scholarly journals Early evolution of the planetary system around PSR B1257+12

2004 ◽  
Vol 202 ◽  
pp. 187-189
Author(s):  
Alexander Gusev ◽  
Irina Kitiashvili
Keyword(s):  
Bifurcation Theory ◽  
Planetary System ◽  
Early Time ◽  
Plane Motion ◽  
Phase Portraits ◽  
Dimensional Parameter ◽  
3 Dimensional ◽  
Resonance Rotation ◽  
Tidal Torques ◽  
Gravitational Capture

For the plane motion we are completely analyzing the differential equations systems of gravitational capture of the exoplanet at the resonance rotation with action of gravitational and tidal torques by qualitative analysis and bifurcation theory of dynamical systems (DS). The separation of 3-dimensional parameter space of dynamical system by bifurcation surfaces is obtained. The gallery of more than thirty phase portraits of gravitational capture extends the known scenario of cosmogonical evolution of the exoplanet on the early time, when the tidal interaction is very important.

Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields

2015 ◽  
Vol 25 (11) ◽  
pp. 1550150 ◽  
Author(s):  
Oxana Cerba Diaconescu ◽  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Parameter Space ◽  
Group Action ◽  
Sufficient Conditions ◽  
Phase Portraits ◽  
Topological Spaces ◽  
Canonical Forms ◽  
Affine Transformations ◽  
Bifurcation Diagrams ◽  
Dimensional Parameter ◽  
Quotient Spaces

In this article, we consider the class [Formula: see text] of all real quadratic differential systems [Formula: see text], [Formula: see text] with gcd (p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class [Formula: see text] so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.

scholarly journals Dynamic Behaviors and Mechanisms of Permanent Magnet Synchronous Motor with Excitation

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Shaohua Liu ◽  
Tingting Shao ◽  
Hai-dong Wu ◽  
Dong-hui Zhang ◽  
Qing-zhen Han
Keyword(s):  
Theoretical Analysis ◽  
Permanent Magnet ◽  
Bifurcation Theory ◽  
Permanent Magnet Synchronous Motor ◽  
External Perturbation ◽  
Synchronous Motor ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Dynamic Behaviors ◽  
Simulation Results

The aim of this study is to make a general exploration of the dynamic characteristics of the permanent magnet synchronous motor (PMSM) with parametric or external perturbation. The pitchfork, fold, and Hopf bifurcations are derived by using bifurcation theory. Simulation results not only confirm the theoretical analysis results but also show the Bogdanov–Takens bifurcation of the equilibrium. Dynamic behaviors, such as period three and chaotic motion of PMSM, are analyzed by using bifurcation diagram and phase portraits. The symmetric fold/fold bursting oscillation as well as two kinds of delayed pitchfork bursting oscillations is obtained, and different mechanisms are presented.

Codimension-Two Bifurcation Analysis on a Discrete Gierer–Meinhardt System

2020 ◽  
Vol 30 (16) ◽  
pp. 2050251
Author(s):  
Xijuan Liu ◽  
Yun Liu
Keyword(s):  
Bifurcation Theory ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Affine Transformations ◽  
Codimension Two ◽  
Dynamical Behaviors ◽  
Theoretical Predictions ◽  
The Stability ◽  
Two Parameter ◽  
Codimension Two Bifurcation

The stability and the two-parameter bifurcation of a two-dimensional discrete Gierer–Meinhardt system are investigated in this paper. The analysis is carried out both theoretically and numerically. It is found that the model can exhibit codimension-two bifurcations ([Formula: see text], [Formula: see text], and [Formula: see text] strong resonances) for certain critical values at the positive fixed point. The normal forms are obtained by using a series of affine transformations and bifurcation theory. Numerical simulations including bifurcation diagrams, phase portraits and basins of attraction are conducted to validate the theoretical predictions, which can also display some interesting and complex dynamical behaviors.

scholarly journals Explicit Wave Solutions and Qualitative Analysis of the (1 + 2)-Dimensional Nonlinear Schrödinger Equation with Dual-Power Law Nonlinearity

2015 ◽  
Vol 2015 ◽  
pp. 1-16
Author(s):  
Qing Meng ◽  
Bin He ◽  
Zhenyang Li
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Power Law ◽  
Bifurcation Theory ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Point Of View ◽  
Phase Portraits ◽  
Nonlinear Schrödinger ◽  
Dual Power

The (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity is studied using the factorization technique, bifurcation theory of dynamical system, and phase portraits analysis. From a dynamic point of view, the existence of smooth solitary wave, and kink and antikink waves is proved and all possible explicit parametric representations of these waves are presented.

scholarly journals Periodic Wave Solutions and Their Limit Forms of the Modified Novikov Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Dynamical System ◽  
Nonlinear Waves ◽  
Bifurcation Theory ◽  
Dynamic Properties ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Novikov Equation

The modified Novikov equationut-utxx+(b+1)u2ux=buuxuxx+u2uxxxis studied by using the bifurcation theory of dynamical system and the method of phase portraits analysis. The existences, dynamic properties, and limit forms of periodic wave solutions forbbeing a negative even are investigated. All possible exact parametric representations of the different kinds of nonlinear waves also are presented.

