scholarly journals Fast phase randomization via two-folds

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150782 ◽  
Author(s):  
D. J. W. Simpson ◽  
M. R. Jeffrey
Keyword(s):  
Vector Field ◽  
Initial Conditions ◽  
Discontinuity Surface ◽  
Control Law ◽  
Smooth Vector ◽  
Small Noise ◽  
Phase Singularities ◽  
Piecewise Smooth Vector Field ◽  
Asymptotic Phase ◽  
Phase Randomization

A two-fold is a singular point on the discontinuity surface of a piecewise-smooth vector field, at which the vector field is tangent to the discontinuity surface on both sides. If an orbit passes through an invisible two-fold (also known as a Teixeira singularity) before settling to regular periodic motion, then the phase of that motion cannot be determined from initial conditions, and, in the presence of small noise, the asymptotic phase of a large number of sample solutions is highly random. In this paper, we show how the probability distribution of the asymptotic phase depends on the global nonlinear dynamics. We also show how the phase of a smooth oscillator can be randomized by applying a simple discontinuous control law that generates an invisible two-fold. We propose that such a control law can be used to desynchronize a collection of oscillators, and that this manner of phase randomization is fast compared with existing methods (which use fixed points as phase singularities), because there is no slowing of the dynamics near a two-fold.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

2014 ◽  
Vol 24 (07) ◽  
pp. 1450090 ◽  
Author(s):  
Tiago de Carvalho ◽  
Durval José Tonon
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Piecewise Smooth ◽  
Topological Equivalence ◽  
Smooth Vector ◽  
Codimension One ◽  
Piecewise Smooth Vector Fields ◽  
Piecewise Smooth Vector Field

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

scholarly journals Tails of exit times from unstable equilibria on the line

2020 ◽  
Vol 57 (2) ◽  
pp. 477-496
Author(s):  
Yuri Bakhtin ◽  
Zsolt Pajor-Gyulai
Keyword(s):  
Vector Field ◽  
Malliavin Calculus ◽  
Elementary Proof ◽  
Exit Time ◽  
Exit Times ◽  
Smooth Vector ◽  
Small Noise ◽  
One Dimensional ◽  
Smooth Vector Field ◽  
Exit Distribution

AbstractFor a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits. We also discuss our program on rare transitions in noisy heteroclinic networks.

scholarly journals Birth of Isolated Nested Cylinders and Limit Cycles in 3D Piecewise Smooth Vector Fields with Symmetry

2020 ◽  
Vol 30 (07) ◽  
pp. 2050098
Author(s):  
Tiago Carvalho ◽  
Bruno Rodrigues de Freitas
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Vector Fields ◽  
Piecewise Smooth ◽  
Initial Model ◽  
Smooth Vector ◽  
Piecewise Smooth Vector Fields ◽  
Piecewise Smooth Vector Field

Our start point is a 3D piecewise smooth vector field defined in two zones and presenting a shared fold curve for the two smooth vector fields considered. Moreover, these smooth vector fields are symmetric relative to the fold curve, giving rise to a continuum of nested topological cylinders such that each orthogonal section of these cylinders is filled by centers. First, we prove that the normal form considered represents a whole class of piecewise smooth vector fields. After we perturb the initial model in order to obtain exactly [Formula: see text] invariant planes containing centers, a second perturbation of the initial model is also considered in order to obtain exactly [Formula: see text] isolated cylinders filled by periodic orbits. Finally, joining the two previous bifurcations we are able to exhibit a model, preserving the symmetry relative to the fold curve, and having exactly [Formula: see text] limit cycles.

scholarly journals On Singular "Semifocus" Type Point Bifurcations of Piecewise Smooth Dynamical System

2018 ◽  
pp. 57-70
Author(s):  
V. Sh. Roitenberg
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Piecewise Smooth ◽  
Dimensional Manifold ◽  
Smooth Vector ◽  
Weak Focus ◽  
Smooth Vector Field ◽  
Small Parameters ◽  
Line Of Discontinuity ◽  
Piecewise Smooth Vector Field

For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

scholarly journals Birth of limit cycles from a 3D triangular center of a piecewise smooth vector field

2017 ◽  
Vol 82 (3) ◽  
pp. 561-578
Author(s):  
Tiago Carvalho ◽  
Rodrigo D. Euzébio ◽  
Marco Antonto Teixeira ◽  
Durval José Tonon
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Piecewise Smooth ◽  
Smooth Vector ◽  
Smooth Vector Field ◽  
Piecewise Smooth Vector Field

scholarly journals Flows of measures generated by vector fields

2018 ◽  
Vol 148 (4) ◽  
pp. 773-818
Author(s):  
Emanuele Paolini ◽  
Eugene Stepanov
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Borel Measure ◽  
Weak Sense ◽  
Positive Borel Measure ◽  
Smooth Structure ◽  
Smooth Vector ◽  
Integral Curves ◽  
The Given ◽  
Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

