Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations

1991 ◽  
Vol 113 (4) ◽  
pp. 299-311 ◽  
Author(s):  
Timothy J. Healey ◽  
Hansj�rg Kielh�fer
Keyword(s):  
Bifurcation Analysis ◽  
Elliptic Equations ◽  
Global Bifurcation ◽  
Nodal Properties

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.

Unilateral Global Bifurcation from Infinity in Nonlinearizable One-Dimensional Dirac Problems

2021 ◽  
Vol 31 (01) ◽  
pp. 2150005
Author(s):  
Ziyatkhan S. Aliyev ◽  
Nazim A. Neymatov ◽  
Humay Sh. Rzayeva
Keyword(s):  
Dirac Equation ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nontrivial Solutions ◽  
One Dimensional ◽  
Bifurcation From Infinity ◽  
The One ◽  
Nodal Properties

In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

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.

scholarly journals Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

2019 ◽  
Vol 19 (2) ◽  
pp. 391-412
Author(s):  
Uriel Kaufmann ◽  
Humberto Ramos Quoirin ◽  
Kenichiro Umezu
Keyword(s):  
Elliptic Equations ◽  
Bifurcation Theory ◽  
A Priori ◽  
Global Bifurcation ◽  
Neumann Boundary ◽  
Zero Solution ◽  
Regularization Scheme ◽  
Convex Problems ◽  
Nonnegative Solutions ◽  
Indefinite Weights

AbstractWe establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a nonregular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn’s topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved by us, and extend previous results established in the powerlike case.

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.

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

scholarly journals Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Fatma Aydin Akgun
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nonlinear Eigenvalue Problems ◽  
Nodal Properties ◽  
Nonlinear Eigenvalue

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

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