scholarly journals Reliability of 3-Dimensional Measures of Single-Leg Drop Landing Across 3 Institutions: Implications for Multicenter Research for Secondary ACL-Injury Prevention

2015 ◽  
Vol 24 (2) ◽  
pp. 198-209 ◽  
Author(s):  
Gregory D. Myer ◽  
Nathaniel A. Bates ◽  
Christopher A. DiCesare ◽  
Kim D. Barber Foss ◽  
Staci M. Thomas ◽  
...  
Keyword(s):  
Lower Extremity ◽  
Injury Prevention ◽  
Outcome Measures ◽  
Sagittal Plane ◽  
Frontal Plane ◽  
Plane Motion ◽  
3 Dimensional ◽  
Drop Landing ◽  
Prevention Studies ◽  
Acl Injury Prevention

Context:Due to the limitations of single-center studies in achieving appropriate sampling with relatively rare disorders, multicenter collaborations have been proposed to achieve desired sampling levels. However, documented reliability of biomechanical data is necessary for multicenter injury-prevention studies and is currently unavailable.Objective:To measure the reliability of 3-dimensional (3D) biomechanical waveforms from kinetic and kinematic variables during a single-leg landing (SLL) performed at 3 separate testing facilities.Design:Multicenter reliability study.Setting:3 laboratories.Patients:25 female junior varsity and varsity high school volleyball players who visited each facility over a 1-mo period.Intervention:Subjects were instrumented with 43 reflective markers to record 3D motion as they performed SLLs. During the SLL the athlete balanced on 1 leg, dropped down off of a 31-cm-high box, and landed on the same leg. Kinematic and kinetic data from both legs were processed from 2 trials across the 3 laboratories.Main Outcome Measures:Coefficients of multiple correlations (CMC) were used to statistically compare each joint angle and moment waveform for the first 500 ms of landing.Results:Average CMC for lower-extremity sagittal-plane motion was excellent between laboratories (hip .98, knee .95, ankle .99). Average CMC for lower-extremity frontal-plane motion was also excellent between laboratories (hip .98, knee .80, ankle .93). Kinetic waveforms were repeatable in each plane of rotation (3-center mean CMC ≥.71), while knee sagittal-plane moments were the most consistent measure across sites (3-center mean CMC ≥.94).Conclusions:CMC waveform comparisons were similar relative to the joint measured to previously published reports of between-sessions reliability of sagittal- and frontal-plane biomechanics performed at a single institution. Continued research is needed to further standardize technology and methods to help ensure that highly reliable results can be achieved with multicenter biomechanical screening models.

scholarly journals BIFURCATIONS OF THE GLOBAL STABLE SET OF A PLANAR ENDOMORPHISM NEAR A CUSP SINGULARITY

2008 ◽  
Vol 18 (08) ◽  
pp. 2207-2222 ◽  
Author(s):  
C. A. HOBBS ◽  
H. M. OSINGA
Keyword(s):  
Bifurcation Theory ◽  
Global Bifurcation ◽  
Phase Portraits ◽  
Stable Set ◽  
Cusp Point ◽  
Local Phase ◽  
Bifurcation Diagrams ◽  
Critical Locus ◽  
Codimension Two ◽  
Cusp Points

The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J0 where the Jacobian of the map is singular. The critical locus, denoted J1, is the image of J0. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set Ws of a fixed point or periodic orbit interacts with J1 near such a cusp point C1. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve Ws is tangent to J1 exactly at C1. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams.

scholarly journals Extra-solar planets: from direct rotation into reverse rotation

2004 ◽  
Vol 202 ◽  
pp. 205-207
Author(s):  
Irina Kitiashvili ◽  
Alexander Gusev
Keyword(s):  
Planetary Systems ◽  
Magnetic Interaction ◽  
Central Star ◽  
Phase Portraits ◽  
Momentum Vector ◽  
Resonance Rotation ◽  
Direct Rotation

We investigate the equation describing the evolution of the kinetic momentum vector for the case of non-resonance rotation of dynamically symmetrical planets by action of gravitational and magnetic interaction with the central star. The obtained gallery of more twenty phase portraits of kinetic momentum evolution illustrates the various regimes of the planetary systems evolution. The analyses of obtained portraits has shown that a direct rotation of the planet may be passed into reverse rotation and vice versa for a rather broad range of the parameters.

A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters

1988 ◽  
Vol 416 (1851) ◽  
pp. 361-389 ◽  
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Ordinary Differential Equations ◽  
Parameter Space ◽  
Bifurcation Theory ◽  
Parameter Plane ◽  
Phase Portraits ◽  
Relevant Parameter ◽  
Bifurcation Diagrams ◽  
Independent Parameters

In this paper, recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations (ODEs) containing at most three independent parameters. For such systems, by plotting in the relevant parameter plane the locus of successively degenerate singular points, a complete description of all the qualitatively distinct behaviour of the system can be obtained. The description is in terms of phase portraits and bifurcation diagrams. Even though much use is made of existing results obtained via local analyses, the results of this technique cover the entire parameter space. Furthermore, because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise. This can mean a major saving in effort, particularly for degenerate Hopf points. Finally if, as is often the case, the parameters appear in the system in a simple way, the procedure can be applied analytically because the variables (which will appear non-linearly) can be used to parametrize the relevant loci.

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