The bifurcation diagram of a model chemical reaction—II. Two dimensional time-periodic patterns

1979 ◽  
Vol 41 (6) ◽  
pp. 767-790 ◽  
Author(s):  
T ERNEUX ◽  
M HERSCHKOWITZKAUFMAN
Keyword(s):  
Bifurcation Diagram ◽  
Chemical Reaction ◽  
Two Dimensional ◽  
Periodic Patterns ◽  
Model Chemical Reaction ◽  
Time Periodic

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

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.

Two-dimensional numerical study of temperature field in an anode supported planar SOFC: Effect of the chemical reaction

2011 ◽  
Vol 36 (6) ◽  
pp. 4228-4235 ◽  
Author(s):  
Bariza Zitouni ◽  
G.M. Andreadis ◽  
Ben Moussa Hocine ◽  
Abdenebi Hafsia ◽  
Haddad Djamel ◽  
...  
Keyword(s):  
Temperature Field ◽  
Chemical Reaction ◽  
Numerical Study ◽  
Two Dimensional

An Intrinsically Three-Dimensional Fractal

2014 ◽  
Vol 24 (06) ◽  
pp. 1430017 ◽  
Author(s):  
M. Fernández-Guasti
Keyword(s):  
Bifurcation Diagram ◽  
Chaotic Behavior ◽  
Logistic Map ◽  
Three Dimensional ◽  
Periodic Points ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Hyperbolic Numbers ◽  
Self Similar ◽  
One To One

The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits self-similar 3D copies of the elementary fractal along the negative scalar axis. These 3D copies correspond to the windows amid the chaotic behavior of the logistic map. Nonetheless, the two-dimensional projection becomes identical to the nonfractal quadratic iteration produced with hyperbolic numbers. Two- and three-dimensional renderings are presented to explore some of the features of this set.

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