Rotation and periodicity in plane separating continua

1991 ◽  
Vol 11 (4) ◽  
pp. 619-631 ◽  
Author(s):  
Marcy Barge ◽  
Richard M. Gillette
Keyword(s):  
Periodic Orbit ◽  
Convex Hull ◽  
Rotation Number ◽  
Closed Interval ◽  
Rotation Numbers ◽  
Indecomposable Continuum ◽  
Rotation Set

AbstractWe prove that ifFis an orientation-preserving homeomorphism of the plane that leaves invariant a continuum Λ which irreducibly separates the plane into exactly two domains, then the convex hull of the rotation set ofFrestricted to Λ is a closed interval and each reduced rational in this interval is the rotation number of a periodic orbit in Λ. We also show that the interior and exterior rotation numbers ofFassociated with Λ are contained in the convex hull of the rotation set ofFrestricted to Λ and that if this set is nondegenerate then Λ is an indecomposable continuum.

Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity

1991 ◽  
Vol 11 (1) ◽  
pp. 115-128 ◽  
Author(s):  
J. Llibre ◽  
R. S. Mackay
Keyword(s):  
Periodic Orbit ◽  
Convex Hull ◽  
Topological Entropy ◽  
Rational Point ◽  
Rotation Vector ◽  
Connected Subset ◽  
Positive Topological Entropy ◽  
Rotation Vectors ◽  
Compact Connected Subset ◽  
Rotation Set

AbstractWe show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ∈ ℤ2, q ∈ ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’.

Influence of Radial Rotation on Heat Transfer in a Rectangular Channel With Two Opposite Walls Roughened by Hemispherical Protrusions at High Rotation Numbers

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Shyy Woei Chang ◽  
Tong-Miin. Liou ◽  
Wei-Chun Chen
Keyword(s):  
Heat Transfer ◽  
Infrared Thermography ◽  
Rotation Number ◽  
Rectangular Channel ◽  
Flow Region ◽  
Transfer Data ◽  
Nusselt Numbers ◽  
Rotation Numbers ◽  
Heat Transfer Correlations ◽  
Rotating Channel

Detailed heat transfer distributions over two opposite leading and trailing walls roughened by hemispherical protrusions were measured from a rotating rectangular channel at rotation number up to 0.6 to examine the effects of Reynolds (Re), rotation (Ro), and buoyancy (Bu) numbers on local and area-averaged Nusselt numbers (Nu and Nu¯) using the infrared thermography. A set of selected heat transfer data illustrates the Coriolis and rotating buoyancy effects on the detailed Nu distributions and the area-averaged heat transfer performances of the rotating channel. The Nu¯ for the developed flow region on the leading and trailing walls are parametrically analyzed to devise the empirical heat transfer correlations that permit the evaluation of the interdependent and individual Re, Ro, and Bu effect on Nu¯.

scholarly journals AN ELEMENTARY PROOF OF A THEOREM BY MATSUMOTO

2016 ◽  
Vol 227 ◽  
pp. 77-85 ◽  
Author(s):  
LUIS HERNÁNDEZ-CORBATO
Keyword(s):  
Elementary Proof ◽  
Alternative Proof ◽  
Rotation Numbers ◽  
Rotation Set ◽  
Prime End

Matsumoto proved in, [Prime end rotation numbers of invariant separating continua of annular homeomorphisms, Proc. Amer. Math. Soc. 140(3) (2012), 839–845.] that the prime end rotation numbers associated to an invariant annular continuum are contained in its rotation set. An alternative proof of this fact using only simple planar topology is presented.

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

Knotted periodic orbits in suspensions of annulus maps

1987 ◽  
Vol 411 (1841) ◽  
pp. 351-378 ◽  
Keyword(s):  
Periodic Orbits ◽  
Rotation Number ◽  
Existence And Uniqueness ◽  
Torus Knots ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Rotation Numbers ◽  
Higher Genus

We consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3-manifold T 2 x I. Using results of Birman & Williams ( Topology 22, 47‒82 (1983); Contemp. Math . 20, 1‒60 (1983)), we construct a knotholder or template that carries the set of periodic orbits of the flow. We define rotation numbers and show that any orbit of period q and rotation number p / q can be arranged as a positive braid on p strands. This yields existence and uniqueness results for families of resonant torus knots ( p -braids that are ( p , q )-torus knots of period q > p which correspond to order-preserving (Birkhoff-) periodic orbits of the diffeomorphism. We show that all other q -periodic p -braids have higher genus, and we establish bounds on the genera of such knots. We obtain existence and uniqueness results for a number of other, non-resonant, torus knots, including non-order-preserving ( q + s , q )-torus knots of rotation number 1.

