Singular Perturbations, Transversality, and Sil'nikov Saddle-Focus Homoclinic Orbits

We consider the singularly perturbed system $\dot x$ =εf(x,y,ε,λ), $\dot y$ =g(x,y,ε,λ). We assume that for small (ε,λ), (0,0) is a hyperbolic equilibrium on the normally hyperbolic centre manifold y=0 and that y ₀(t) is a homoclinic solution of $\dot y$ =g(0,y,0,0). Under an additional condition, we show that there is a curve in the (ε,λ) parameter space on which the perturbed system has a homoclinic orbit also. We investigate the transversality properties of this orbit and use our results to give examples of 4 dimensional systems with Sil'nikov saddle-focus homoclinic orbits.     

Singular Perturbations and Torsional Wrinkling in a Truncated Hemispherical Thin Elastic Shell

Journal of Elasticity - The work described in this paper is concerned with providing a rational asymptotic analysis for the partial wrinkling bifurcation of a thin elastic hemispherical segment in...     

Geometric Singular Perturbations for Multiple Turning Points: Invariant Manifolds and Exchange Lemmas

In this paper, we consider singularly perturbed systems with multiple sets of turning points. Two types of related results contributing to geometric singular perturbation theory are obtained. The first result establishes a new class of invariant manifolds. The second result consists of several exchange lemmas that characterize the evolution of an invariant manifold passing through the vicinity of the slow manifold.Partially supported by NSF Grants DMS-0071931 and DMS-0406998.     

Periodic Solutions of Pendulum-Like Perturbations of Singular and Bounded $${\phi}$$-Laplacians

In this paper we study the existence and multiplicity of periodic solutions of pendulum-like perturbations of bounded or singular f{\phi}-Laplacians. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.     

Order and Pattern Formation in Psychotherapy

In an empirical analysis of the dynamics of psychotherapy processes, concepts and tools from self-organization theory are applied. Our focus is on pattern formation in the therapy system, i.e. the system constituted by the interaction dynamics of therapist and patient. The hypothesis of pattern formation explored is based on data sets of 28 psychotherapy courses (10 behavioral, 3 client-centered, 9 heuristic, 6 schema-oriented; 40 to 90 weekly sessions). Patients' and therapists' therapy session records are analyzed (33 variables addressing various aspects of the therapeutic relationship, of progress within and outside the therapy setting). Multivariate methods are implemented to test the key hypothesis of self-organization theory, namely the reduction of degrees of freedom of a system. Consistent with this hypothesis, a significant reduction of degrees of freedom is found when the initial and final sessions of the therapies are compared. Correspondingly, Landsberg order increased significantly in the course of therapies. Given these findings, the explorative question of how this self-organizing property relates to therapy outcome is investigated. We find a significant positive association with various outcome measures, such as: therapist's and client's evaluations of success (direct change measures), feelings of guilt, anxiety, social potence, depression (pre-post effect sizes), and others. These results suggest that order is related to therapy outcome. In our interpretation, order is a dynamical attribute of the therapeutic alliance to be considered as a promising therapy process variable.     

The Effect of Growth and Curvature on Pattern Formation

Based on first principles, we derive a general model to describe the spatio-temporal dynamics of two morphogens. The diffusive part of the model incorporates the dynamics, growth and curvature of one- and two-dimensional domains embedded in ³. Our generalized diffusion process includes spatio-temporal varying diffusion coefficients, advection, and dilution terms. We present specific examples by analyzing a third order activator--inhibitor mechanism for the kinetic part. We carry out illustrative numerical simulations on two-dimensional growing domains having different geometries. Comparisons with former results on fixed domains show the crucial role of growth and curvature of pattern selection. Evidence is given that both effects might be biologically relevant in explaining the selection of some observed patterns and in changing or enhancing their stability.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Singular Kernels,Multiscale Decomposition of Microstructure,and Dislocation Models

We consider a model for dislocations in crystals introduced by Koslowski, Cuitiño and Ortiz, which includes elastic interactions via a singular kernel behaving as the H ^1/2 norm of the slip. We obtain a sharp-interface limit of the model within the framework of Γ-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitté and Seppecher to which their rearrangement argument no longer applies. Instead, we show that the microstructure must be approximately one-dimensional on most length scales and we exploit this property to derive a sharp lower bound.     

