scholarly journals Bifurcation diagrams and Fomenko's surgery on Liouville tori of the Kolossoff potential $U=\rho+(1/\rho)-k\cos\phi$

1993 ◽  
Vol 26 (5) ◽  
pp. 545-564 ◽  
Author(s):  
Ljubomir Gavrilov ◽  
Mohammed Ouazzani-Jamil ◽  
Régis Caboz
Keyword(s):  
Bifurcation Diagrams ◽  
Liouville Tori

Bifurcation diagrams and Fomenko's Surgery on Liouville tori of a rigid body in the Goryachev-Chaplygin case on sokolov terms

2018 ◽  
Vol 7 (4) ◽  
pp. 437-449
Author(s):  
Jaouad Kharbach ◽  
Mohammed Benkhali ◽  
Walid Chatar ◽  
Ahmed Sali ◽  
Abdellah Rezzouk ◽  
...  
Keyword(s):  
Rigid Body ◽  
Bifurcation Diagrams ◽  
Liouville Tori

scholarly journals Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image Encryption

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 474 ◽  
Author(s):  
Lazaros Moysis ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Jesus M. Munoz-Pacheco ◽  
Jacques Kengne ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Image Encryption ◽  
Fuzzy Numbers ◽  
Statistical Tests ◽  
Logistic Map ◽  
Simple Rule ◽  
Pseudorandom Number ◽  
Bifurcation Diagrams ◽  
Triangular Numbers ◽  
Pseudorandom Number Generation

A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence. The resulting random bit generator (RBG) successfully passes the National Institute of Standards and Technology (NIST) statistical tests, and it is then successfully applied to the problem of image encryption.

scholarly journals Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 876
Author(s):  
Wieslaw Marszalek ◽  
Jan Sadecki ◽  
Maciej Walczak
Keyword(s):  
Equilibrium Points ◽  
Autonomous Systems ◽  
Cytosolic Calcium ◽  
Calcium Oscillations ◽  
Dynamical Model ◽  
Bifurcation Diagrams ◽  
Step Size ◽  
Oscillatory Systems ◽  
Two Parameter ◽  
Parameter Bifurcation

Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, Ls diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the Ca2+ signal regulation.

scholarly journals Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

2021 ◽  
Vol 11 (15) ◽  
pp. 6955
Author(s):  
Andrzej Rysak ◽  
Magdalena Gregorczyk
Keyword(s):  
Optimal Parameter ◽  
Differential Transform Method ◽  
Bifurcation Diagrams ◽  
Transform Method ◽  
Rossler System ◽  
Differential Transform ◽  
Rössler System ◽  
Fractional System ◽  
Parameter Values

This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.

scholarly journals Dynamic ensembles with variable structure and friction simulation

1998 ◽  
Vol 2 (3) ◽  
pp. 167-172
Author(s):  
I. V. Feldstein ◽  
N. N. Kuzmin
Keyword(s):  
Normal Load ◽  
Logistic Map ◽  
Variable Structure ◽  
Experimental Result ◽  
Bifurcation Diagrams ◽  
Continuous Contact ◽  
Real Contact ◽  
Simple Physical Interpretation ◽  
Friction Interaction ◽  
Dynamic Ensembles

The paper presents an approach to the simulation of friction interaction. The model does not use any physical descriptions of the processes in the system, but it has simple physical interpretation. It is based on one qualitative experimental result – the value of first Lyapunov exponent drops with normal load. It is shown that the logistic map could be considered as the simplest model of continuous contact. The generalization of the model (which takes into account the discreteness of the real contact) gives results very similar to the experimental ones. It is in the form of a dynamic ensemble with variable structure (DEVS), which has some interesting properties – particularly bifurcation diagrams.

scholarly journals Random perturbations of bifurcation diagrams

1994 ◽  
Vol 5 (3) ◽  
pp. 353-373 ◽  
Author(s):  
Fritz Colonius ◽  
Wolfgang Kliemann
Keyword(s):  
Random Perturbations ◽  
Bifurcation Diagrams

scholarly journals Chaotic Behavior in Gear System. Adaptation of Bifurcation Diagrams and Method of Statistical Mechanics.

1995 ◽  
Vol 61 (587) ◽  
pp. 3108-3115
Author(s):  
Keijin Sato ◽  
Sumio Yamamoto ◽  
Kazutaka Yokota ◽  
Toshihiro Aoki ◽  
Shu Karube
Keyword(s):  
Statistical Mechanics ◽  
Chaotic Behavior ◽  
Gear System ◽  
Bifurcation Diagrams ◽  
System Adaptation

scholarly journals Bifurcations of Oscillatory and Rotational Solutions of Double Pendulum with Parametric Vertical Excitation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Michal Marszal ◽  
Krzysztof Jankowski ◽  
Przemyslaw Perlikowski ◽  
Tomasz Kapitaniak
Keyword(s):  
Periodic Solutions ◽  
Parameter Space ◽  
Double Pendulum ◽  
Bifurcation Diagrams ◽  
Kinematic Excitation ◽  
Vertical Excitation ◽  
Two Parameter ◽  
Domain Of Periodic Solutions

This paper investigates dynamics of double pendulum subjected to vertical parametric kinematic excitation. It includes detailed bifurcation diagrams in two-parameter space (excitation’s frequency and amplitude) for both oscillations and rotations in the domain of periodic solutions.

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