Local chaos induced by spatial oscillations of a perturbation

We study motion of an one-dimensional Hamiltonian oscillator driven by an external force which is periodic in time and in coordinate as well. It is shown that dynamics of the oscillator is strongly affected by the resonance between spatial and temporal oscillations of the perturbation imposed. In particular, this resonance can induce strong but bounded chaotic diffusion in certain areas of phase space. The model of the Duffing oscillator is used as an example for the numerical simulation.     

Capture into resonance and escape from it in a forced nonlinear pendulum

We study the dynamics of a nonlinear pendulum under a periodic force with small amplitude and slowly decreasing frequency. It is well known that when the frequency of the external force passes through the value of the frequency of the unperturbed pendulum’s oscillations, the pendulum can be captured into resonance. The captured pendulum oscillates in such a way that the resonance is preserved, and the amplitude of the oscillations accordingly grows. We consider this problem in the frames of a standard Hamiltonian approach to resonant phenomena in slow-fast Hamiltonian systems developed earlier, and evaluate the probability of capture into resonance. If the system passes through resonance at small enough initial amplitudes of the pendulum, the capture occurs with necessity (so-called autoresonance). In general, the probability of capture varies between one and zero, depending on the initial amplitude. We demonstrate that a pendulum captured at small values of its amplitude escapes from resonance in the domain of oscillations close to the separatrix of the pendulum, and evaluate the amplitude of the oscillations at the escape.     

Synchronization of heteroclinic circuits through learning in coupled neural networks

The synchronization of oscillatory activity in neural networks is usually implemented by coupling the state variables describing neuronal dynamics. Here we study another, but complementary mechanism based on a learning process with memory. A driver network, acting as a teacher, exhibits winner-less competition (WLC) dynamics, while a driven network, a learner, tunes its internal couplings according to the oscillations observed in the teacher. We show that under appropriate training the learner can “copy” the coupling structure and thus synchronize oscillations with the teacher. The replication of the WLC dynamics occurs for intermediate memory lengths only, consequently, the learner network exhibits a phenomenon of learning resonance.     

Spiral waves in excitable media due to noise and periodic forcing

We investigate the jointly driven effects of external periodic forcing and Gaussian white noise on meandering spiral waves in excitable media with FitzHugh-Nagumo local dynamics. Interesting phenomena resulted from various forcing periods are found, for example, piece-wise line drift, intermittent straight-line drift and so on. We also observe new type of breakup of spiral wave between entrainment bands with 1:1 and 2:1. It is believed that the occurrence of the new type is relevant to the appearance of local bidirectional propagation window. There exist optimized noise intensities which can induce the broadest entrainments and Arnold tongues. Such a phenomenon is referred to as stochastic resonance. It is also observed that the noise makes significant effects on the spiral wave with straight-line drift. Via the tip Fourier spectrum, the varying of tip motion with external periods on the resonance band is interpreted.     

Parametric and Autoparametric Resonance

A large number of mathematical questions are related to problems of parametric and autoparametric resonance in engineering models. The linearized problems generally produce systems of differential equations with periodic coefficients with special stability and genericity questions. We start by reviewing linear systems while discussing normal form techniques and bifurcation results. The linear and nonlinear analysis is illustrated in three cases: rotor dynamics, autoparametric resonance of a parametric oscillator and autoparametric resonance of a self-excited oscillator. In all cases bifurcations, symmetry considerations and attraction to nonclassical limit sets play a part.     

Sub-Laplacian Eigenvalue Bounds on CR Manifolds

We prove upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. These bounds are compatible with the Menikoff-Sjöstrand asymptotic law, and can be viewed as a CR version of Korevaar's bounds for Laplace eigenvalues of conformal metrics.     

Magnetic Elements at Finite Temperature and Large Deviation Theory

We investigate thermally activated phenomena in micromagnetics using large deviation theory and concepts from stochastic resonance. We give a natural mathematical definition of finite-temperature astroids, finite-temperature hysteresis loops, etc. Generically, these objects emerge when the (generalized) Arrhenius timescale governing the thermally activated barrier crossing event of magnetic switching matches the timescale at which the magnetic element is pulsed or ramped by an external field; in the special and physically relevant case of multiple-pulse experiments, on the other hand, short-time switching can lead to non-Arrhenius behavior. We show how large deviation theory can be used to explain some properties of the astroids, like their shrinking and sharpening as the number of applied pulses is increased. We also investigate the influence of the dynamics, in particular the relative importance of the gyromagnetic and the damping terms. Finally, we discuss some issues and open questions regarding spatially nonuniform magnetization.     

