Poincare Rotation Number Research Articles

Imperfect chimera states for coupled pendula.

The phenomenon of chimera states in the systems of coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interest. In such a state, different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Here, considering the coupled pendula, we find another pattern, the so-called imperfect chimera state, which is characterized by a certain number of oscillators which escape from the synchronized chimera's cluster or behave differently than most of uncorrelated pendula. The escaped elements oscillate with different average frequencies (Poincare rotation number). We show that imperfect chimera can be realized in simple experiments with mechanical oscillators, namely Huygens clock. The mathematical model of our experiment shows that the observed chimera states are controlled by elementary dynamical equations derived from Newton's laws that are ubiquitous in many physical and engineering systems.

On the simple isotopy class of a source-sink diffeomorphism on the 3-sphere

The results obtained in this paper are related to the Palis-Pugh problem on the existence of an arc with finitely or countably many bifurcations which joins two Morse-Smale systems on a closed smooth manifold M n . Newhouse and Peixoto showed that such an arc joining flows exists for any n and, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. For n = 1, this is related to the presence of the Poincare rotation number, and for n = 2, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimension n = 3, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse-Smale diffeomorphism on the 3-sphere without heteroclinic intersections to be joined by a simple arc with a “source-sink” diffeomorphism are also found.

Parallel chip-firing on the complete graph: Devil’s staircase and Poincaré rotation number

We study how parallel chip-firing on the complete graph K_n changes behavior as we vary the total number of chips. Surprisingly, the activity of the system, defined as the average number of firings per time step, does not increase smoothly in the number of chips; instead it remains constant over long intervals, punctuated by sudden jumps. In the large n limit we find a "devil's staircase" dependence of activity on the number of chips. The proof proceeds by reducing the chip-firing dynamics to iteration of a self-map of the circle S^1, in such a way that the activity of the chip-firing state equals the Poincare rotation number of the circle map. The stairs of the devil's staircase correspond to periodic chip-firing states of small period.

Wandering sets for a class of borel isomorphisms of [0,1)

A wandering set for a map ϕ is a set containing precisely one element from each orbit of ϕ. We study the existence of Borel wandering sets for piecewise linear isomorphisms. Such sets need not exist even when the parameters involved are rational, but they do exist if in addition all the slopes are powers of 2. For ϕ having at most one discontinuity, the existence of a Borel wandering set is equivalent to rationality of the Poincare rotation number. We compute the rotation numbers for a special class of such functions. The main result provides a concrete method of connecting certain pairs of wavelet sets.

On continuity of geodesic frameworks of flows on surfaces

For flows on an orientable closed surface of larger genus (that is, of genus ) a special geodesic distribution (the geodesic framework of the flow) is constructed that consists of geodesics with the same asymptotic directions as the trajectories of the flow and that is a complete topological invariant of the irrational flows on such surfaces. The problem of the dependence of the geodesic framework on a perturbation of the flow (or on the parameter of a family of flows) is considered. It is shown that an irreducible elementary irrational geodesic framework of a flow depends continuously on the perturbation of the flow (which is analogous to the continuous dependence of an irrational Poincare rotation number on a perturbation of a flow).

Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane

A survey of works concerning a kind of generalization of the Poincare rotation number for a rather general class of flows on closed surfaces is given. The idea (going back to A. Weil) is to replace this number by the asymptotic direction at infinity of the trajectory lifted to the universal covering plane. Other works on flows on surfaces are also mentioned, but this is done primarily for comparison in order to shed light on the subject.

Poincare's Rotation Number and Morse's Type Number

Poincare's Rotation Number and Morse's Type Number