On the local stability of limit cycles

Orbital stability of limit cycles is the result of the competing local tendencies of perturbations from the cycle to decay (during phases of local stability) and to grow (during phases of local instability), averaged over a cycle. We examine this coexistence of attractive and repulsive phases on limit cycles, including the local rates of expansion and contraction of phase space volumes. This is done in a frame of reference that moves along the orbit, to partially decouple motions tangential and perpendicular to the cycle. Dynamical systems used for illustration are the generalized Bonhoeffer-van-der-Pol and Rossler models, both far from and near to different types of bifurcations. Finally, it is shown that the nonuniformity of local stability in phase space affects the response of limit cycle oscillators to perturbations and gives rise to their phase-dependent response. (c) 1999 American Institute of Physics.     

Nonlinear time series analysis of normal and pathological human walking

Characterizing locomotor dynamics is essential for understanding the neuromuscular control of locomotion. In particular, quantifying dynamic stability during walking is important for assessing people who have a greater risk of falling. However, traditional biomechanical methods of defining stability have not quantified the resistance of the neuromuscular system to perturbations, suggesting that more precise definitions are required. For the present study, average maximum finite-time Lyapunov exponents were estimated to quantify the local dynamic stability of human walking kinematics. Local scaling exponents, defined as the local slopes of the correlation sum curves, were also calculated to quantify the local scaling structure of each embedded time series. Comparisons were made between overground and motorized treadmill walking in young healthy subjects and between diabetic neuropathic (NP) patients and healthy controls (CO) during overground walking. A modification of the method of surrogate data was developed to examine the stochastic nature of the fluctuations overlying the nominally periodic patterns in these data sets. Results demonstrated that having subjects walk on a motorized treadmill artificially stabilized their natural locomotor kinematics by small but statistically significant amounts. Furthermore, a paradox previously present in the biomechanical literature that resulted from mistakenly equating variability with dynamic stability was resolved. By slowing their self-selected walking speeds, NP patients adopted more locally stable gait patterns, even though they simultaneously exhibited greater kinematic variability than CO subjects. Additionally, the loss of peripheral sensation in NP patients was associated with statistically significant differences in the local scaling structure of their walking kinematics at those length scales where it was anticipated that sensory feedback would play the greatest role. Lastly, stride-to-stride fluctuations in the walking patterns of all three subject groups were clearly distinguishable from linearly autocorrelated Gaussian noise. As a collateral benefit of the methodological approach taken in this study, some of the first steps at characterizing the underlying structure of human locomotor dynamics have been taken. Implications for understanding the neuromuscular control of locomotion are discussed. (c) 2000 American Institute of Physics.     

Sufficient conditions for linear long-wave instability of steady-state axisymmetric flows of an ideal liquid with a free boundary in an azimuthal magnetic field

The problem of linear stability of steady-state axisymmetric shear jet flows of an inviscid ideally conducting incompressible liquid with a free surface and “frozen-in” azimuthal magnetic field is analyzed. The sufficient conditions for theoretical (on semi-infinite time intervals) and practical (on finite time intervals) instability of these flows relative to small axisymmetric long-wave perturbations are obtained by the direct Lyapunov method. An a priori lower estimate indicating (at least) an exponential increase with time of small perturbations under investigation is constructed in the case when these conditions are valid for theoretical as well as practical instability. In addition, an illustrative analytic example of steady-state flows under investigation and small axisymmetric long-wave perturbations superimposed on them is constructed (according to our estimate, these perturbations increase with time).     

