scholarly journals Operating with External Arguments of Douady and Hubbard

2007 ◽  
Vol 2007 ◽  
pp. 1-17 ◽  
Author(s):  
G. Pastor ◽  
M. Romera ◽  
G. Alvarez ◽  
J. Nunez ◽  
D. Arroyo ◽  
...  
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Typical Case ◽  
Mandelbrot Set ◽  
Nonlinear Phenomena ◽  
External Rays ◽  
General Method

The external arguments of the external rays theory of Douady and Hubbard is a valuable tool in order to analyze the Mandelbrot set, a typical case of discrete dynamical system used to study nonlinear phenomena. We suggest here a general method for the calculation of the external arguments of external rays landing at the hyperbolic components root points of the Mandelbrot set. Likewise, we present a general method for the calculation of the external arguments of external rays landing at Misiurewicz points.

Drawing and computing external rays in the multiple-spiral medallions of the Mandelbrot set

2008 ◽  
Vol 32 (5) ◽  
pp. 597-610 ◽  
Author(s):  
M. Romera ◽  
G. Alvarez ◽  
D. Arroyo ◽  
A.B. Orue ◽  
V. Fernandez ◽  
...  
Keyword(s):  
Mandelbrot Set ◽  
External Rays

Identification of gene interactions in fungal–plant symbiosis through discrete dynamical system modelling

2009 ◽  
Vol 3 (5) ◽  
pp. 414-428 ◽  
Author(s):  
J.G.C. Angeles ◽  
Z. Ouyang ◽  
A.M. Aguirre ◽  
P.J. Lammers ◽  
M. Song
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Gene Interactions ◽  
System Modelling

scholarly journals $(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables

10.37236/7188 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bolor Turmunkh
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Cluster Algebra ◽  
Quiver Varieties ◽  
Quantum Cluster ◽  
T System

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.

Geometry and Kinematics of Cylindrical Waterbomb Tessellation

2021 ◽  
Author(s):  
Rinki Imada ◽  
Tomohiro Tachi
Keyword(s):  
Dynamical System ◽  
Recurrence Relation ◽  
Discrete Dynamical System ◽  
Kinematic Model ◽  
Theoretical Reason ◽  
Finite Solution ◽  
Geometry And Kinematics ◽  
Folded Surfaces

Abstract Folded surfaces of origami tessellations have attracted much attention because they sometimes exhibit non-trivial behaviors. It is known that cylindrical folded surfaces of waterbomb tessellation called waterbomb tube can transform into wave-like surfaces, which is a unique phenomenon not observed on other tessellations. However, the theoretical reason why wave-like surfaces arise has been unclear. In this paper, we provide a kinematic model of waterbomb tube by parameterizing the geometry of a module of waterbomb tessellation and derive a recurrence relation between the modules. Through the visualization of the configurations of waterbomb tubes under the proposed kinematic model, we classify solutions into three classes: cylinder solution, wave-like solution, and finite solution. Furthermore, we give proof of the existence of a wave-like solution around one of the cylinder solutions by applying the knowledge of the discrete dynamical system to the recurrence relation.

scholarly journals The Limit Point of the Pentagram Map

2018 ◽  
Vol 2020 (9) ◽  
pp. 2818-2831 ◽  
Author(s):  
Max Glick
Keyword(s):  
Dynamical System ◽  
Limit Point ◽  
Convex Polygon ◽  
Discrete Dynamical System ◽  
Explicit Description ◽  
Cartesian Coordinates ◽  
The Subject ◽  
Pentagram Map

Abstract The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the 1st paper on the subject, Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials.

EFFECT OF STEP SIZE ON BIFURCATIONS AND CHAOS OF A MAP-BASED BVP OSCILLATOR

2010 ◽  
Vol 20 (06) ◽  
pp. 1789-1795 ◽  
Author(s):  
HONGJUN CAO ◽  
CAIXIA WANG ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Dynamical System ◽  
Neuron Model ◽  
Discrete Dynamical System ◽  
Period Doubling ◽  
Vital Role ◽  
Bifurcation Parameter ◽  
Decomposition Technique ◽  
Step Size ◽  
Discrete Step ◽  
Slow Decomposition

The continuous Bonhoeffer–van der Pol (BVP for short) oscillator is transformed into a map-based BVP model by using the forward Euler scheme. At first, the bifurcations and chaos of the map-based BVP model are investigated when the step size varies as a bifurcation parameter. By using the fast-slow decomposition technique, a two-parameter bifurcation diagram is obtained to give insight into the effect of the step size on bifurcations and chaos of the map-based BVP model. The investigation shows that the period-doubling bifurcation is dependent on the step size, while the saddle-node bifurcation is independent of the step size. Second, when the fast–slow decomposition technique cannot be used, we rigorously prove that in the map-based BVP model there exists chaos in the sense of Marotto when the discrete step size varies as a bifurcation parameter. These results show that the discrete step sizes play a vital role between the continuous-time dynamical system and the corresponding discrete dynamical system. Much attention should be paid on the step size when a map-based neuron model is used as an alternative to a continuous neuron model.

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