Curve In Parameter Space Research Articles

High-accuracy Approximation to the Integrated Length of Toroidal/Helical Orbits

The new innovative concept of High Power Microwave (HPM) Amplifier recently introduced, combines Multi-Beam Klystron (MBK) and Electron Storage-Ring (ESR) technologies, by using closed, multi-turn Toroidal/Helical electron-orbits. The selected toroidal/helical orbit-configuration was defined as a parametric space-curve in three dimensional space R. We get the parametric equations of that orbit. A high-accuracy approximation has now been obtained for the total length of the Toroidal/Helical orbit, that attains much faster numerical computation. Higher accuracy could be attained by using a higher-order expansion of the orbit-length rate-of-increase.

Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit

We analyze homoclinic orbits near codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions. The main motivation for this study is a self-organized periodic replication process of travelling pulses which has been observed in reaction–diffusion equations. We establish conditions for existence and uniqueness of countably infinite families of curve segments of 1-homoclinic orbits which accumulate at codimension-1 or -2 heteroclinic cycles. The main result shows the bifurcation of a number of curves of 1-homoclinic orbits from such codimension-2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points. In addition, a leading order expansion of the associated curves in parameter space is derived. Its coefficients are periodic with one frequency from the imaginary part of the leading stable Floquet exponents of the periodic orbit and one from the winding number.

Geometric Singular Perturbation Analysis of the Yamada Model

The Yamada model for lasers with saturable absorber is known to display interesting dynamics on widely different time scales. Methods from geometric singular perturbation theory are used to analyze the dynamics. On the slow scale solutions drift along a slow manifold which is attracting on one side of a nonhyperbolic line and repelling at the other side. On the fast scale solutions jump from the repelling part of the slow manifold back to the attracting part. This mechanism generates periodic and homoclinic orbits of relaxation type. A detailed analysis of the complicated behavior of the homoclinic orbits in the singular limit is given. The recently developed blow-up method for systems of singularly perturbed ordinary differential equations is used to analyze the dynamics near the nonhyperbolic line where an essential part of the dynamics takes place. In the blown-up system a singular homoclinic orbit interacting with a transcritical bifurcation of equilibria is identified. A return map along the singular homoclinic orbit is studied to prove that the homoclinic orbit persists along a smooth curve in parameter space which is shown to be exponentially close to the transcritical bifurcation curve.

Reconstruction of the chaotic set from classical cross section data

We study a paradigm of a chaotic scattering system with one open and one closed degree of freedom. We show how the development scenario of the chaotic invariant set is reflected in the changes of the rainbow singularities in the classical cross section. This allows us to reconstruct the actual development stage of the chaotic set from the knowledge of the cross section. We do this by following the rainbow structure of the cross section along a curve in parameter space which connects the case we are interested in with a known case, for example, the case of a complete horseshoe. We conclude with a discussion about the generality of our procedure and of its application to various systems.

Existence and Stability of Localized Oscillations in 1-Dimensional Lattices With Soft-Spring and Hard-Spring Potentials

In this paper, we use the method of homoclinic orbits to study the existence and stability of discrete breathers, i.e., spatially localized and time-periodic oscillations of a class of one-dimensional (1D) nonlinear lattices. The localization can be at one or several sites and the 1D lattices we investigate here have linear interaction between nearest neighbors and a quartic on-site potential Vu=12Ku2±14u4, where the (+) sign corresponds to “hard spring” and (−) to “soft spring” interactions. These localized oscillations—when they are stable under small perturbations—are very important for physical systems because they seriously affect the energy transport properties of the lattice. Discrete breathers have recently been created and observed in many experiments, as, e.g., in the Josephson junction arrays, optical waveguides, and low-dimensional surfaces. After showing how to construct them, we use Floquet theory to analyze their linear (local) stability, along certain curves in parameter space (α,ω), where α is the coupling constant and ω the frequency of the breather. We then apply the Smaller Alignment Index method (SALI) to investigate more globally their stability properties in phase space. Comparing our results for the ± cases of Vu, we find that the regions of existence and stability of breathers of the “hard spring” lattice are considerably larger than those of the “soft spring” system. This is mainly due to the fact that the conditions for resonances between breathers and linear modes are much less restrictive in the former than the latter case. Furthermore, the bifurcation properties are quite different in the two cases: For example, the phenomenon of complex instability, observed only for the “soft spring” system, destabilizes breathers without giving rise to new ones, while the system with “hard springs” exhibits curves in parameter space along which the number of monodromy matrix eigenvalues on the unit circle is constant and hence breather solutions preserve their stability character.