Direct numerical simulations of rotating turbulent channel flow

2008 ◽  
Vol 598 ◽  
pp. 177-199 ◽  
Author(s):  
OLOF GRUNDESTAM ◽  
STEFAN WALLIN ◽  
ARNE V. JOHANSSON
Keyword(s):  
Numerical Simulations ◽  
Channel Flow ◽  
Rotation Number ◽  
Turbulent Channel Flow ◽  
Shear Velocity ◽  
Direct Numerical Simulations ◽  
A Value ◽  
Rotation Numbers ◽  
Wall Shear ◽  
Two Sides

Fully developed rotating turbulent channel flow has been studied, through direct numerical simulations, for the complete range of rotation numbers for which the flow is turbulent. The present investigation suggests that complete flow laminarization occurs at a rotation number Ro = 2Ωδ/Ub ≤ 3.0, where Ω denotes the system rotation, Ub is the mean bulk velocity and δ is the half-width of the channel. Simulations were performed for ten different rotation numbers in the range 0.98 to 2.49 and complemented with earlier simulations (done in our group) for lower values of Ro. The friction Reynolds number Reτ = uτδ/ν (where uτ is the wall-shear velocity and ν is the kinematic viscosity) was chosen as 180 for these simulations. A striking feature of rotating channel flow is the division into a turbulent (unstable) and an almost laminarized (stable) side. The relatively distinct interface between these two regions was found to be maintained by a balance where negative turbulence production plays an important role. The maximum difference in wall-shear stress between the two sides was found to occur for a rotation number of about 0.5. The bulk flow was found to monotonically increase with increasing rotation number and reach a value (for Reτ = 180) at the laminar limit (Ro = 3.0) four times that of the non-rotating case.

Effect of Rib Size on Heat (Mass) Transfer Distribution in a Rotation Channel

1997 ◽  
Author(s):  
C. W. Park ◽  
R. T. Kukreja ◽  
S. C. Lau
Keyword(s):  
Mass Transfer ◽  
Rotation Number ◽  
Heat Mass Transfer ◽  
Reynolds Numbers ◽  
Local Heat ◽  
Test Channel ◽  
Turbine Blade Cooling ◽  
Blade Cooling ◽  
Rotation Numbers ◽  
Better Than

Experiments have been conducted to study the effect of rib size on the local heat (mass) transfer distribution for radial outward flow in a rotating channel with transverse ribs on the leading and trailing walls. The test channel modeled internal turbine blade cooling passages. Results were obtained for Reynolds numbers of 5,500 and 10,000, rotation numbers of 0.09 and 0.24, and for a fixed rib pitch that was equal to the channel hydraulic diameter. For a fixed rib configuration on the leading wall, increasing the size of the ribs on the trailing wall increased the heat (mass) transfer on the leading wall. Ribs with D/e = p/e = 16 on the trailing wall performed better than ribs with D/e = p/e = 10. When the rotation number was large, the heat (mass) transfer on the leading wall was quite low, regardless of the sizes of the ribs on the leading and trailing walls. There was very little spanwise variation of the local heat (mass) transfer between the transverse ribs on the trailing wall. When the rotation number was large, however, there was a significant spanwise variation of the local heat (mass) transfer between ribs on the leading wall.

OBSERVED ROTATION NUMBERS IN FAMILIES OF CIRCLE MAPS

2001 ◽  
Vol 11 (01) ◽  
pp. 73-89 ◽  
Author(s):  
MICHAEL A. SAUM ◽  
TODD R. YOUNG
Keyword(s):  
Lebesgue Measure ◽  
Rotation Number ◽  
Narrow Band ◽  
Initial Conditions ◽  
Positive Probability ◽  
Circle Maps ◽  
Numerical Evidence ◽  
Rotation Interval ◽  
Large Numbers ◽  
Rotation Numbers

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the effects of various types of saddle-node bifurcations on the observed rotation numbers.

scholarly journals Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

ROTATION VECTORS FOR TORAL MAPS AND FLOWS: A TUTORIAL

1995 ◽  
Vol 05 (02) ◽  
pp. 321-348 ◽  
Author(s):  
JAMES A. WALSH
Keyword(s):  
Dynamical Systems ◽  
Rotation Number ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Mode Locking ◽  
Rotation Vector ◽  
Higher Genus ◽  
Rotation Vectors ◽  
Dissipative Dynamical Systems ◽  
Rotation Set

This paper is an introduction to the concept of rotation vector defined for maps and flows on the m-torus. The rotation vector plays an important role in understanding mode locking and chaos in dissipative dynamical systems, and in understanding the transition from quasiperiodic motion on attracting invariant tori in phase space to chaotic behavior on strange attractors. Throughout this article the connection between the rotation vector and the dynamics of the map or flow is emphasized. We begin with a brief introduction to the dimension one setting, in which case the rotation vector reduces to the well known rotation number of H. Poincaré. A survey of the main results concerning the rotation number and bifurcations of circle maps is presented. The various definitions of rotation vector in the higher dimensional setting are then introduced with emphasis again placed on how certain properties of the rotation set relate to the dynamics of the map or flow. The dramatic differences between results in dimension two and results in higher dimensions are also presented. The tutorial concludes with a brief introduction to extensions of the concept of rotation vector to the setting of dynamical systems defined on surfaces of higher genus.

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