scholarly journals Examples of a complex hyperpolar action without singular orbit

Cubo (Temuco) ◽  
2010 ◽  
Vol 12 (2) ◽  
pp. 127-143 ◽  
Author(s):  
Naoyuki Koike
Keyword(s):  
Singular Orbit

Multiple-S-Shaped Critical Manifold and Jump Phenomena in Low Frequency Forced Vibration with Amplitude Modulation

2019 ◽  
Vol 29 (05) ◽  
pp. 1930012 ◽  
Author(s):  
Yue Yu ◽  
Qianqian Wang ◽  
Qinsheng Bi ◽  
C. W. Lim
Keyword(s):  
Amplitude Modulation ◽  
Large Amplitude ◽  
Low Frequency ◽  
Slow Variable ◽  
Critical Manifold ◽  
Singular Orbit ◽  
Modulation Control ◽  
Cycle Oscillation ◽  
Tuning Frequency ◽  
Complex Mechanical Systems

Motivated by the forced harmonic vibration of complex mechanical systems, we analyze the dynamics involving different waves in a double-well potential oscillator coupling amplitude modulation control of low frequency. The combination of amplitude modulation factor significantly enriches the dynamical behaviors on the formation of multiple-S-shaped manifold and multiple jumping phenomena that alternate between epochs of slow and fast motion. We can conduct bifurcation analysis to identify two harmonic vibrations. One is that the singular orbit makes multiple jumps to a fast trajectory segment from one attracting equilibrium to another as the expression of slow variable by using the DeMoivre formula. With the increase of tuning frequency, the system exhibits relaxation-type oscillations whose small amplitude oscillations are produced by nonlinear local cycles together with a distinct large amplitude cycle oscillation accounting for the Melnikov threshold values. The tuning frequency may not only affect the asymptotic expressions for the solution curves near fold singularities but also allow for the large amplitude orbit vibrations near fold-cycle singularities. Numerical analysis for computing critical manifolds and their intersections is used to detect the dynamical features in this paper.

scholarly journals SMOOTHNESS CONDITIONS IN COHOMOGENEITY ONE MANIFOLDS

2020 ◽  
Author(s):  
L. VERDIANI ◽  
W. ZILLER
Keyword(s):  
Efficient Method ◽  
Regular Part ◽  
Cohomogeneity One ◽  
Singular Orbit ◽  
Cohomogeneity One Manifolds

Abstract We present an efficient method for determining the conditions that a metric on a cohomogeneity one manifold, defined in terms of functions on the regular part, needs to satisfy in order to extend smoothly to the singular orbit.

Parameter rays in the space of exponential maps

2009 ◽  
Vol 29 (2) ◽  
pp. 515-544 ◽  
Author(s):  
MARKUS FÖRSTER ◽  
DIERK SCHLEICHER
Keyword(s):  
Parameter Space ◽  
Complete Classification ◽  
Detailed Structure ◽  
Exponential Parameter ◽  
Singular Orbit ◽  
Exponential Maps

AbstractWe investigate the setIof parametersκ∈ℂ for which the singular orbit (0,eκ,…) ofEκ(z):=exp (z+κ) converges to$\infty $. These parameters are organized in curves in parameter space calledparameter rays, together with endpoints of certain rays. Parameter rays are an important tool to understand the detailed structure of exponential parameter space. In this paper, we construct and investigate these parameter rays. Based on these results, a complete classification of the setIis given in the following paper [M. Förster, L. Rempe and D. Schleicher. Classification of escaping exponential maps.Proc. Amer. Math. Soc.136(2008), 651–663].

A non-singular inverse Vitali lemma with applications

2000 ◽  
Vol 20 (4) ◽  
pp. 1215-1229 ◽  
Author(s):  
GENEVIEVE MORTISS
Keyword(s):  
Ergodic Theorem ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Singular Orbit

A proof for a non-singular version of the inverse Vitali lemma is given. The result is used to describe non-singular orbit equivalence within the framework of Rudolph's restricted orbit equivalence and in the construction of an alternative proof of the Hurewicz ergodic theorem.

Some unexpectedly non-singular orbit spaces

1977 ◽  
Vol 82 (1) ◽  
pp. 35-37
Author(s):  
H. R. Morton
Keyword(s):  
Lie Group ◽  
Orbit Space ◽  
Compact Lie Group ◽  
Orbit Spaces ◽  
Singular Orbit ◽  
Smooth Action ◽  
The Matrix ◽  
Orbit Types ◽  
Image Position

The orbit space of a manifold under the smooth action of a compact Lie group is typically a ‘manifold with singularities’. The orbits of a given type each form a mani-fold, but these ‘strata’ generally do not give a manifold when pieced together. For example, the orbit space of S3 under the action of given by the matrix.is the suspension of P(2), which fails to be a manifold at the suspension points.In this paper I shall give examples where, in spite of having many orbit types, the orbit space is a manifold, with or without boundary.

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