“Syncope Implies Chaos” in Walking-stick Maps

1977 ◽  
Vol 32 (6) ◽  
pp. 607-613 ◽  
Author(s):  
Otto E. Rössler
Keyword(s):  
Cross Sections ◽  
Monotone Sequence ◽  
Homoclinic Point ◽  
One Dimensional ◽  
Variable Chemical ◽  
The One ◽  
Diagnostic Criterion ◽  
Special Case ◽  
Walking Stick ◽  
Modest Degree

Abstract A number of 3-variable chemical and other systems capable of showing 'nonperiodic' oscillations are governed by walking-stick shaped maps as Poincare cross-sections in state space. The 2-dimensional simple walking-stick diffeomorphism contains the one-dimensional 'single-humped' Li-Yorke map (known to be chaos producing) as a 'degenerate' special case. To prove that chaos is possible also in strictly 2-dimensional walking-stick maps, it suffices to show that a homoclinic point (and hence an in­ finite number of periodic solutions) is possible in these maps. Such a point occurs in the second iterate at a certain (modest) 'degree of overlap' of the walking-stick map. At a slightly larger degree, a 'nonlinear horseshoe map' is formed in the second iterate. It implies presence of periodic trajectories of all even periodicities (at least) in the walking-stick map. At the same time, two major, formerly disconnected, chaotic subregimes merge into one. Diagnostic criterion: presence of 'syncopes' in an otherwise non-monotone sequence of amplitudes.

scholarly journals Impact of Energy Slope Averaging Methods on Numerical Solution of 1D Steady Gradually Varied Flow

2015 ◽  
Vol 62 (3-4) ◽  
pp. 101-119 ◽  
Author(s):  
Wojciech Artichowicz ◽  
Dzmitry Prybytak
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Cross Sections ◽  
Flow Model ◽  
One Dimensional ◽  
Averaging Methods ◽  
Integration Formulas ◽  
Different Types ◽  
The One

AbstractIn this paper, energy slope averaging in the one-dimensional steady gradually varied flow model is considered. For this purpose, different methods of averaging the energy slope between cross-sections are used. The most popular are arithmetic, geometric, harmonic and hydraulic means. However, from the formal viewpoint, the application of different averaging formulas results in different numerical integration formulas. This study examines the basic properties of numerical methods resulting from different types of averaging.

Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers

1999 ◽  
Vol 51 (5) ◽  
pp. 915-935 ◽  
Author(s):  
Zoltán M. Balogh ◽  
Christoph Leuenberger
Keyword(s):  
Unit Circle ◽  
Complex Flow ◽  
One Dimensional ◽  
Holomorphic Motions ◽  
Strictly Convex ◽  
Fixed Domain ◽  
Polynomial Hulls ◽  
Polynomial Hull ◽  
The One ◽  
Special Case

AbstractConsider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in C2 is studied in details.

Analysis of Thin-Walled Closed Beams With General Quadrilateral Cross Sections

1999 ◽  
Vol 66 (4) ◽  
pp. 904-912 ◽  
Author(s):  
J. H. Kim ◽  
Y. Y. Kim
Keyword(s):  
Finite Element ◽  
Cross Sections ◽  
Distortion Function ◽  
Element Formulation ◽  
Numerical Work ◽  
New Approach ◽  
One Dimensional ◽  
Static And Dynamic Analysis ◽  
Thin Walled ◽  
The One

This paper deals with the one-dimensional static and dynamic analysis of thin-walled closed beams with general quadrilateral cross sections. The coupled deformations of distortion as well as torsion and warping are investigated in this work. A new approach to determine the functions describing section deformations is proposed. In particular, the present distortion function satisfies all the necessary continuity conditions unlike Vlasov's distortion function. Based on these section deformation functions, a one-dimensional theory dealing with the coupled deformations is presented. The actual numerical work is carried out using two-node C0 finite element formulation. The present one-dimensional results for some static and free-vibration problems are compared with the existing and the plate finite element results.

Unified geometric and analytical treatment of magneto– gasdynamic shocks. Part 2. The shock polars

1973 ◽  
Vol 10 (3) ◽  
pp. 397-423 ◽  
Author(s):  
Lee A. Bertram
Keyword(s):  
Particle Velocity ◽  
Two Dimensional ◽  
Perfect Gas ◽  
Analytical Treatment ◽  
Classification Schemes ◽  
One Dimensional ◽  
The One ◽  
Shock Solution ◽  
Special Case ◽  
Alfven Velocity

Previously derived shock solutions for a perfectly conducting perfect gas are used to compute shock polars for the one-dimensional unsteady and two- dimensional non-aligned shock representations. A new special-case shock solution, having a downstream particle velocity relative to the shock equal to upstream Alfvén velocity, is obtained, in addition to exhaustive analytical classification schemes for the shock polars. Eight classes of one-dimensional polars and twelve classes of two-dimensional polars are identified.

