Rotated Vector Fields Research Articles

Global Behavior of Limit Cycles in Rotated Vector Fields

We give a general definition for rotated vector fields and establish certain new theorems for the global behavior of limit cycles in the family of rotated vector fields which generalize, improve, or correct some known results.

Two-Dimensional Phase Unwrapping by Direct Elimination of Rotational Vector Fields from Phase Gradients Obtained by Heterodyne Techniques

Noting the analogy between electro-magnetic fields and the vector fields of phase gradients obtained by heterodyne techniques, we propose a new algorithm for least-squares phase unwrapping. Instead of solving a Poisson equation for the unwrapped phase map or using Green’s formulation, we directly remove singular points or residues from the measured phase map by superposing sign-reversed vortex fields which cancel out the rotational vector fields associated with the phase singular points.

COMPUTER-GENERATED PROOFS OF PHASE PORTRAITS FOR PLANAR SYSTEMS

This paper introduces new algorithms for automatically constructing proofs for the correctness of phase portraits obtained from numerical integration of structurally stable planar polynomial vector fields. These algorithms are based upon the concept of rotated vector fields and the use of interval arithmetic to obtain rigorous bounds on the accuracy of floating point arithmetic calculations. Phase portrait features such as the existence or non-existence of limit cycles in particular regions are proved while avoiding the awkwardness associated with error estimates for the accuracy of approximate trajectories obtained from numerical integration. In some cases this method extends local analysis to obtain a complete, rigorous description of the flow structure of a system.

On Quadratic Systems with a Degenerate Critical Point

We study phase portraits of quadratic systems with exactly two critical points, one of them degenerate. This problem has already been considered in [10], where part of the results are obtained by computer. Here we deal with these systems in terms of semi-complete families of rotated vector fields. This new approach allows us to prove most of the bifurcation diagrams that we obtain.

A kinematically exact space finite strain beam model — finite element formulation by generalized virtual work principle

The present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by Simo and Vu-Quoc [A geometrically-exact rod model incorporating shear and torsion-warping deformation, Int. J. Solids Structures 27 (3) (1991) 371–393], along the lines of the work on the planar beam theory presented by Saje [A variational principle for finite planar deformation of straight slender elastic beams, Internat. J. Solids and Structures 26 (1990) 887–900]. We apply exact non-linear kinematic relationships of the space finite-strain beam theory, assuming the Bernoulli hypothesis and neglecting the warping deformations of the cross-section. Finite displacements and rotations as well as finite extensional, shear, torsional and bending strains are accounted for in the formulation. A deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame, rigidly attached to the cross-section of the beam. The position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix, parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step. Also, the incremental rotational vector is introduced, which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam. Its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points. Because in 3-D space both the axial and the follower moments are non-conservative, not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization. Here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure, similar to that of Lagrangian multipliers. This makes possible the elimination of the displacement vector field from the principle, so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle. Other researchers, on the other hand, employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field. As a result, more accurate and efficient family of beam finite elements for the non-linear analysis of space frames has been obtained. A one-field formulation results in the fact that in the present finite elements the locking never occurs. Any combination of deformation states is described equally precisely. This is in contrast with the elements developed in literature, where, in order to avoid the locking, a reduced numerical integration has to be applied, which unfortunately, diminishes the accuracy of the solution. Polynomials have been chosen for the approximation of the components of the rotational vector. In this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value. The Newton method is used for the iterative solution of the non-linear equilibrium equations. In an non-equilibrium configuration, the tangent stiffness matrix, obtained by the linearization of governing equations using the directional derivative, is non-symmetric even for conservative loadings. Only upon achieving an equilibrium state, the tangent stiffness matrix becomes symmetric. Thus, obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the Newton method. For non-conservative loadings, however, the tangent stiffness matrix is always non-symmetric. The numerical examples demonstrate capability of the present formulation to determine accurately the non-linear behaviour of space frames. In numerical examples the out-of-plane buckling loads are determined and the whole pre-and post-critical load-displacement paths of a cantilever and a right-angle frame are traced. These, in the analysis of space beams standard verification example problems, show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself, the rotations of 2π, and extensional strains much beyond the realistic values of linear elastic material.

Rotated Vector Fields

Rotated Vector Fields

Global Families of Limit Cycles of Planar Analytic Systems

The global behavior of any one-parameter family of limit cycles of a planar analytic system $\dot x = f(x,\lambda )$ depending on a parameter $\lambda \in R$ is determined. It is shown that any one-parameter family of limit cycles belongs to a maximal one-parameter family which is either open or cyclic. If the family is open, then it terminates as the parameter or the orbits become unbounded, or it terminates at a critical point or on a (compound) separatrix cycle of the system. This implies that the periods in a one-parameter family of limit cycles can become unbounded only if the orbits become unbounded or if they approach a degenerate critical point or (compound) separatrix cycle of the system. This is a more specific result for planar analytic systems than Wintner’s principle of natural termination for $n$-dimensional systems where the periods can become unbounded in strange ways. This work generalizes Duffs results for one-parameter families of limit cycles generated by a one-parameter family of rotated vector fields. In particular, it is shown that the behavior at a nonsingular, multiple limit cycle of any one-parameter family of limit cycles is exactly the same as the behavior at a multiple limit cycle of a one-parameter family of limit cycles generated by a one-parameter family of rotated vector fields.

Global families of limit cycles of planar analytic systems

The global behavior of any one-parameter family of limit cycles of a planar analytic system x ˙ = f ( x , λ ) \dot x = f(x,\lambda ) depending on a parameter λ ∈ R \lambda \in R is determined. It is shown that any one-parameter family of limit cycles belongs to a maximal one-parameter family which is either open or cyclic. If the family is open, then it terminates as the parameter or the orbits become unbounded, or it terminates at a critical point or on a (compound) separatrix cycle of the system. This implies that the periods in a one-parameter family of limit cycles can become unbounded only if the orbits become unbounded or if they approach a degenerate critical point or (compound) separatrix cycle of the system. This is a more specific result for planar analytic systems than Wintner’s principle of natural termination for n n -dimensional systems where the periods can become unbounded in strange ways. This work generalizes Duffs results for one-parameter families of limit cycles generated by a one-parameter family of rotated vector fields. In particular, it is shown that the behavior at a nonsingular, multiple limit cycle of any one-parameter family of limit cycles is exactly the same as the behavior at a multiple limit cycle of a one-parameter family of limit cycles generated by a one-parameter family of rotated vector fields.

Analyzing orthographic projection of multiple 3D velocity vector fields in optical flow

This paper describes a method for analyzing multiple optical flow fields in optical flow and reconstructing 3D velocity vector fields. First, the discussion is concentrated on purely rotational motion. This method estimates a set of descriptive parameters of a purely rotational optical flow field such as projected rotational axis and projected angular velocity of a purely rotational optical flow field and extracts rotational optical flow fields specified by the parameters from input optical flow. This process based on the Hough transform utilizes constraints holding between parallel optical flow vectors to construct the parameter space. Taking correspondence between two rotational optical flow fields in two different views is profitable to decompose the descriptive parameters and reconstruct the 3D rotational velocity vector fields. The method proposed is applied to computer generated optical flow fields. Second, a discussion suggests that the method is applicable to analyzing general motion, from the fact that its optical flow field is essentially the same as that of purely rotational motion, though the test for the general motion is not presented here.

Some theorems on the existence, uniqueness, and nonexistence of limit cycles for quadratic systems

Given a quadratic system (QS) with a focus or a center at the origin we write it in the form ẋ = y + P 2(x, y) , ẏ = − x + dy + Q 2(x, y) where P 2 and Q 2 are homogeneous polynomials of degree 2. If we define F( x, y) = ( x − dy) P 2( x, y) + yQ 2( x, y) and g( x, y) = xQ 2( x, y) − yP 2( x, y) we give results of existence, nonexistence, and uniqueness of limit cycles if F( x, y) g( x, y) does not change of sign. Then, by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of QS, i.e., the QS with a unique finite singularity and the bounded QS with either one or two finite singularities.

Rotated vector fields and the global behavior of limit cycles for a class of quadratic systems in the plane

Rotated vector fields and the global behavior of limit cycles for a class of quadratic systems in the plane

Limit-Cycles and Rotated Vector Fields

The problem of limit-cycles in the plane was formulated by Poincar6 (7), who pointed out the existence and importance of these special solutions. The question at issue is to find a systematic method for determining the existence and location of the cycles of a given second-order system of ordinary differential equations. The importance of the problem lies in the fact that its solution would enable us to determine the limiting sets of non-periodic motions. This paper contains a contribution to the solution of the general problem, based on a method of complete families of rotated vector fields. In this method the vector field is varied so that the qualitative properties change in a controlled manner. The theory of complete families is first developed, and the limitcycle problem is then treated as an application. For this purpose the cycles of a given vector field are classified according to the critical points they enclose. In the simplest case, namely when the cycles in question enclose only one critical point, which is elementary, a general test is given by which the existence and location problem can be solved. More general cases can be solved if a Bendixson curve enclosing the critical points in question can be found. The results of this paper were submitted in the author's doctoral thesis, Princeton 1951. Acknowledgement is due the Faculty of Princeton University for the Proctor Fellowship held at that time. The author wishes also to thank Professors S. Lefschetz, M. Schiffer, and S. Diliberto for guidance and advice.