scholarly journals Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis

1994 ◽  
Vol 56 (2) ◽  
pp. 295-321 ◽  
Author(s):  
Vladimir A. Kuznetsov ◽  
Iliya A. Makalkin ◽  
Mark A. Taylor ◽  
Alan S. Perelson
Keyword(s):  
Nonlinear Dynamics ◽  
Parameter Estimation ◽  
Bifurcation Analysis ◽  
Global Bifurcation

BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (02) ◽  
pp. 399-404 ◽  
Author(s):  
T. SÜNNER ◽  
H. SAUERMANN
Keyword(s):  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Dynamical Behaviour ◽  
Van Der Pol Oscillator ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

Global Dynamics in the Leslie–Gower Model with the Allee Effect

2018 ◽  
Vol 28 (12) ◽  
pp. 1850151 ◽  
Author(s):  
Valery A. Gaiko ◽  
Cornelis Vuik
Keyword(s):  
Singular Point ◽  
Bifurcation Analysis ◽  
Limit Cycles ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Biomedical System

We complete the global bifurcation analysis of the Leslie–Gower system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations of limit cycles, we prove that such a system can have at most two limit cycles surrounding one singular point.

Global Bifurcation Analysis of a Controlled Underactuated Mechanical System

2005 ◽  
Vol 40 (3) ◽  
pp. 205-225 ◽  
Author(s):  
Diego M. Alonso ◽  
Eduardo E. Paolini ◽  
Jorge L. Moiola
Keyword(s):  
Mechanical System ◽  
Bifurcation Analysis ◽  
Global Bifurcation ◽  
Underactuated Mechanical System

Global Bifurcation Analysis of a Population Model with Stage Structure and Beverton–Holt Saturation Function

2015 ◽  
Vol 25 (12) ◽  
pp. 1550170 ◽  
Author(s):  
Li Fan ◽  
Sanyi Tang
Keyword(s):  
Bifurcation Analysis ◽  
Population Model ◽  
Birth Rate ◽  
Global Bifurcation ◽  
Stage Structure ◽  
Phase Portraits ◽  
Approximation Techniques ◽  
Codimension Two ◽  
Codimension One ◽  
Stage Transition

In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton–Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov–Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically.

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