BIFURCATION STRUCTURES GENERATED BY THE NONAUTONOMOUS DUFFING EQUATION

1999 ◽  
Vol 09 (07) ◽  
pp. 1363-1379 ◽  
Author(s):  
C. MIRA ◽  
M. TOUZANI-QRIOUET ◽  
H. KAWAKAMI
Keyword(s):  
Parameter Space ◽  
Global Bifurcation ◽  
Duffing Equation ◽  
Damping Term ◽  
Bifurcation Structure ◽  
The Past ◽  
Spring Area ◽  
Bifurcation Structures ◽  
Duffing Model ◽  
Qualitative Changes

This paper deals with some properties of bifurcation structures in the parameter space related to the Duffing equation in the presence of an external periodical excitation B + B0 cos t. So global qualitative modifications of structures in the parameter plane (B, B0) are considered, when a third parameter ε (the damping term of the equation) varies. Complex sets of bifurcation curves are defined. They are based on the global bifurcation structure: crossroad area, saddle area, spring area, lip, quasi-lip, identified in the past with their "foliated representation", and their qualitative changes. For the Duffing model, special associations of the above areas, giving typical patterns called islands, are described with their qualitative modifications.

Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics

2015 ◽  
Vol 25 (03) ◽  
pp. 1530006 ◽  
Author(s):  
Anastasiia Panchuk ◽  
Iryna Sushko ◽  
Viktor Avrutin
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Chaotic Attractors ◽  
Bifurcation Structure ◽  
Tent Map ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Replacement Technique ◽  
Bifurcation Structures

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.

Bifurcation in the Duffing equation with independent parameters, II

1978 ◽  
Vol 79 (3-4) ◽  
pp. 317-326 ◽  
Author(s):  
Jack K. Hale ◽  
Hildebrando M. Rodrigues
Keyword(s):  
Parameter Space ◽  
Duffing Equation ◽  
Damping Term ◽  
Duffing's Equation ◽  
Duffing’S Equation ◽  
All Solutions ◽  
Independent Parameters

SynopsisIn a previous paper, the authors gave a complete description of the number of even harmonic solutions of Duffing's equation without damping for the parameters varying in a full neighbourhood of the origin in the parameter space. In this paper, the analysis is extended to the case of an independent small damping term. It is also shown that all solutions of the undamped equation are even functions of time.

THE DOVETAIL BIFURCATION STRUCTURE AND ITS QUALITATIVE CHANGES

1993 ◽  
Vol 03 (04) ◽  
pp. 903-919 ◽  
Author(s):  
C. MIRA ◽  
H. KAWAKAMI ◽  
R. ALLAM
Keyword(s):  
Three Dimensional ◽  
Parameter Plane ◽  
Bifurcation Curve ◽  
Bifurcation Structure ◽  
Fold Bifurcation ◽  
Spring Area ◽  
Two Parameters ◽  
Qualitative Changes

In a parameter plane defined by two parameters of a map, the dovetail bifurcation structure is a bifurcation curve organization, for which two fold cusps have a fold bifurcation segment in common. This gives rise to the association of either a crossroad area with a saddle area or a spring area with a saddle area. This paper describes this structure in the parameter plane for different configurations, and in the corresponding three-dimensional representations of the plane sheets. Then it identifies six different qualitative changes of this structure when a third parameter varies.

BIFURCATION STRUCTURE IN A MODEL OF A FREQUENCY MODULATED CO2 LASER

1993 ◽  
Vol 03 (01) ◽  
pp. 97-111 ◽  
Author(s):  
C. MIRA ◽  
I. DJELLIT
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Organization Structure ◽  
The Other ◽  
Two Dimensional ◽  
Periodic Excitation ◽  
Bifurcation Structure ◽  
Higher Harmonic ◽  
Bifurcation Structures ◽  
Qualitative Changes

This paper concerns the bifurcation properties of a model of a frequency modulated CO 2 laser in the form of a two-dimensional ordinary differential equation with a parametric periodic excitation. These properties are related to the bifurcation curves organization (structure) in a parameter plane (amplitude, frequency of the modulation). Two basic bifurcation structures appear, one concerning the higher harmonic solutions, the other the subharmonic solutions. Qualitative changes of these structures are considered when a third parameter (pump parameter) is varied.

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.

"CROSSROAD AREA–SPRING AREA" TRANSITION (II) FOLIATED PARAMETRIC REPRESENTATION

1991 ◽  
Vol 01 (02) ◽  
pp. 339-348 ◽  
Author(s):  
C. MIRA ◽  
J. P. CARCASSÈS ◽  
M. BOSCH ◽  
C. SIMÓ ◽  
J. C. TATJER
Keyword(s):  
Complete Information ◽  
Parametric Representation ◽  
Periodic Points ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Cusp Point ◽  
Dimensional Representation ◽  
Bifurcation Structure ◽  
Spring Area

The areas considered are related to two different configurations of fold and flip bifurcation curves of maps, centred at a cusp point of a fold curve. This paper is a continuation of an earlier one devoted to parameter plane representation. Now the transition is studied in a thee-dimensional representation by introducing a norm associated with fixed or periodic points. This gives rise to complete information on the map bifurcation structure.

Global bifurcation structure of a Bonhoeffer-van der Pol oscillator driven by periodic pulse trains

1994 ◽  
Vol 72 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Taishin Nomura ◽  
Shunsuke Sato ◽  
Shinji Doi ◽  
Jose P. Segundo ◽  
Michael D. Stiber
Keyword(s):  
Global Bifurcation ◽  
Van Der Pol Oscillator ◽  
Bifurcation Structure ◽  
Van Der Pol ◽  
Pulse Trains ◽  
Periodic Pulse

RESONANT GLUING BIFURCATIONS

2000 ◽  
Vol 10 (09) ◽  
pp. 2141-2160 ◽  
Author(s):  
ROBERT W. GHRIST
Keyword(s):  
Saddle Point ◽  
Parameter Space ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Centre Manifold ◽  
Principal Eigenvalues ◽  
Homoclinic Connections ◽  
Homoclinic Bifurcations ◽  
Zero Entropy

We consider the codimension-three phenomenon of homoclinic bifurcations of flows containing a pair of orbits homoclinic to a saddle point whose principal eigenvalues are in resonance. We concentrate upon the simplest possible configuration, the so-called "figure-of-eight," and reduce the dynamics near the homoclinic connections to those on a two-dimensional locally invariant centre manifold. The ensuing resonant gluing bifurcations exhibit features of both gluing bifurcations and resonant homoclinic bifurcations. Under certain twist conditions, the bifurcation structure is extremely rich, although describing zero-entropy flows. The analysis carefully exploits the topology of the orbits, the centre manifold and the parameter space.

Sign in / Sign up

Export Citation Format

Share Document