The Diffusion Kernel Filter

A particle filter method is presented for the discrete-time filtering problem with nonlinear Itô stochastic ordinary differential equations (SODE) with additive noise supposed to be analytically integrable as a function of the underlying vector-Wiener process and time. The Diffusion Kernel Filter is arrived at by a parametrization of small noise-driven state fluctuations within branches of prediction and a local use of this parametrization in the Bootstrap Filter. The method applies for small noise and short prediction steps. With explicit numerical integrators, the operations count in the Diffusion Kernel Filter is shown to be smaller than in the Bootstrap Filter whenever the initial state for the prediction step has sufficiently few moments. The established parametrization is a dual-formula for the analysis of sensitivity to gaussian-initial perturbations and the analysis of sensitivity to noise-perturbations, in deterministic models, showing in particular how the stability of a deterministic dynamics is modeled by noise on short times and how the diffusion matrix of an SODE should be modeled (i.e. defined) for a gaussian-initial deterministic problem to be cast into an SODE problem. From it, a novel definition of prediction may be proposed that coincides with the deterministic path within the branch of prediction whose information entropy at the end of the prediction step is closest to the average information entropy over all branches. Tests are made with the Lorenz-63 equations, showing good results both for the filter and the definition of prediction.     

A primary mechanism for spiral wave meandering

The stability and dynamics of spiral wave meandering were studied by examining the behavior of small perturbations to a steadily rotating action potential wave. The disturbances responsible for meandering were found to be generated through an interaction between the unstable local linear dynamics characteristic of the action potential trailing edge near the core and perturbations existing in the region immediately behind this edge. Significantly, for the cases studied, neither wavefront curvature nor head-tail interactions were involved in this process. Study of the generation mechanism using a series of representative mathematical models and computer experiments led to the prediction that the following features of rotating action potentials render them more susceptible to meandering: (1) proximity of the wave tip to the center of rotation, (2) wider action potential leading and trailing edges, and (3) slower wave rotation speeds. Variation of basic tissue properties, including firing threshold potentials and excitability above threshold, affected these properties, and those of the perturbation dynamics, in several ways, producing both stabilizing and destabilizing effects. The nature of the involvement of various tissue and membrane electrical properties is therefore complex, affecting several factors relevant to meandering at once. (c) 2002 American Institute of Physics.     

Nuclear spin relaxation and molecular motions in sebacate polyesters

In this study, pulsed NMR techniques were used to probe the molecular motions occurring in poly(hexamethylenesebacate) (HMS), and its amorphous isomer poly(2-methyl-2-ethyl-1,3-propylenesebacate (MEPS), and in particular, to examine the perturbations in the molecular motions of HMS and MEPS that arise as a result of block copolymerization. The results show a low-temperature β-relaxation due to local motion of the methylenes of the chain backbone, an α-relaxation associated with the glass transition, and an α_c-relaxation due to the melting of the crystalline phase. Both α-and β-relaxations in the block copolymer are only slightly perturbed, and this suggests that the blocks are incompatible. The β-relaxation of HMS is confined to the amorphous regions of the polymer. Although sample preparation had little effect on the degree of crystallinity, large differences in the relaxation behavior were observed when the sample preparation was varied.     

Coupling and stability of bright holographic soliton pairs

The coupling effect and stability property of symmetric bright holographic soliton pairs have been investigated numerically. Results show that when any one of the two solitary beams from a pair is perturbed in amplitude or width, both beams will be affected by such a perturbation via the coupling effect between the beams, thus resulting in both beams propagating in the medium without a constant shape; however, these two solitary beams are still stable against small perturbations. When both solitary beams from a pair are perturbed simultaneously in amplitude, for some given absolute values of the perturbations, the two beams are stable against these perturbations if both beams are perturbed with the same sign, whereas are unstable with the different signs. When the two beams are simultaneously perturbed in width, both beams exhibit their stability property similar to that when only one beam is perturbed no matter whether both perturbations have the same or different signs.     

Impact of perturbations on watersheds

We find that watersheds in real and artificial landscapes can be strongly affected by small, local perturbations like landslides or tectonic motions. We observe power-law scaling behavior for both the distribution of areas enclosed by the original and the displaced watershed as well as the probability density to induce, after perturbation, a change at a given distance. Scaling exponents for real and artificial landscapes are determined, where in the latter case the exponents depend linearly on the Hurst exponent of the applied fractional Brownian noise. The obtained power laws are shown to be independent on the strength of perturbation. Theoretical arguments relate our scaling laws for uncorrelated landscapes to properties of invasion percolation.     

