5,564 research outputs found
Coexistence of periods in a bisecting bifurcation
The inner structure of the attractor appearing when the
Varley-Gradwell-Hassell population model bifurcates from regular to chaotic
behaviour is studied. By algebraic and geometric arguments the coexistence of a
continuum of neutrally stable limit cycles with different periods in the
attractor is explained.Comment: 13 pages, 5 figure
On organizing principles of Discrete Differential Geometry. Geometry of spheres
Discrete differential geometry aims to develop discrete equivalents of the
geometric notions and methods of classical differential geometry. In this
survey we discuss the following two fundamental Discretization Principles: the
transformation group principle (smooth geometric objects and their
discretizations are invariant with respect to the same transformation group)
and the consistency principle (discretizations of smooth parametrized
geometries can be extended to multidimensional consistent nets). The main
concrete geometric problem discussed in this survey is a discretization of
curvature line parametrized surfaces in Lie geometry. We find a discretization
of curvature line parametrization which unifies the circular and conical nets
by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is
slightly changed and umbilic points are discusse
Balanced partitions of 3-colored geometric sets in the plane
Let SS be a finite set of geometric objects partitioned into classes or colors . A subset S'¿SS'¿S is said to be balanced if S'S' contains the same amount of elements of SS from each of the colors. We study several problems on partitioning 33-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m2m lines of each color, there is a segment intercepting mm lines of each color. (b) Given nn red points, nn blue points and nn green points on any closed Jordan curve ¿¿, we show that for every integer kk with 0=k=n0=k=n there is a pair of disjoint intervals on ¿¿ whose union contains exactly kk points of each color. (c) Given a set SS of nn red points, nn blue points and nn green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of SS.Peer ReviewedPostprint (published version
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