interface motion on the viscous instability in fluid flow past flexible surfaces

The stability of linear shear flow of a Newtonian fluid past a flexible membrane is analysed in the limit of low Reynolds number as well as in the intermediate Reynolds number regime for two different membrane models. The objective of this paper is to demonstrate the importance of tangential motion in the membrane on the stability characteristics of the shear flow. The first model assumes the wall to be a “spring-backed” plate membrane, and the displacement of the wall is phenomenologically related in a linear manner to the change in the fluid stresses at the wall. In the second model, the membrane is assumed to be a two-dimensional compressible viscoelastic sheet of infinitesimal thickness, in which the constitutive relation for the shear stress contains an elastic part that depends on the local displacement field and a viscous component that depends on the local velocity in the membrane. The stability characteristics of the laminar flow in the limit of low are crucially dependent on the tangential motion in the membrane wall. In both cases, the flow is stable in the low Reynolds number limit in the absence of tangential motion in the membrane. However, the presence of tangential motion in the membrane destabilises the shear flow even in the absence of fluid inertia. In this case, the non-dimensional velocity (Λ_t) required for unstable fluctuations is proportional to the wavenumber k ( Λ _t∼k) in the plate membrane type of wall while it scales as k² in the viscoelastic membrane type of wall ( Λ _t∼k ²) in the limit k→ 0. The results of the low Reynolds number analysis are extended numerically to the intermediate Reynolds number regime for the case of a viscoelastic membrane. The numerical results show that for a given set of wall parameters, the flow is unstable only in a finite range of Reynolds number, and it is stable in the limit of large Reynolds number. Received 8 November 2000 and Received in final form 20 March 2001     

Dynamics of a single ion in a perturbed Penning trap. Sextupolar perturbation

In the frame work of classical mechanics, we study the nonlinear dynamics of a single ion trapped in a Penning trap perturbed by an electrostatic sextupolar perturbation. The perturbation is caused by a deformation in the configuration of the electrodes. By using a Hamiltonian formulation, we obtain that the system is governed by three parameters: the z-component of the canonical angular momentum P _φ - which is a constant of the motion because the perturbation we assume is axial-symmetric -, the parameter δ that determines the ratio between the axial and the cyclotron frequencies, and the parameter a which indicates how far from the ideal design the electrodes are. We study the case P _φ = 0. By means of surfaces of section, we show that the phase space structure is made of three fundamental families of orbits: arch, loop and box orbits. The coexistence of these kinds of orbits depends on the parameter δ. The escape is also explained on the basis of the shape of the potential energy surface as well as of the phase space structure. Received 6 September 2001 / Received in final form 19 March 2002 Published online 28 June 2002     

Spatial variations of the mean and statistical quantities in the thermal boundary layers of turbulent convection

We have studied the dynamics of the contact line of a viscous liquid on a solid substrate with macroscopic random defects. We have first characterized the friction force f₀ at microscopic scale for a substrate without defects; f₀ is found to be a strongly nonlinear function of the velocity U of the contact line. In presence of macroscopic defects, we find that the applied force F(U) is simply shifted with respect to f₀(U) by a constant: we do not observe any critical behavior at the depinning transition. The only observable effect of the substrate disorder is to increase the hysteresis. We have also performed realistic numerical simulation of the motion of the contact line. Using the same values of the parameters as in the experiment, we find that the experimental data is qualitatively well reproduced. In light of experimental and numerical results, we discuss the possibility of measuring a true critical behavior.Received: 6 October 2003, Published online: 19 February 2004PACS: 46.65. + g Random phenomena and media - 64.60.Ht Dynamic critical phenomena - 68.08.Bc Wetting     

A limited resource model of fault-tolerant capability against cascading failure of complex network

We propose a novel capacity model for complex networks against cascading failure. In this model, vertices with both higher loads and larger degrees should be paid more extra capacities, i.e. the allocation of extra capacity on vertex i will be proportional to k_i ^γ , where k_i is the degree of vertex i and γ > 0 is a free parameter. We have applied this model on Barabási-Albert network as well as two real transportation networks, and found that under the same amount of available resource, this model can achieve better network robustness than previous models.