Crystals as dynamical systems : A new class of models

2008 ◽  
pp. 365-373
Author(s):  
Peter Kasperkovitz
Keyword(s):  
Dynamical Systems ◽  
New Class

scholarly journals Global Dynamical Systems Involving Generalized -Projection Operators and Set-Valued Perturbation in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yun-zhi Zou ◽  
Xi Li ◽  
Nan-jing Huang ◽  
Chang-yin Sun
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Banach Spaces ◽  
Equilibrium Points ◽  
Projection Operators ◽  
Initial Value ◽  
New Class ◽  
The Fixed Point Theorem ◽  
Generalized Projection Operators ◽  
Generalized Dynamical Systems

A new class of generalized dynamical systems involving generalizedf-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.

CHAOS, BIFURCATION AND ROBUSTNESS OF A CLASS OF HOPFIELD NEURAL NETWORKS

2011 ◽  
Vol 21 (03) ◽  
pp. 885-895 ◽  
Author(s):  
WEN-ZHI HUANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Chaotic Behavior ◽  
Robustness Analysis ◽  
Original System ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Computer Assisted ◽  
New Class

Chaos, bifurcation and robustness of a new class of Hopfield neural networks are investigated. Numerical simulations show that the simple Hopfield neural networks can display chaotic attractors and limit cycles for different parameters. The Lyapunov exponents are calculated, the bifurcation plot and several important phase portraits are presented as well. By virtue of horseshoes theory in dynamical systems, rigorous computer-assisted verifications for chaotic behavior of the system with certain parameters are given, and here also presents a discussion on the robustness of the original system. Besides this, quantitative descriptions of the complexity of these systems are also given, and a robustness analysis of the system is presented too.

Bekenstein–Hawking entropy for discrete dynamical systems on sites

2019 ◽  
Vol 34 (32) ◽  
pp. 1950265
Author(s):  
Sh. Najmizadeh ◽  
M. Toomanian ◽  
M. R. Molaei ◽  
T. Nasirzade
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Upper Bound ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
New Class ◽  
A Site

In this paper, we extend the notion of Bekenstein–Hawking entropy for a cover of a site. We deduce a new class of discrete dynamical system on a site and we introduce the Bekenstein–Hawking entropy for each member of it. We present an upper bound for the Bekenstein–Hawking entropy of the iterations of a dynamical system. We define a conjugate relation on the set of dynamical systems on a site and we prove that the Bekenstein–Hawking entropy preserves under this relation. We also prove that the twistor correspondence preserves the Bekenstein–Hawking entropy.

The Stabilization Problem for Certain Class of Ecological Systems

1997 ◽  
Vol 07 (02) ◽  
pp. 247-251
Author(s):  
A. N. Kirillov
Keyword(s):  
Dynamical Systems ◽  
Ecological Systems ◽  
Constructive Method ◽  
Global Stabilization ◽  
Stabilization Problem ◽  
Piecewise Constant ◽  
New Class ◽  
Constant Control ◽  
Phase Coordinates ◽  
Piecewise Constant Control

The stabilization problem for dynamical systems is to find control providing stability of some sets in phase space. This problem is not solved in general for nonlinear systems. The ecological systems are nonlinear, as usual, with restrictions on phase coordinates and controls. In this paper we consider piecewise constant control stabilization. Some general results are obtained. A new class of population dynamical systems is introduced, for which the constructive method of global stabilization and controllability is given. The set of attraction is found.

Neutral networks: a combinatorial perspective

1999 ◽  
Vol 02 (03) ◽  
pp. 283-301 ◽  
Author(s):  
Stephan Kopp ◽  
Christian M. Reidys
Keyword(s):  
Combinatorial Optimization ◽  
Dynamical Systems ◽  
Computer Simulations ◽  
Evolutionary Change ◽  
Rna Molecules ◽  
New Class ◽  
Neutral Networks ◽  
Threshold Phenomenon ◽  
Theoretical Investigations ◽  
Sequential Dynamical Systems

The existence of neutral networks in genotype-phenotype maps has provided significant insight in theoretical investigations of evolutionary change and combinatorial optimization. In this paper we will consider neutral networks of two particular genotype-phenotype maps from a combinatorial perspective. The first map occurs in the context of folding RNA molecules into their secondary structures and the second map occurs in the study of sequential dynamical systems, a new class of dynamical systems designed for the analysis of computer simulations. We will prove basic properties of neutral nets and present an error threshold phenomenon for evolving populations of simulation schedules.

scholarly journals Chaos in Essentially Singular 3D Dynamical Systems with Two Quadratic Nonlinearities

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Vasiliy Belozyorov
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Chaotic Behavior ◽  
Chaotic Attractors ◽  
Lorenz Attractor ◽  
Quadratic Systems ◽  
New Class ◽  
Quadratic Nonlinearities ◽  
Existence Conditions

A new class of 3D autonomous quadratic systems, the dynamics of which demonstrate a chaotic behavior, is found. This class is a generalization of the well-known class of Lorenz-like systems. The existence conditions of limit cycles in systems of the mentioned class are found. In addition, it is shown that, with the change of the appropriate parameters of systems of the indicated class, chaotic attractors different from the Lorenz attractor can be generated (these attractors are the result of the cascade of limit cycles bifurcations). Examples are given.

scholarly journals Existence of Perpetual Points in Nonlinear Dynamical Systems and Its Applications

2015 ◽  
Vol 25 (02) ◽  
pp. 1530005 ◽  
Author(s):  
Awadhesh Prasad
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Critical Points ◽  
Nonlinear Dynamical Systems ◽  
Transient Dynamics ◽  
Coexisting Attractors ◽  
Nonlinear Dynamical ◽  
New Class ◽  
Perpetual Points ◽  
Bifurcation Behavior

A new class of critical points, termed as perpetual points, where acceleration becomes zero but the velocity remains nonzero, are observed in dynamical systems. The velocity at these points is either maximum or minimum or of inflection behavior. These points also show the bifurcation behavior as the parameters of the system vary. These perpetual points are useful for locating the hidden oscillating attractors as well as coexisting attractors. Results show that these points are important for a better understanding of transient dynamics in the phase space. The existence of these points confirms whether a system is dissipative or not. Various examples are presented, and the results are discussed analytically as well as numerically.

Introducing projective billiards

1997 ◽  
Vol 17 (4) ◽  
pp. 957-976 ◽  
Author(s):  
SERGE TABACHNIKOV
Keyword(s):  
Dynamical Systems ◽  
Plane Curve ◽  
New Class ◽  
Convex Plane ◽  
Outer Billiards

We introduce and study a new class of dynamical systems, the projective billiards, associated with a smooth closed convex plane curve equipped with a smooth field of transverse directions. Projective billiards include the usual billiards along with the dual, or outer, billiards.

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