2018 ◽  
Vol 18 (3) ◽  
pp. 337-344 ◽  
Author(s):  
Ju Tan ◽  
Shaoqiang Deng
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Algebras ◽  
Critical Points ◽  
Vector Fields ◽  
Energy Functional ◽  
Smooth Vector ◽  
Solvable Lie Groups ◽  
Invariant Vector ◽  
Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

GENERAL RELATIVITY AND SECTIONAL CURVATURE

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1077-1087
Author(s):  
G. S. HALL
Keyword(s):  
General Relativity ◽  
Vector Field ◽  
Sectional Curvature ◽  
Relativity Theory ◽  
Curvature Function ◽  
Smooth Vector ◽  
General Relativity Theory ◽  
Metric Tensor ◽  
Algebraic Properties ◽  
Einstein’S General Relativity

A discussion is given of the sectional curvature function on a four-dimensional Lorentz manifold and, in particular, on the space–time of Einstein's general relativity theory. Its tight relationship to the metric tensor is demonstrated and some of its geometrical and algebraic properties evaluated. The concept of a sectional curvature preserving symmetry, in the form of a certain smooth vector field, is introduced and discussed.

Combined DAE and Sliding Mode Control Methods for Simulation of Constrained Mechanical Systems

2000 ◽  
Vol 122 (4) ◽  
pp. 691-698 ◽  
Author(s):  
M. D. Compere ◽  
R. G. Longoria
Keyword(s):  
Sliding Mode Control ◽  
Numerical Method ◽  
Hybrid Method ◽  
Sliding Mode ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Stabilization Method ◽  
Mode Control ◽  
Hybrid Numerical Method

In dynamic analysis of constrained multibody systems (MBS), the computer simulation problem essentially reduces to finding a numerical solution to higher-index differential-algebraic equations (DAE). This paper presents a hybrid method composed of multi-input multi-output (MIMO), nonlinear, variable-structure control (VSC) theory and post-stabilization from DAE solution theory for the computer solution of constrained MBS equations. The primary contributions of this paper are: (1) explicit transformation of constrained MBS DAE into a general nonlinear MIMO control problem in canonical form; (2) development of a hybrid numerical method that incorporates benefits of both Sliding Mode Control (SMC) and DAE stabilization methods for the solution of index-2 or index-3 MBS DAE; (3) development of an acceleration-level stabilization method that draws from SMC’s boundary layer dynamics and the DAE literature’s post-stabilization; and (4) presentation of the hybrid numerical method as one way to eliminate chattering commonly found in simulation of SMC systems. The hybrid method presented can be used to simulate constrained MBS systems with either holonomic, nonholonomic, or both types of constraints. In addition, the initial conditions (ICs) may either be consistent or inconsistent. In this paper, MIMO SMC is used to find the control law that will provide two guarantees. First, if the constraints are initially not satisfied (i.e., for inconsistent ICs) the constraints will be driven to satisfaction within finite time using SMC’s stabilization method, urobust,i=−ηisgnsi. Second, once the constraints have been satisfied, the control law, ueq and hybrid stabilization techniques guarantee surface attractiveness and satisfaction for all time. For inconsistent ICs, Hermite-Birkhoff interpolants accurately locate when each surface reaches zero, indicating the transition time from SMC’s stabilization method to those in the DAE literature. [S0022-0434(00)02404-7]

Active Wave Cancellation of the Axially Moving String

2001 ◽  
Author(s):  
Chin An Tan ◽  
Shenger Ying
Keyword(s):  
Boundary Conditions ◽  
Initial Conditions ◽  
General Boundary ◽  
Closed Form Expression ◽  
Control Law ◽  
General Boundary Conditions ◽  
Axially Moving String ◽  
Axially Moving ◽  
Active Wave ◽  
Control Forces

Abstract The active wave control of the linear, axially moving string with general boundary conditions is presented in this paper. Considerations of general boundary conditions are important from both practical and experimental viewpoints. The active control law is established by employing the idea of wave cancellation. An exact, closed-form expression for the transverse response of the controlled system, consisting of the flexible structure, the wave controller, and the sensing and actuation devices, is derived in the frequency domain. Two actuation forces, one upstream and one downstream of an excitation force, are applied. The proposed control law shows that all modes of the string are controlled and the vibration in the regions upstream and downstream of the control forces can be cancelled. However, these results are based on ideal conditions and the assumption of zero initial conditions at the non-fixed boundaries. Effects of non-zero boundary motions at the instant of application of the control forces are examined and the control is shown to be effective under these conditions. The stability and robustness of the control forces are improved by the introduction of a stabilization coefficient in the control law. The effectiveness, robustness and stability of the control forces are demonstrated by simulations and verified by experiments on axially moving belt drive and chain drive systems.

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