Alder's topological entropy of generalized discrete dynamical system

2012 ◽  
Author(s):  
Xiaoqian Song ◽  
Liangwei Wang
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Discrete Dynamical System

ON TOPOLOGICAL DYNAMICS OF SEQUENCES OF CONTINUOUS MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1437-1438 ◽  
Author(s):  
SERGIĬ KOLYADA ◽  
LUBOMÍR SNOHA
Keyword(s):  
Dynamical System ◽  
Topological Space ◽  
Topological Entropy ◽  
Discrete Dynamical System ◽  
Topological Dynamics ◽  
Limit Sets ◽  
Compact Topological Space ◽  
Continuous Maps ◽  
Uniformly Convergent

We define and study ω-limit sets and topological entropy for a nonautonomous discrete dynamical system given by a sequence [Formula: see text] of continuous selfmaps of a compact topological space. A special attention is paid to the case when the space is metric and the sequence [Formula: see text] either forms an equicontinuous family of maps or is uniformly convergent. We also show that for any continuous maps f and g from a compact topological space into itself the topological entropies h(f ◦ g) and h(g ◦ f) are equal.

scholarly journals Kinematics in Biology: Symbolic Dynamics Approach

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 339
Author(s):  
Carlos Correia Ramos
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Empirical Data ◽  
Symbolic Dynamics ◽  
Discrete Dynamical System ◽  
Building Blocks ◽  
Geometrical Method ◽  
Two Parameters ◽  
Complex Trajectories

Motion in biology is studied through a descriptive geometrical method. We consider a deterministic discrete dynamical system used to simulate and classify a variety of types of movements which can be seen as templates and building blocks of more complex trajectories. The dynamical system is determined by the iteration of a bimodal interval map dependent on two parameters, up to scaling, generalizing a previous work. The characterization of the trajectories uses the classifying tools from symbolic dynamics—kneading sequences, topological entropy and growth number. We consider also the isentropic trajectories, trajectories with constant topological entropy, which are related with the possible existence of a constant drift. We introduce the concepts of pure and mixed bimodal trajectories which give much more flexibility to the model, maintaining it simple. We discuss several procedures that may allow the use of the model to characterize empirical data.

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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