CR singular images of generic submanifolds under holomorphic maps

The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in dimension 2) condition for the diffeomorphism to extend to a finite holomorphic map. The multiplicity of this map is a biholomorphic invariant that is precisely the Moser invariant of the image, when it is a Bishop surface with vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR singular images and we prove that the set of CR singular points must be large, and in the case of codimension 2, necessarily Levi-flat or complex. We also show that there exist real-analytic CR functions on such images that satisfy the tangential CR conditions at the singular points, yet fail to extend to holomorphic functions in a neighborhood. We provide many examples to illustrate the phenomena that arise.     

Throughput-oriented channel assignment for opportunistic spectrum access networks

Cognitive radio (CR) is a revolutionary technology in wireless communications that enhances spectrum utilization by allowing opportunistic and dynamic spectrum access. One of the key challenges in this domain is how CR users cooperate to dynamically access the available spectrum opportunities in order to maximize the overall perceived throughput. In this paper, we consider the coordinated spectrum access problem in a multi-user single-transceiver CR network (CRN), where each CR user is equipped with only one half-duplex transceiver. We first formulate the dynamic spectrum access as a rate/power control and channel assignment optimization problem. Our objective is to maximize the sum-rate achieved by all contending CR users over all available spectrum opportunities under interference and hardware constraints. We first show that this problem can be formulated as a mixed integer nonlinear programming (MINLP) problem that is NP-hard, in general. By exploiting the fact that actual communication systems have a finite number of available channels, each with a given maximum transmission power, we transfer this MINLP into a binary linear programming problem (BLP). Due to its integrality nature, this BLP is expected to be NP-hard. However, we show that its constraint matrix satisfies the total unimodularity property, and hence our problem can be optimally solved in polynomial time using linear programming (LP). To execute the optimal assignment in a distributed manner, we then present a distributed CSMA/CA-based random access mechanism for CRNs. We compare the performance of our proposed mechanism with reference CSMA/CA channel access mechanisms designed for CRNs. Simulation results show that our proposed mechanism significantly improves the overall network throughput and preserves fairness.     

Resonance layers in the interactions of rapidly oscillating wave fields

Effects pertaining to weak resonance interactions of rapidly oscillating wave fields are investigated. In the general case, there exist narrow transient layers where the wave field dynamics can be described in terms of the nonlinear Schrödinger equation.     

Exact solution of a nonresonance Zwanzig model and “quantum chaos”

We consider the quantum dynamics of the Zwanzig model in the absence of resonance of an impurity site with a reservoir and obtain analytic expressions for the evolution of the impurity site population under different initial conditions. The agreement with numerical calculations is excellent. We analyze the possibility of the existence of quantum chaos in the model.     

Regularity of b complex on nondegenerate Cauchy-Riemann manifolds

For each point ξ in a CR manifold M of codimension greater than 1, the CR structure of M can be approximated by the CR structure of a nilpotent Lie group Gξ of step two near ξ. Gξ varies with ξ.□b and b on M can be approximated by □b and b on the nilpotent Lie group Gξ. We can construct the parametrix of □b on M by using the parametrix of □b on nilpotent group of step two, and define a quasidistance on M by the approximation. The regularity of □b and b follows from the Harmonic analysis on M.     

Chaotic dynamics in classical nuclear billiards

We consider several noninteracting nucleons moving in a 2D Woods–Saxon type potential well and hitting the vibrating surface. The Hamiltonian has a coupling term between the particle motion and the collective coordinate which generates a self-consistent dynamics. The numerical simulation is based on the solutions of the Hamilton equations which was solved using an algorithm of Runge–Kutta type (order 4–5) having an optimized step size, taking into account that the absolute error for each variable is less than 10⁻⁶. Total energy is conserved with high accuracy, i.e., approx. 10⁻⁶ in absolute value. We analyze the chaotic behavior of the nonlinear dynamics system using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov–Sinai entropies. A qualitative and quantitative picture of the achievement of soft chaos is shown for a comparative study between the adiabatic and the resonance stage of nuclear interaction. We consider that the onset of chaos would be linked to the resonance stage of interaction. This assumption is argued in [1].     