Singular solitons,shock waves,and other solutions to potential KdV equation

This paper addresses the potential Korteweg–de Vries equation. The singular 1-soliton solution is obtained by the aid of ansatz method. Subsequently, the $G^{\prime }/G$ -expansion method and the exp-function approach also gives a few more interesting solutions. Finally, the Lie symmetry analysis leads to another plethora of solution to the equation. These results are going to be extremely useful and applicable in applied mathematics and theoretical physics.     

Synthesis of iron oxide nanoparticles via sonochemical method and their characterization

Preparation of iron oxide ( -Fe 2 O 3 ) nanoparticles was carried out via a sonochemical process. The process parameters such as temperature, sonication time and power of ultrasonication play important roles in the size and morphology of the final products. The iron oxide nanoparticles were characterized by transmission electron microscopy, X-ray powder diffraction, and thermogravimetric and differential thermal analyses. From transmission electron microscopy observations, the size of the iron oxide nanopar...     

Characterizing JONSWAP rogue waves and their statistics via inverse spectral data

Rogue waves in random sea states modeled by the JONSWAP power spectrum are high amplitude waves arising over non-uniform backgrounds that cannot be viewed as small amplitude modulations of Stokes waves. In the context of Nonlinear Schrödinger (NLS) models for waves in deep water, this poses the challenge of identifying appropriate analytical solutions for JONSWAP rogue waves, investigating possible mechanisms for their formation, and examining the validity of the NLS models in these more realistic settings. In this work we investigate JONSWAP rogue waves using the inverse spectral theory of the periodic NLS equation for moderate values of the period. For typical JONSWAP initial data, numerical experiments show that the developing sea state is well approximated by the first few dominant modes of the nonlinear spectrum and can be described in terms of a 2- or 3-phase periodic NLS solution. As for the case of uniform backgrounds, proximity to instabilities of the underlying 2-phase solution appears to be the main predictor of rogue wave occurrence, suggesting that the modulational instability of 2-phase solutions of the NLS is a main mechanism for rogue wave formation and that heteroclinic orbits of unstable 2-phase solutions are plausible models of JONSWAP rogue waves. To support this claim, we correlate the maximum wave strength as well as the higher statistical moments with elements of the nonlinear spectrum. The result is a diagnostic tool widely applicable to both model or field data for predicting the likelihood of rogue waves. Finally, we examine the validity of NLS models for JONSWAP data, and show that NLS solutions with JONSWAP initial data are described by non-Gaussian statistics, in agreement with the TOPEX field studies of sea surface height variability.     

Singular,hypersingular and singular free electromagnetic fields at wedge tips in metamaterials

The engineering response of metamaterials has a dramatic impact on the physics, optics and engineering communities, because they offer electromagnetic properties that are difficult or impossible to achieve with conventional materials. In this paper, an asymptotic analysis of the electromagnetic fields at multi-material wedges composed of metamaterials is proposed. This is made possible by removing the assumption of positive electric permittivities and magnetic permeabilities, an hypothesis which usually applies to conventional materials. Exploring the whole range of variability of these electromagnetic properties, it is shown that, in addition to the classical real eigenvalues 0 ? λ < 1 leading to power-law singularities of the type O(r^λ?1) as r → 0, it is also possible to find imaginary eigenvalues leading to hypersingular solutions, as well as nonsingular configurations for a suitable choice of the negative electric permittivities and magnetic permeabilities of the media. Moreover, to fully characterize the asymptotic fields, the analysis is not only restricted to the determination of the lowest real and complex eigenvalues, but is also extended to the evaluation of the higher-order nonsingular ones. The obtained analytical results collected in synthetic diagrams are expected to have impact on the design of micro- and nano-electro-mechanical systems.     

切割式多爆炸成形弹丸成形及对钢靶的穿甲效应

基于60mm 弧锥结合罩EFP装药,设计了一种在药型罩前适当位置安装可抛掷的十字形网栅的 切割式多爆炸成形弹丸战斗部结构,并进行了靶场静爆实验。由实验结果可知,该战斗部经网栅切割后能形 成5枚具有一定质量和方向性、可贯穿6mm 厚45钢靶的弹丸,有效提高了毁伤元的数量和毁伤面积。利用 LS-DYNA程序对弹丸侵彻45钢靶过程进行了数值模拟,分析了弹丸侵彻钢靶过程。通过对中心弹丸穿靶 模拟数据的处理,得到了其余尺寸弹丸侵彻不同厚度45钢靶的极限穿透速度计算公式,该公式可对相关切割 式多爆炸成形弹丸战斗部威力优化设计和评估提供依据。     

Fish and Plankton Interplay Determines Both Plankton Spatio-Temporal Pattern Formation and Fish School Walks: A Theoretical Study

The fascinating variety of spatio-temporal patterns in aquatic ecosystems and the understanding of the governing mechanisms of its generation and further dynamics requires ongoing experimental and theoretical studies. After introducing a certain hybrid mathematical model, this paper makes an attempt to demonstrate that the predation of a mobile planktivorous fish school on zooplankton can initiate both plankton pattern formation and fish school walks. Nonlinear interactions in the model of a fish-zooplankton-algae trophic chain prevent a simple intuitive understanding of the system dynamics. It is shown that the fish school predation and motion can give rise to plankton spiral waves. In the course of the spiral wave formation, the amplitudes of the spatially averaged plankton density oscillations are decreasing dramatically. Fish school walks are shown to resemble a fractional Brownian motions with a Hurst exponent depending on the fish predation rate.     

The Riemann Function,Singular Entropies,and the Structure of Oscillations in Systems of Two Conservation Laws

We use singular entropies, and the connection with the fundamental solution of the entropy equation and its adjoint operator, in order to derive a new formula describing the coupling of oscillations between the two characteristic fields in systems of two conservation laws.     

Elastic silicon-film-based nanoshells: Formation, properties, and applications

Controllable formation and properties of solid single-crystal micro-and nanoshells of various shapes (tubes and spirals, vertically positioned rings and cylinders, and bent and trough-shaped cantilevers) are briefly reviewed, and new results are given. The shells and complicated structures of prescribed size and shape are formed with the use of elastic energy of initial strained SiGe/Si films of nanometer thickness and methods of highly selective and directed detachment of the films from the silicon substrates. It is experimentally demonstrated that the diameters of the fabricated SiGe/Si nanotubes are several times smaller than the values predicted by the continuum elasticity theory. The properties of the shells made of semiconductor and hybrid (metal-semiconductor and metal-dielectric-semiconductor) films and their applications in micro-and nanoscale electrical engineering are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 114–128, November–December, 2006.     

Exactly solvable SIR models,their extensions and their application to sensitive pandemic forecasting

Nonlinear Dynamics - The classic SIR model of epidemic dynamics is solved completely by quadratures, including a time integral transform expanded in a series of incomplete gamma functions. The...     

Formation and interaction of intense vortex sheets in 3-dimensional,incompressible, hydrodynamics

Various questions related to the physics of inviscid flows are reviewed. The emergence of strong vortex sheets has repeatedly been observed in the simulation of the 3-dimensional equations, with a variety of initial conditions. In the case of axisymmetric Euler flows, the origin of these sheets can be easily understood with the help of an analogy with thermally driven flows. A more general mechanism to explain these sheets is proposed. Questions of singularities are briefly reviewed. Lastly, preliminary results on the connection between the sheets forming in inviscid flows, and the vortex tubes observed in high Reynolds number flows are presented.

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Sommario Sono considerate varie questioni correlate con la fisica dei flussi non-viscosi. La nascita di strati di forti vortici è stata ripetutamente osservata nella simulazione delle equazioni tridimensionali, per diverse condizioni iniziali. Nel caso di flussi di Eulero assialsimmetrici, l'origine di questi piani può essere facilmente compresa con l'aiuto di una analogia con i flussi guidati termicamente. Viene inoltre proposto un più generale meccanismo per giustificare questi strati e si passano in rassegna brevemente questioni riguardanti le singolarità. Infine, vengono presentati alcuni risultati preliminari sulla connessione tra i piani formantisi in flussi non viscosi ed i vortici tubolari osservati nei flussi ad alti numeri di Reynolds.

     

A note on divergence,rotation, gradient and their associated theorems

In this note, the essence and some supplements for the unified definition of divergence, rotation and gradient advanced by Tai have been presented based on the method of exterior differential form with an expression of vectors of tensors. The main purpose of this note is to introduce the useful expressions and their applications, and to simplify the proofs of many theorems in various field theories, and they are also important because of their utility for establishing a wide class of principles.