planar map
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2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Jérémie Bouttier ◽  
Ariane Carrance

A planar hypermap with a boundary is defined as a planar map with a boundary, endowed with a proper bicoloring of the inner faces. The boundary is said alternating if the colors of the incident inner faces alternate along its contour. In this paper we consider the problem of counting planar hypermaps with an alternating boundary, according to the perimeter and to the degree distribution of inner faces of each color. The problem is translated into a functional equation with a catalytic variable determining the corresponding generating function. In the case of constellations—hypermaps whose all inner faces of a given color have degree $m\geq 2$, and whose all other inner faces have a degree multiple of $m$—we completely solve the functional equation, and show that the generating function is algebraic and admits an explicit rational parametrization. We finally specialize to the case of Eulerian triangulations—hypermaps whose all inner faces have degree $3$—and compute asymptotics which are needed in another work by the second author, to prove the convergence of rescaled planar Eulerian triangulations to the Brownian map.





Author(s):  
GIORGIO GUBBIOTTI ◽  
NALINI JOSHI

Abstract We compactify and regularise the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system has certain unusual properties, including a sequence of points of indeterminacy in $\mathbb {P}^{1}\!\times \mathbb {P}^{1}$ . These indeterminacy points lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions.



2019 ◽  
Vol 1324 ◽  
pp. 012064
Author(s):  
Yu-Kun Huang
Keyword(s):  






2017 ◽  
Vol 27 (03) ◽  
pp. 1730012
Author(s):  
Kyohei Kamiyama ◽  
Motomasa Komuro ◽  
Kazuyuki Aihara

In this study, bifurcations of an invariant closed curve (ICC) generated from a homoclinic connection of a saddle fixed point are analyzed in a planar map. Such bifurcations are called homoclinic cycle (HCC) bifurcations of the saddle fixed point. We examine the HCC bifurcation structure and the properties of the generated ICC. A planar map that can accurately control the stable and unstable manifolds of the saddle fixed point is designed for this analysis and the results indicate that the HCC bifurcation depends upon a product of two eigenvalues of the saddle fixed point, and the generated ICC is a chaotic attractor with a positive Lyapunov exponent.



2013 ◽  
Vol 23 (06) ◽  
pp. 1330021
Author(s):  
CHRISTOFOROS SOMARAKIS ◽  
JOHN S. BARAS

The dynamics of the map [Formula: see text] are discussed for various values of its parameters. Despite the simple algebraic structure, this map, recently introduced in the literature, is very rich in nonlinear phenomena. Multiple strange attractors, transitions to chaos via period-doubling bifurcations, quasiperiodicity as well as intermittency, interior crisis, hyperchaos are only a few. In this work, strange attractors, bifurcation diagrams, periodic windows, invariant characteristics are investigated both analytically and numerically.



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