Magic-Angle Spinning NMR and Molecular Mobility in Heterogeneous Systems

Magic-angle sample spinning (MAS) nuclear magnetic resonance (NMR) measurements are treated for heterogeneous systems with nanometer dimensions. An appreciable line narrowing in the MAS NMR spectra of the embedded molecules may be achieved also in the cases when the molecules still possess an appreciable local mobility. It appears that the MAS frequencies are of comparable order of magnitude as the frequencies which characterize the random molecular motional processes and which compete with MAS. It will be shown that this behavior may occur if inhomogeneous local magnetic fields due to susceptibility effects have a dominating influence on the widths and shapes of the resonance NMR lines. Properties of these local fields are described. Spectra simulations are carried for molecules embedded in these heterogeneous systems when the coherent averaging by MAS is superimposed by random local motions. This situation may occur for molecules contained in nanoporous solids and also for heterogeneous systems like membranes and biological tissues with flexible components like water, lipids, and small peptides. Several examples are treated which reveal advantages and limitations of these experiments and their theoretical interpretation.     

Topological origin of global attractors in gene regulatory networks

Fixed-point attractors with global stability manifest themselves in a number of gene regulatory networks. This property indicates the stability of regulatory networks against small state perturbations and is closely related to other complex dynamics. In this paper, we aim to reveal the core modules in regulatory networks that determine their global attractors and the relationship between these core modules and other motifs. This work has been done via three steps. Firstly, inspired by the signal transmission in the regulation process, we extract the model of chain-like network from regulation networks. We propose a module of “ideal transmission chain(ITC)”, which is proved sufficient and necessary(under certain condition) to form a global fixed-point in the context of chain-like network. Secondly, by examining two well-studied regulatory networks(i.e., the cell-cycle regulatory networks of Budding yeast and Fission yeast), we identify the ideal modules in true regulation networks and demonstrate that the modules have a superior contribution to network stability(quantified by the relative size of the biggest attraction basin). Thirdly, in these two regulation networks, we find that the double negative feedback loops, which are the key motifs of forming bistability in regulation, are connected to these core modules with high network stability. These results have shed new light on the connection between the topological feature and the dynamic property of regulatory networks.     

Stability of self-gravitating astrophysical disks

Criteria which guarantee the stability of self-gravitating gaseous and stellar disks toward any localized small perturbations are obtained. These criteria are formulated as inequalities of the form Q>Q _c (separately for gas and stars). The latter should be satisfied by the “stability parameter” Q, which is equal, by definition, to unity on the stability boundary of radial perturbations. The critical value of the stability parameter Q _c is appreciably greater than (although of the order of) unity, attesting to the great instability of nonaxially symmetric perturbations. It is shown that the stability criterion derived for gaseous disks is valid for disks rotating within a spheroidal component (as in spiral galaxies) or in the field of a central mass (planetary rings and accretion disks). Stellar disks are stabilized with significantly greater difficulty. This is attributable mainly to the anisotropy of the velocity distribution inherent to them, which is favorable for instability. Zh. éksp. Teor. Fiz. 112, 771–795 (September 1997)     

Local 3D perturbation experiments for probing the ELM stability

Empirical estimates indicate that the transient divertor power load induced by type-I ELMs in the standard H-mode ITER scenario might result in intolerably short target life times. Significant experimental effort has been put on the exploration of recipes to avoid or at least mitigate the ELM size by external intervention. Experimental ELM mitigation studies (and related modeling) are of interest from different points of views. First, the active intervention gives access to new, externally controlled ELMy discharge scenarios and ELM diagnostic techniques. Second, the ELM response on the type and amplitude of the applied perturbation provides additional hints on the ELM stability. Third, mitigated ELMs, possibly different from spontaneous ones, and their scaling are of interest ‘per se’, in case that mitigation is to be finally applied in ITER. In this contribution we report on ELM mitigation results in ASDEX Upgrade, where we further restrict the scope to those methods involving 3D, local non-axisymmetric perturbations. Perturbations applied were cryogenic pellet injection, localized supersonic pulsed injection of D gas and laser blow of from targets carrying C or Al micro pellets. The corresponding perturbations differ strongly both in the space and time domain and provide an instrument to probe the local stability. Varying the parameters of the external perturbations by e.g. changing the amount of injected material or the injection speed we mapped out the ELM trigger threshold with high spatial and temporal resolutions. Presented at the Workshop “Electric Fields, Structures and Relaxation in Edge Plasmas”, Tarragona, Spain, July 3–4, 2005.     