Dislocation Dynamics in Semiconductor Thin Film-Substrate Systems

ABSTRACTA discrete dislocation dynamics model is developed to establish the equations of motion for three-dimensional interacting dislocation loops in the semiconductor thin film – substrate system. The film is assumed to be an elastic layer and is perfectly bonded with another elastic substrate. Dislocation loops are discretized into segments, each of which is represented by a parametric space curve of specific shape functions and associated degree of freedom. The dislocation stress field is calculated as an essential ingredient in the dislocation dynamics method. Dislocation dynamics and interaction with film surface/interface are simulated.

Accuracy and convergence of parametric dislocation dynamics

In the parametric dislocation dynamics (PDD), closed dislocation loops are described as an assembly of segments, each represented by a parametric space curve. Their equations of motion are derived from an energy variational principle, thus allowing large-scale computer simulations of plastic deformation. We investigate here the limits of temporal and spatial resolution of strong dislocation interactions. The method is demonstrated to be highly accurate, with unconditional spatial convergence that is limited to distances of the order of interatomic dimensions. It is shown that stability of dislocation line shape evolution requires very short time steps for explicit integration schemes, or can be unconditionally stable for implicit time integration schemes. Limitations of the method in resolving strong dislocation interactions are established for the following mechanisms: dislocation generation, annihilation, dipole and junction formation, pileup evolution.

Optimal pose trajectory planning for robot manipulators

Planning of robot pose trajectory is a very complex task that plays a crucial role in design and application of robots in task space. This paper presents a unified approach to optimal pose trajectory planning for robot manipulators in Cartesian space through a genetic algorithm (GA) enhanced optimization of the pose ruled surface. The robot pose ruled surface is formed as a motion locus of its configuration vector. The unified treatment of the end-effector positions and orientations is based on the robot pose ruled surface concept and used in trajectory interpolations. The planning of robot pose trajectories is accomplished through the way of generating and optimizing robot pose ruled surfaces under the constraints of kinematics, dynamics and control performances. The optimization model is established based on functional analysis and dynamics planning, and instantiated by using high-order parametric space curves as position and orientation trajectories. A GA enhanced optimization method is used to choose the parameters (coefficients) of space curves such that the pose ruled surface and its area and area change ratios are optimized for achieving good kinematics and dynamics performances. With robot's dynamics constraints taken into account, the optimized trajectories are realizable and robust for control. Two cases are studied to verify the feasibility of the proposed approach.

Bifurcations of invariant curves of a difference equation

Bifurcation of the invariant curves of a difference equation is studied. The system defined by the difference equation is integrable, so the study of the invariant curves of the difference system can become the study of topological classification of the planar phase portraits defined by a planar Hamiltonian system. By strict qualitative analysis, the classification of the invariant curves in parameter space can be obtained.

An existence theory for three-dimensional periodic travelling gravity–capillary water waves with bounded transverse profiles

This article presents a rigorous existence theory for three-dimensional gravity–capillary water waves which are uniformly translating and periodic in one horizontal spatial direction x and have a nontrivial transverse profile in the other z. The hydrodynamic equations are formulated as an infinite-dimensional Hamiltonian system in which z is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system of ordinary differential equations. A family of straight lines C 1, C 2,… in an appropriate two-dimensional parameter space is identified at which the number of purely imaginary eigenvalues of the linear problem changes: at each point on one of these lines two real eigenvalues become purely imaginary by passing through zero. There are also codimension-two points: the line C k intersects each of the lines C k+1 , C k+2 ,… in precisely one point. General statements concerning the existence of waves which are periodic or quasiperiodic in z are made by applying standard tools in Hamiltonian-systems theory to the reduced equations. Moreover, a critical curve in parameter space is found at which a two-dimensional Stokes wave and a three-dimensional wave with a spatially localised and exponentially decaying transverse profile simultaneously bifurcate from the uniform flow. This curve is piecewise linear: it contains one line segment from each of C 1, C 2,… .

The screensaver map: Dynamics on elliptic curves arising from polygonal folding

We study the repeated folding of a two-parameter family of quadrilaterals about their successively transformed diagonals by examining the evolution of the diagonal lengths. Successively mapped pairs of squared lengths lie on an elliptic curve on which folding acts as translation under the group law. We prove the rotation number attains all possible values and any value determines a unique curve in parameter space. For rational parameters we give an algorithm to determine if the folding map is periodic. This gives a partial explanation for the diversity and intricacy of the curves traced out by the paths of the vertices of the transformed quadrilaterals.

Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation

A thermodynamics-based variational method is developed to establish the equations of motion for three-dimensional (3D) interacting dislocation loops. The approach is appropriate for investigations of plastic deformation at the mesoscopic scale by direct numerical simulations. A fast sum technique for determination of elastic field variables of dislocation ensembles is utilized to calculate forces acting on generalized coordinates of arbitrarily curved loop segments. Each dislocation segment is represented by a parametric space curve of specified shape functions and associated degrees of freedom. Kinetic equations for the time evolution of generalized coordinates are derived for general 3D climb/glide motion of curved dislocation loops. It is shown that the evolution equations for the position $(\mathbf{P}),$ tangent $(\mathbf{T}),$ and normal $(\mathbf{N})$ vectors at segment nodes are sufficient to describe general 3D dislocation motion. When crystal structure constraints are invoked, only two degrees of freedom per node are adequate for constrained glide motion. A selected number of applications are given for: (1) adaptive node generation on interacting segments, (2) variable time-step determination for integration of the equations of motion, (3) dislocation generation by the Frank-Read mechanism in fcc, bcc, and dc crystals, (4) loop-loop deformation and interaction, and (5) formation of dislocation junctions.

Decomposing trimmed surfaces using the Voronoï diagram and a scan line algorithm

Many applications deal with the rendering of trimmed surfaces and the generation of grids for trimmed surfaces. Usually, a structured or unstructured grid must be constructed in the parameter space of the trimmed surface. Trimmed surfaces not only cause problems in the context of grid generation but also when exchanging data between different CAD systems. This paper describes a new approach for decomposing the valid part of the parameter space of a trimmed surface into a set of four-sided surfaces. The boundaries of these four-sided surfaces are line segments, segments of the trimming curves themselves, and segments of bisecting curves that are defined by a generalized Voronoï diagram implied by the trimming curves in parameter space. We use a triangular background mesh for the computation of the bisecting curves of the generalized Voronoï diagram.

Approximations to solitary waves on lattices, III: the monatomic lattice with second-neighbour interactions

We find new behaviour in a lattice with second-neighbour interactions. First we find solitary waves with oscillatory spatial decay, this is something that cannot occur in lattices with nearest-neighbour interactions alone. This is found by standard asymptotic analysis, which leads to an exact curve in parameter space where this behaviour starts. A number of highly accurate quasi-continuum approximations are derived and solved. One of these suggests a possible method for subsonic solitary waves to cease existing. An alternative method of approximation elegantly reveals the differences in shape between subsonic and supersonic solitary waves. This is based on the weak form of the differential - delay equation for the travelling wave, and is derived using the calculus of variations.

A tessellation algorithm for the representation of trimmed nurbs surfaces with arbitrary trimming curves

In most cad systems, a trimmed parametric surface is defined by two things: the control information of the surface itself and the control information of its trimming curves, which are usually defined as parametric curves in the parameter space of the surface. Often, trimmed parametric surfaces cause problems in the context of data exchange between cad systems, surface evaluation/rendering, and grid/mesh generation. This paper describes a new approach for representing a trimmed surface. The approach is based on the idea of decomposing the valid part of the parameter space of a trimmed surface by a set of planar, ruled surfaces, which are defined in parameter space and whose union defines the entire valid part. A generalized Voronoï diagram is constructed for the set of trimming curves in parameter space. This diagram defines tiles around each trimming curve, which are further tessellated by utilizing a scan line technique for identifying and filling the interior of polygons. Here, the scan line technique is used to extract non-horizontal segments of the given trimming curves. Pairs of these segments are then linearly interpolated in horizontal parameter direction, thus defining a set of planar, ruled surfaces. The set of all these planar, ruled surfaces defines the entire valid part of a trimmed surface exactly.

Doubling bifurcations of destroyed [formula omitted]2 tori

Two Duffing oscillators driven with incommensurate frequencies and coupled additively are investigated. With this system we observe a host of complex behavior which compete with each other in parameter space, in particular the period doubling of an attracting T 2 torus and the destruction of a T 2 torus. As the parameters are varied along a curve in parameter space, we observe period doubling bifurcations of a destroyed T 2 torus. This phenomenon is studied in greater detail via a simple recursive map.