Multi-way decomposition of projected spectra obtained in protein NMR

Multidimensional nuclear magnetic resonance spectroscopy is a highly versatile tool in protein research. Experimental limitations prohibit spectra with ≥ 5 dimensions. However, similar information can be obtained by recording 2-dimensional projections. A decomposition-based approach for analyzing projections is presented and applied to a protein. The resulting resonance identification is a prerequisite for further characterization of structure, dynamics and interactions of proteins. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial coherence resonance in neuronal media with discrete local dynamics

We study effects of spatiotemporal additive noise on the spatial dynamics of excitable neuronal media that is locally modelled by a two-dimensional map. We focus on the ability of noise to enhance a particular spatial frequency of the media in a resonant manner. We show that there exists an optimal noise intensity for which the inherent spatial periodicity of the media is resonantly pronounced, thus marking the existence of spatial coherence resonance in the studied system. Additionally, results are discussed in view of their possible biological importance.     

Vitushkin’s germ theorem for engel-type CR manifolds

We study real analytic CR manifolds of CR dimension 1 and codimension 2 in the three-dimensional complex space. We prove that the germ of a holomorphic mapping between “nonspherical” manifolds can be extended along any path (this is an analog of Vitushkin’s germ theorem). For a cubic model surface (“sphere”), we prove an analog of the Poincaré theorem on the mappings of spheres into ?². We construct an example of a compact “spherical” submanifold in a compact complex 3-space such that the germ of a mapping of the “sphere” into this submanifold cannot be extended to a certain point of the “sphere.”     

Compactness of pseudohermitian structures with integral bounds on curvature

In this paper, we show a compactness criterion for contact forms in a fixed CR structure(i.e., conformal pseudohermitian structures), assuming a volume bound and L^p bounds, p>2, on the Tanaka-Webster curvature, the pseudohermitian torsion and its covariant derivative. We also need the L² bound on the derivative of the Tanaka-Webster curvature along the characteristic vector field. As an application, we can show that the CR automorphism group is compact if M is not CR spherical or the CR Yamabe constant is negative.     

Sub-harmonic resonance and multi-annual oscillations in northern mammals: a non-linear dynamical systems perspective

We conjecture that the well-known oscillations (3- to 5-yr and 10-yr cycles) of northern mammals are examples of subharmonic resonance which obtains when ecological oscillators (predator-prey interactions) are subject to periodic forcing by the annual march of the seasons. The implications of this hypothesis are examined through analysis of a bare-bones, Hamiltonian model which, despite its simplicity, nonetheless exhibits the principal dynamical features of more realistic schemes. Specifically, we describe the genesis and destruction of resonant oscillations in response to variation in the intrinsic time scales of predator and prey. Our analysis suggests that cycle period should scale allometrically with body size, a fact first commented upon in the empirical literature some years ago. Our calculations further suggest that the dynamics of cyclic species should be phase coherent, i.e., that the intervals between successive maxima in the corresponding time series should be more nearly constant than their amplitude—a prediction which is also consistent with observation. We conclude by observing that complex dynamics in more realistic models can often be continued back to Hamiltonian limits of the sort here considered.     

Modeling and control of a novel pressure regulation mechanism for common rail fuel injection systems

This paper presents the modeling and control of a novel pressure regulation mechanism for the common rail (CR) fuel injection system of internal combustion engines (ICE). The pressure pulsations inside the common rail caused by the incoming and outgoing flows negatively affect the accuracy of both injected fuel quantities and flow rates. The objective of this work is to design a new regulating mechanism to suppress the pressure pulsation in the rail. We first present the one-dimensional distributed model for the common rail developed by using fluid flow equations, which can capture the distributed dynamics of the pressure pulsations in the rail and validating it with a physics based model developed in AMESim®. We then propose the concept of an active fluid storage device like a piezoelectric actuator (PZT) to minimize the pressure fluctuations. The location of the actuator on the common rail has also been evaluated to maximize its effect. The periodic nature of the injection event due to the stroke by stroke engine operation generates pressure pulsations in the rail which are periodic when represented in the rotational angle domain. To leverage this unique dynamic phenomenon we design a time-varying internal model-based controller to compensate the pressure pulsations.