Conformal transformations in cosmology of modified gravity: the covariant approach perspective

The 1+3 covariant approach and the covariant gauge-invariant approach to perturbations are used to analyze in depth conformal transformations in cosmology. Such techniques allow us to obtain insights on the physical meaning of these transformations when applied to non-standard gravity. The results obtained lead to a number of general conclusions on the change of some key quantities describing any two conformally related cosmological models. For example, even if some of the geometrical properties of the cosmology are preserved (homogeneous and isotropic Universes are mapped into homogeneous and isotropic universes), it can happen that decelerating cosmologies can be mapped into accelerated ones. From the point of view of the cosmological perturbations it is shown how these fluctuation transform. We find that first-order vector and tensor perturbations equations are left unchanged in their structure by the conformal transformation, but this cannot be said of the scalar perturbations, which present differences in their evolutionary features. The results obtained are then explicitly interpreted and verified with the help of some clarifying examples based on f(R)-gravity cosmologies.     

On the stability of Kolmogorov spectra of a weak turbulence

The stability problem of Kolmogorov spectra of a weak turbulence is analytically solved for the first time in the framework of a three-wave kinetic equation. The spectrum of isotropic perturbations of a stationary not-in-equilibrium distribution is found for the capillary waves on a shallow water surface. It is shown, in the isotropic case, that the Kolmogorov solution is stable with respect to excitations local in k-space. The perturbations drift to the damping region without growth of the magnitude. The structural instability of the isotropic spectrum is found by computer simulation: a small pumping anisotropy causes the spectrum to be essentially anisotropic within the inertial range.     

Causality and Cauchy horizons

LetS be a partial Cauchy surface for (M, go) which remains a partial Cauchy surface under small metric perturbations. In general, the Cauchy horizon H⁺(go, S) may be unstable to small changes in the metric. Points of the horizon may move by large amounts and even the topological type of the horizon may change under arbitrarily small changes in the metric tensor. In this paper, we investigate sufficient conditions for existential, locational, and topological stability of Cauchy horizons under metric changes which perturb the light cones by small amounts.     

The problem of stability of the homogeneous and isotropic universe in the relativistic theory of gravitation

The problem of stability of the homogeneous and isotropic Universe with respect to small perturbations in the gravitational field and matter characteristics is studied in the framework of the relativistic theory of gravitation. The equations for small perturbations of the metric tensor g _μν, energy density ρ, and pressure p are obtained in the linear approximation. The solutions to these equations are found when perturbations depend only on time. The physical character of the obtained solutions is analyzed. A comparison with the results of General Relativity yields the conclusion that all differences are due to the graviton mass.     

Breakdown of a topological phase: quantum phase transition in a loop gas model with tension

We study the stability of topological order against local perturbations by considering the effect of a magnetic field on a spin model--the toric code--which is in a topological phase. The model can be mapped onto a quantum loop gas where the perturbation introduces a bare loop tension. When the loop tension is small, the topological order survives. When it is large, it drives a continuous quantum phase transition into a magnetic state. The transition can be understood as the condensation of "magnetic" vortices, leading to confinement of the elementary "charge" excitations. We also show how the topological order breaks down when the system is coupled to an Ohmic heat bath and relate our results to error rates for topological quantum computations.     

Stability of Semiclassical Gravity Solutions with Respect to Quantum Metric Fluctuations

We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized 2-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein–Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized 2-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski space-time as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations.