Stability and bifurcations in projective holomorphic dynamics

2018 ◽  
Vol 115 ◽  
pp. 37-71
Author(s):  
François Berteloot ◽  
Fabrizio Bianchi
Keyword(s):  
Holomorphic Dynamics ◽  
Stability And Bifurcations

Natural convection in a rectangular cavity with piece-wise heated vertical walls: multiple states, stability and bifurcations

2002 ◽  
Author(s):  
Vladimir Erenburg ◽  
Alexander Gelfgat ◽  
Eliezer Kit ◽  
Pinhas Z. Bar-Yoseph ◽  
Alexander Solan
Keyword(s):  
Natural Convection ◽  
Rectangular Cavity ◽  
Vertical Walls ◽  
Stability And Bifurcations

Frontiers in Complex Dynamics

2014 ◽  
Author(s):  
Araceli Bonifant ◽  
Misha Lyubich ◽  
Scott Sutherland
Keyword(s):  
Historical Perspective ◽  
Complex Dynamics ◽  
Short Paper ◽  
Combinatorial Group Theory ◽  
Entropy Theory ◽  
Holomorphic Dynamics ◽  
Eightieth Birthday ◽  
Several Variables ◽  
Fields Medal ◽  
Combinatorial Group

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.

scholarly journals Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 237
Author(s):  
Rostislav Grigorchuk ◽  
Supun Samarakoon
Keyword(s):  
Operator Algebras ◽  
Probabilistic Approach ◽  
Schur Complement ◽  
Automata Theory ◽  
Random Schrödinger Operators ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Von Neumann ◽  
Intermediate Growth ◽  
Quasi Crystals

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

scholarly journals Global Stability Analysis of a Bioreactor Model for Phenol and Cresol Mixture Degradation

Processes ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 124
Author(s):  
Neli Dimitrova ◽  
Plamena Zlateva
Keyword(s):  
Specific Growth ◽  
Equilibrium Points ◽  
Nonlinear Ordinary Differential Equations ◽  
The Novel ◽  
Model Design ◽  
Stirred Bioreactor ◽  
Global Stability Analysis ◽  
Local Asymptotic Stability ◽  
Model Dynamics ◽  
Stability And Bifurcations

We propose a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The model is described by three nonlinear ordinary differential equations. The novel idea in the model design is the biomass specific growth rate, known as sum kinetics with interaction parameters (SKIP) and involving inhibition effects. We determine the equilibrium points of the model and study their local asymptotic stability and bifurcations with respect to a practically important parameter. Existence and uniqueness of positive solutions are proved. Global stabilizability of the model dynamics towards equilibrium points is established. The dynamic behavior of the solutions is demonstrated on some numerical examples.

scholarly journals Effects of Maximal Sodium and Potassium Conductance on the Stability of Hodgkin-Huxley Model

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yue Zhang ◽  
Kuanquan Wang ◽  
Yongfeng Yuan ◽  
Dong Sui ◽  
Henggui Zhang
Keyword(s):  
Experimental Data ◽  
Mathematical Method ◽  
Bifurcation Point ◽  
Potassium Conductance ◽  
Maximal Conductance ◽  
Computing Model ◽  
The World ◽  
Stability And Bifurcations ◽  
The Stability ◽  
Sodium And Potassium

Hodgkin-Huxley (HH) equation is the first cell computing model in the world and pioneered the use of model to study electrophysiological problems. The model consists of four differential equations which are based on the experimental data of ion channels. Maximal conductance is an important characteristic of different channels. In this study, mathematical method is used to investigate the importance of maximal sodium conductanceg-Naand maximal potassium conductanceg-K. Applying stability theory, and takingg-Naandg-Kas variables, we analyze the stability and bifurcations of the model. Bifurcations are found when the variables change, and bifurcation points and boundary are also calculated. There is only one bifurcation point wheng-Nais the variable, while there are two points wheng-Kis variable. The (g-Na,  g-K) plane is partitioned into two regions and the upper bifurcation boundary is similar to a line when bothg-Naandg-Kare variables. Numerical simulations illustrate the validity of the analysis. The results obtained could be helpful in studying relevant diseases caused by maximal conductance anomaly.

Stability and bifurcations of even periodic orbits in the Sitnikov problem

2018 ◽  
Vol 130 (12) ◽  
Author(s):  
Jorge Galán-Vioque ◽  
Daniel Nuñez ◽  
Andrés Rivera ◽  
Camila Riccio
Keyword(s):  
Periodic Orbits ◽  
Sitnikov Problem ◽  
Stability And Bifurcations

scholarly journals A Comparison of Classical Holomorphic Dynamics and Holomorphic Semigroup Dynamics-II

2020 ◽  
Vol 37 (1-2) ◽  
pp. 86-95
Author(s):  
Bishnu Hari Subedi ◽  
Ajaya Singh
Keyword(s):  
Group Dynamics ◽  
Dynamical Behavior ◽  
Holomorphic Dynamics ◽  
Semi Group ◽  
Holomorphic Semigroup

In this paper, we prove that the escaping set of a transcendental semi group is S-forward invari-ant. We also prove that if a holomorphic semi group is a belian, then the Fatou, Julia, and escaping sets are S-completely invariant. We also investigate certain cases and conditions that the holomorphic semi group dynamics exhibits the similar dynamical behavior just like a classical holomorphic dynamics. Frequently, we also examine certain amount of connections and contrasts between classical holomorphic dynamics and holomorphic semi group dynamics.

scholarly journals Local Stability and Bifurcations in Kaldor Model

2005 ◽  
Vol 13 (1) ◽  
pp. 10-20
Author(s):  
Roman Binter ◽  
Lukáš Vácha
Keyword(s):  
Local Stability ◽  
Stability And Bifurcations
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