scholarly journals Chaotic Behaviour in Two-parameter Family of Transcendental Functions Associated with Exponential Map

2020 ◽  
Vol 5 (6) ◽  
pp. 1191-1209
Author(s):  
Mohammad Sajid ◽  
Abdullah S. Alsuwaiyan
Keyword(s):  
Numerical Simulations ◽  
Period Doubling ◽  
Periodic Points ◽  
Parameter Family ◽  
Exponential Map ◽  
Bifurcation Diagrams ◽  
The Real ◽  
Transcendental Functions ◽  
Parameter Values ◽  
Two Parameter

This article is devoted to the study of chaos and bifurcation in the real dynamics of a newly proposed two-parameter family of transcendental functions. We assume that one parameter is continuous and other parameter is discrete. For certain parameters, the theoretical computations of the real fixed points of a family of functions are given. The numerical simulations of the real periodic points of functions are described. The bifurcation diagrams of the real dynamics of these functions for some selected parameter values are provided. In these bifurcation diagrams, the period-doubling occurs which proceeds to a pathway toward chaos in the dynamics of functions. Further, the periodic-three window is visible in the bifurcation diagrams which implies chaos. Lastly, chaos is quantified in the dynamics of functions by calculating Lyapunov exponents.

COMPLEXITY OF AN IVLEV'S PREDATOR-PREY MODEL WITH PULSE

2007 ◽  
Vol 10 (02) ◽  
pp. 217-231 ◽  
Author(s):  
GUOPING PANG ◽  
LANSUN CHEN
Keyword(s):  
Numerical Simulations ◽  
Functional Response ◽  
Period Doubling ◽  
Impulsive System ◽  
Bifurcation Diagrams ◽  
Bifurcation And Chaos ◽  
Dynamic Complexity ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Predator System

In this paper, we investigate the extinction, permanence and dynamic complexity of the two-prey, one-predator system with Ivlev's functional response and impulsive perturbation on the predator at fixed moments. Conditions for the extinction and permanence of the system are established via the comparison theorem. Numerical simulations are carried out to explain the conclusions we obtain. Furthermore, the resulting bifurcation diagrams clearly show that the impulsive system takes on many forms of complexity including period-doubling bifurcation, period-halving bifurcation, and chaos.

scholarly journals Bifurcation of an attracting invariant circle: a Denjoy attractor

1983 ◽  
Vol 3 (1) ◽  
pp. 87-118 ◽  
Author(s):  
Glen R. Hall
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Periodic Points ◽  
Invariant Circle ◽  
Worst Case ◽  
Dense Orbits ◽  
Parameter Values ◽  
Two Parameter

AbstractWe construct an example of a C∞ diffeomorphism of an annulus into itself which has an attracting invariant circle such that the map restricted to this circle has no periodic points and no dense orbits. By studying two parameter families of maps of the plane which undergo Hopf bifurcation, particularly the set of parameter values for which the rotation number is irrational, we see that the above example can be considered as a ‘worst case’ of the loss of smoothness of an attracting invariant circle without periodic orbits.

Dynamics and bifurcations of a host–parasite model

2017 ◽  
Vol 10 (06) ◽  
pp. 1750089 ◽  
Author(s):  
Ali Atabaigi ◽  
Mohammad Hossein Akrami
Keyword(s):  
Fixed Point ◽  
Difference Equations ◽  
Parameter Family ◽  
Invariant Curve ◽  
Discrete Models ◽  
Nonlinear Difference Equations ◽  
Host Parasite ◽  
Parameter Values ◽  
Two Parameter

A two-parameter family of discrete models, consisting of two coupled nonlinear difference equations, describing a host–parasite interaction is considered. In particular, we prove that the model has at most one nontrivial interior fixed point which is stable for a certain range of parameter values and also undergoes a Neimark–Sacker bifurcation that produces an attracting invariant curve in some areas of the parameter.

scholarly journals Non-expansive derived horseshoes

1998 ◽  
Vol 65 (3) ◽  
pp. 405-415
Author(s):  
Kazuhiro Sakai
Keyword(s):  
Hausdorff Dimension ◽  
Homoclinic Orbit ◽  
Invariant Set ◽  
Parameter Family ◽  
Simple Application ◽  
Parameter Values ◽  
The Relationship ◽  
Two Parameter

AbstractFor a two parameter family of C3 diffeomorphisms having a homoclinic orbit of tangency derived from a horseshoe, the relationship between the measure of the parameter values at which the diffeomorphism (restricted to a certain compact invariant set containing the horseshoe) is not expansive and the Hausdorff dimension of the horseshoe associated to the homoclinic orbit of tangency is investigated. This is a simple application of the Newhouse-Palis-Takens-Yoccoz theory.

NONINVERTIBLE TWO-DIMENSIONAL MAPS ARISING IN RADIOPHYSICS

1994 ◽  
Vol 04 (02) ◽  
pp. 383-400 ◽  
Author(s):  
VLADIMIR MAISTRENKO ◽  
YURI MAISTRENKO ◽  
IRINA SUSHKO
Keyword(s):  
Period Doubling ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
One Dimensional ◽  
Transition To Chaos ◽  
Nonlinear Amplifiers ◽  
Noninvertible Maps ◽  
Triangular Maps ◽  
Two Parameter

We study a two-parameter family of noninvertible maps modeling a generator which consists of two identical nonlinear amplifiers and two delay circuits. The ratio of the delays determines the dimension of the map and our attention is mainly on the two-dimensional case. The mechanism of transition to chaos appears to be one-dimensional and is realized through a period-doubling cascade. To get a more complete description we suggest the use of so-called triangular maps. Phase portraits are constructed for some types of model triangular maps. Also we get one- and two-dimensional bifurcation diagrams for the maps considered and attractor basins in the case of multistability using computer simulation.

scholarly journals Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammad Sajid
Keyword(s):  
Fixed Points ◽  
Research Work ◽  
Period Doubling ◽  
Periodic Points ◽  
Real Parameter ◽  
Chaotic Behaviour ◽  
Positive Real ◽  
The Real ◽  
Positive Real Parameter ◽  
Two Parameters

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

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 A Uniqueness Result for a Simple Superlinear Eigenvalue Problem

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Michael Herrmann ◽  
Karsten Matthies
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Simulations ◽  
Convolution Operator ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Constitutive Laws ◽  
Parameter Family ◽  
Shape Constraint ◽  
Special Case ◽  
Nonlinear Eigenfunctions

AbstractWe study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein–Rutman arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.

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.

Sign in / Sign up

Export Citation Format

Share Document