scholarly journals Explicit transversality conditions and local bifurcation diagrams for Bogdanov–Takens bifurcation on center manifolds

2019 ◽  
Vol 391 ◽  
pp. 52-65 ◽  
Author(s):  
Yang Li ◽  
Hiroshi Kokubu ◽  
Kazuyuki Aihara
Keyword(s):  
Local Bifurcation ◽  
Center Manifolds ◽  
Bifurcation Diagrams ◽  
Transversality Conditions

CONNECTIONS BETWEEN SADDLES FOR THE FITZHUGH–NAGUMO SYSTEM

2001 ◽  
Vol 11 (02) ◽  
pp. 533-540 ◽  
Author(s):  
CARMEN ROCŞOREANU ◽  
NICOLAIE GIURGIŢEANU ◽  
ADELINA GEORGESCU
Keyword(s):  
Dynamical System ◽  
Homoclinic Bifurcation ◽  
Parameter Plane ◽  
Saddle Connection ◽  
Local Bifurcation ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Numerical Investigations ◽  
Codimension One ◽  
Saddle Node

By studying the two-dimensional FitzHugh–Nagumo (F–N) dynamical system, points of Bogdanov–Takens bifurcation were detected (Sec. 1). Two of the curves of homoclinic bifurcation emerging from these points intersect each other at a point of double breaking saddle connection bifurcation (Sec. 2). Numerical investigations of the bifurcation curves emerging from this point, in the parameter plane, allowed us to find other types of codimension-one and -two bifurcations concerning the connections between saddles and saddle-nodes, referred to as saddle-node–saddle connection bifurcation and saddle-node–saddle with separatrix connection bifurcation, respectively. The local bifurcation diagrams corresponding to these bifurcations are presented in Sec. 3. An analogy between the bifurcation corresponding to the point of double homoclinic bifurcation and the point of double breaking saddle connection bifurcation is also presented in Sec. 3.

scholarly journals DISCRETIZING BIFURCATION DIAGRAMS NEAR CODIMENSION TWO SINGULARITIES

2010 ◽  
Vol 20 (05) ◽  
pp. 1391-1403 ◽  
Author(s):  
PÁEZ CHÁVEZ JOSEPH
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Diagram ◽  
Continuous Time ◽  
Local Bifurcation ◽  
Bifurcation Diagrams ◽  
Codimension Two ◽  
Numerical Example ◽  
Codimension One ◽  
One Step ◽  
Parameter Dependent

We consider parameter-dependent, continuous-time dynamical systems under discretizations. It is shown that fold-Hopf singularities are O(hp)-shifted and turned into fold-Neimark–Sacker points by one-step methods of order p. Then we analyze the effect of discretizations methods on the local bifurcation diagram near Bogdanov–Takens and fold-Hopf singularities. In particular, we prove that the discretized codimension one curves intersect at the singularities in a generic manner. The results are illustrated by a numerical example.

ON THE COMPUTATION OF LOCAL BIFURCATION DIAGRAMS NEAR DEGENERATE HOPF BIFURCATIONS OF CERTAIN TYPES

1993 ◽  
Vol 03 (05) ◽  
pp. 1103-1122 ◽  
Author(s):  
JORGE LUIS MOIOLA
Keyword(s):  
System Theory ◽  
Phase Portrait ◽  
Harmonic Balance ◽  
Graphical Method ◽  
Feedback System ◽  
Hopf Bifurcations ◽  
Local Bifurcation ◽  
Bifurcation Diagrams ◽  
Nondegeneracy Conditions ◽  
Analytical Expressions

The computation of local bifurcation diagrams near degenerate Hopf bifurcations of certain types using feedback system theory and harmonic balance techniques is presented. This approach also provides the analytical expressions for the defining and the nondegeneracy conditions in the so-called frequency domain counterpart. A classical graphical method is easily adapted to carry on the continuation of the oscillatory branches to depict the local bifurcation diagrams. Moreover, several higher-order harmonic balance approximations are implemented to compare the accuracy of the computed solutions. The results are presented using local bifurcation diagrams, phase portrait plots and period diagrams, with similar ones obtained by using AUTO.

scholarly journals Bifurcation study on fractional non-smooth oscillator containing clearance constraints

2020 ◽  
pp. 146134842096095
Author(s):  
Jun Wang ◽  
Yongjun Shen ◽  
Shaopu Yang ◽  
Jianchao Zhang
Keyword(s):  
Periodic Motion ◽  
Singularity Theory ◽  
Local Bifurcation ◽  
Bifurcation Diagrams ◽  
Transition Set ◽  
Response Equation ◽  
The Stability ◽  
Fractional Terms ◽  
Smooth System ◽  
Sinusoidal Excitation

Bifurcation characteristics of a fractional non-smooth oscillator containing clearance constraints under sinusoidal excitation are investigated. First, the bifurcation response equation of the fractional non-smooth system is obtained via the K–B method. Second, the stability of the bifurcation response equation is analyzed, and parametric conditions for stability are acquired. The bifurcation characteristics of the fractional non-smooth system are then studied using singularity theory, and the transition set and bifurcation diagram under six different constrained parameters are acquired. Finally, the analysis of the influence of fractional terms on the dynamic characteristics of the system is emphasized through numerical simulation. Local bifurcation diagrams of the system under different fractional coefficients and orders verify that the system will present various motions, such as periodic motion, multiple periodic motion, and chaos, with the change in fractional coefficient and order. This manifestation indicates that fractional parameters have a direct effect on the motion form of this non-smooth system. Thus, these results provide a theoretical reference for investigating and repressing oscillation problems of similar systems.

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.

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