scholarly journals Non uniform warping for beams

2008 ◽  
pp. 933-944
Author(s):  
Rached El Fatmi
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Beam Theory ◽  
Elastic Behavior ◽  
Cantilever Beams ◽  
Shear Forces ◽  
One Dimensional ◽  
The One ◽  
Displacement Model ◽  
Shear Bending

A non-uniform warping beam theory including the effects of torsion and shear forces is presented. Based on a displacement model using three warping parameters associated to the three St Venant warping functions corresponding to torsion and shear forces, this theory is free from the classical assumptions on the warpings or on the shears, and valid for any kind of homogeneous elastic and isotropic cross-section. This general theory is applied to analyze, for a representative set of cross-sections, the elastic behavior of cantilever beams subjected to torsion or shear-bending. Numerical results are given for the one-dimensional structural behavior and the three-dimensional stresses distributions; for the stresses in the critical region of the built-in section, comparisons with three-dimensional finite elements computations are presented. The study clearly shows when the effect of the restrained warping is localized or not.

Exact damped sinusoidal electric field of nonlinear one-dimensional Vlasov-Maxwell equations

1984 ◽  
Vol 32 (2) ◽  
pp. 197-205 ◽  
Author(s):  
B. Abraham-Shrauner
Keyword(s):  
Distribution Function ◽  
Electric Field ◽  
Maxwell Equations ◽  
Angular Frequency ◽  
Wave Number ◽  
One Dimensional ◽  
Damping Decrement ◽  
Sinusoidal Electric Field ◽  
The One ◽  
Special Case

An exact solution for a temporally damped sinusoidal electric field which obeys the nonlinear, one-dimensional Vlasov-Maxwell equations is given. The electric field is a generalization of the O'Neil model electric field for Landau damping of plasma oscillations. The electric field is a special case of the form found from the invariance of the one-dimensional Vlasov equation under infinitesimal Lie group transformations. The time dependences of the damping decrement, of the wave-number and of the angular frequency are derived. Use of a time-dependent BGK one-particle distribution function is justified for weak damping where, in general, it is necessary to carry out a numerical calculation of the invariant of which the distribution function is a functional.

scholarly journals PHASE-DRIVEN INTERACTION OF WIDELY SEPARATED NONLINEAR SCHRÖDINGER SOLITONS

2012 ◽  
Vol 09 (03) ◽  
pp. 511-543 ◽  
Author(s):  
JUSTIN HOLMER ◽  
QUANHUI LIN
Keyword(s):  
Scattering Theory ◽  
General Framework ◽  
Nls Equation ◽  
One Dimensional ◽  
Initial Separation ◽  
The One ◽  
External Forces ◽  
Special Case ◽  
Equal Amplitude ◽  
Inverse Scattering Theory

We show that, for the one-dimensional cubic NLS equation, widely separated equal amplitude in-phase solitons attract and opposite-phase solitons repel. Our result gives an exact description of the evolution of the two solitons valid until the solitons have moved a distance comparable to the logarithm of the initial separation. Our method does not use the inverse scattering theory and should be applicable to nonintegrable equations with local nonlinearities that support solitons with exponentially decaying tails. The result is presented as a special case of a general framework which also addresses, for example, the dynamics of single solitons subject to external forces.

scholarly journals Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases

1983 ◽  
Vol 24 (4) ◽  
pp. 392-416 ◽  
Author(s):  
J. G. Burnell ◽  
A. A. Lacey ◽  
G. C. Wake
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Nonlinear Elliptic ◽  
Special Cases ◽  
Exothermic Chemical Reaction ◽  
Dimensional Shape ◽  
The One ◽  
Special Case

AbstractIn an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.

scholarly journals A theory of spectral partitions of metric graphs

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
James B. Kennedy ◽  
Pavel Kurasov ◽  
Corentin Léna ◽  
Delio Mugnolo
Keyword(s):  
Spectral Gaps ◽  
Qualitative Properties ◽  
Metric Graphs ◽  
One Dimensional ◽  
Graph Partitions ◽  
Planar Domains ◽  
Open Questions ◽  
The One ◽  
Abstract Framework ◽  
Special Case

AbstractWe introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

scholarly journals DYADIC TRIANGULAR HILBERT TRANSFORM OF TWO GENERAL FUNCTIONS AND ONE NOT TOO GENERAL FUNCTION

2015 ◽  
Vol 3 ◽  
Author(s):  
VJEKOSLAV KOVAČ ◽  
CHRISTOPH THIELE ◽  
PAVEL ZORIN-KRANICH
Keyword(s):  
Singular Integral ◽  
Hilbert Transform ◽  
Integral Form ◽  
General Function ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Bilinear Hilbert Transform ◽  
The One ◽  
Special Case ◽  
Dyadic Model

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate $L^{p}$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

Sign in / Sign up

Export Citation Format

Share Document