A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field

Axisymmetric steady flows driven by an electric field about a deformable fluid drop suspended in an immiscible fluid are studied within the framework of the leaky dielectric model. Deformations of the drop and the flow fields are determined by solving the nonlinear free-boundary problem composed of the Navier-Stokes system governing the flow field and Laplace's system governing the electric field. The solutions are obtained by using the Galerkin finite-element method with an elliptic mesh generation scheme. Under conditions of creeping flow and vanishingly small drop deformations, the results of finite-element computations recover the asymptotic results. When drop deformations become noticeable, the asymptotic results are often found to underestimate both the flow intensity and drop deformation. By tracking solution branches in parameter space with an arc-length continuation method, curves in parameter space of the drop deformation parameter D versus the square of the dimensionless field strength E usually exhibit a turning point when E reaches a critical value Ec. Along such a family of drop shapes, steady solutions do not exist for E > Ec. The nonlinear relationship revealed computationally between D and E2 appears to be capable of providing insight into discrepancies reported in the literature between experiments and predictions based on the asymptotic theory. In some special cases with fluid conductivities closely matched, however, drop deformations are found to grow with E2 indefinitely and no critical value Ec is encountered by the corresponding solution branches. For most cases with realistic values of physical properties, the overall electrohydrodynamic behaviour is relatively insensitive to effects of finite-Reynolds-number flow. However, under extreme conditions when fluids of very low viscosities are involved, computational results illustrate a remarkable shape turnaround phenomenon: a drop with oblate deformation at low field strength can evolve into a prolate-like drop shape as the field strength is increased.

Curves of Constant Precession

1. INTRODUCTION. Given initial position and direction, the flight-path of a ship in Euclidean space is completely determined by how much it turns and how much it twists at each odometer reading. This is an intuitive interpretation of the Fundamental Theorem for Space Curves, which states that curvature K and torsion , as functions of arclength s, determine a space curve uniquely up to rigid motion. This statement of the Fundamental Theorem ([14], §1-8) should be tempered with the reservations expressed by Nomizu [12] and Wong & Lai [15]. Given a parametric space curve, there are well-known formulae for the arclength, curvature, and torsion (as functions of the parameter). Given two functions of one parameter (potentially curvature and torsion parametrized by arc-length) one might like to find a parametrized space curve for which the two functions are the curvature and torsion. This activity, called natural ([14], §1-10), is generally achieved by solving Riccati equations like dw/ds = -iz/2 iKW + i7W /2. Although the solution generally exists, it usually cannot be obtained explicitly. Euler [6] found explicit integral formulae for plane curves (where z - O) through direct geometric analysis. Hoppe [9] developed a general method for solving the natural equations for space curves by solving Riccati equations through a complicated sequence of integral transformations. He digressed to obtain formulae for the tangent, normal, and binormal indicatrices for general helices and essentially for curves of constant precession. Enneper [5] obtained explicit closed-form solutions for helices on revolved conic sections through direct geometric analysis. A curve of constant precession is defined by the property that as the curve is traversed with unit speed, its centrode revolves about a fixed axis with constant angle and constant speed. In this paper we obtain an arclength-parametrized closed-form solution of the natural equations for curves of constant precession through direct geometric analysis. As part of this analysis, we obtain a new theorem for curves of constant precession analogous with Lancret's Theorem for general helices. We provide the first rendering of a curve of constant precession. We also note for the first time that curves of constant precession lie on circular hyperboloids of one sheet and have closure conditions that are simply related to their arclength, curvature, and torsion. These are 3-type curves, except one family of closed 2-type curves (when Z = 4,u; see [2], [3], and [1]). Given a closed C3 curve in space, it is rather obvious that the curvature and torsion functions will be periodic functions of the arclength, with period equal the total arclength. This is a necessary condition but, as the circular helices (K and z both constant) show, not a sufficient condition that integral curves be closed. Efimov [4] and Fenchel [7] independently formulated The Closed Curve Problem. Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions K(S) and z(s) with the same period L, the integral curve is closed.

Geometric phases and anholonomy for a class of chaotic classical systems.

Berry's phase may be viewed as arising from the parallel transport of a quantal state around a loop in parameter space. In this Letter, the classical limit of this transport is obtained for a particular class of chaotic systems. It is shown that this ``classical parallel transport'' is anholonomic --- transport around a closed curve in parameter space does not bring a point in phase space back to itself --- and is intimately related to the Robbins-Berry classical two-form.

Bifurcation analysis of two coupled periodically driven Duffing oscillators.

Bifurcation diagrams and phase diagrams of two coupled periodically driven identical Duffing oscillators are presented. It is shown that the global pattern of bifurcation curves in parameter space consists of repeated subpatterns similar to the superstructure observed for single, periodically driven, strictly dissipative oscillators The subpattern itself, however, is different from that of a single Duffing oscillator due, in particular, to Hopf bifurcations that are newly added to the bifurcation scenario.