Global bifurcations and single-pulse homoclinic orbits of a plate subjected to the transverse and in-plane excitations

2017 ◽  
Vol 40 (12) ◽  
pp. 4338-4349 ◽  
Author(s):  
Dongmei Zhang ◽  
Fangqi Chen
Keyword(s):  
Single Pulse ◽  
Homoclinic Orbits ◽  
Global Bifurcations

MULTIPARAMETRIC BIFURCATIONS IN AN ENZYME-CATALYZED REACTION MODEL

2005 ◽  
Vol 15 (03) ◽  
pp. 905-947 ◽  
Author(s):  
E. FREIRE ◽  
L. PIZARRO ◽  
A. J. RODRÍGUEZ-LUIS ◽  
F. FERNÁNDEZ-SÁNCHEZ
Keyword(s):  
Periodic Orbits ◽  
Dynamical Behavior ◽  
Homoclinic Orbits ◽  
Reaction Model ◽  
Numerical Continuation ◽  
Global Bifurcations ◽  
Codimension One ◽  
Local Bifurcations ◽  
Homoclinic Connections ◽  
Local And Global Bifurcations

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system. First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.

QUADRATIC VECTOR FIELDS EQUIVARIANT UNDER THE D2 SYMMETRY GROUP

2013 ◽  
Vol 23 (01) ◽  
pp. 1350017 ◽  
Author(s):  
STAVROS ANASTASSIOU ◽  
SPYROS PNEVMATIKOS ◽  
TASSOS BOUNTIS
Keyword(s):  
Symmetry Group ◽  
Parameter Space ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Global Behavior ◽  
Global Bifurcations ◽  
Heteroclinic Cycles ◽  
Different Types ◽  
Parameter Values ◽  
Two Parameter

Symmetry often plays an important role in the formation of complicated structures in the dynamics of vector fields. Here, we study a specific family of systems defined on ℝ3, which are invariant under the D2 symmetry group. Under the assumption that they are polynomial of degree at most two, they belong to a two-parameter family of vector fields, called the D2 model. We describe the global behavior of the system, for most parameter values, and locate a region of parameter space where complicated structures occur. The existence of heteroclinic and homoclinic orbits is shown, as well as of heteroclinic cycles (for other parameter values), implying the presence of (different types of) Shil'nikov type of chaos in the D2 systems. We then employ Poincaré maps to illustrate the bifurcations leading to this behavior. The global bifurcations exhibited by its strange attractors are explained as an effect of symmetry. We conclude by describing the behavior of the system at infinity.

scholarly journals Asset price dynamics in a financial market with fundamentalists and chartists

2001 ◽  
Vol 6 (2) ◽  
pp. 69-99 ◽  
Author(s):  
Carl Chairella ◽  
Roberto Dieci ◽  
Laura Gardini
Keyword(s):  
Financial Market ◽  
Asset Price ◽  
Homoclinic Orbits ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Critical Curves ◽  
Qualitative Structure ◽  
Local And Global Bifurcations ◽  
Noninvertible Maps

In this paper we consider a model of the dynamics of speculative markets involving the interaction of fundamentalists and chartists. The dynamics of the model are driven by a two-dimensional map that in the space of the parameters displays regions of invertibility and noninvertibility. The paper focuses on a study of local and global bifurcations which drastically change the qualitative structure of the basins of attraction of several, often coexistent, attracting sets. We make use of the theory of critical curves associated with noninvertible maps, as well as of homoclinic bifurcations and homoclinic orbits of saddles in regimes of invertibility.

GLOBAL ANALYSIS AND CHAOTIC DYNAMICS OF SIX-DIMENSIONAL NONLINEAR SYSTEM FOR AN AXIALLY MOVING VISCOELASTIC BELT

2011 ◽  
Vol 25 (17) ◽  
pp. 2299-2322 ◽  
Author(s):  
W. ZHANG ◽  
M. J. GAO ◽  
M. H. YAO
Keyword(s):  
Nonlinear System ◽  
Normal Form ◽  
Perturbation Method ◽  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Global Analysis ◽  
Single Pulse ◽  
Global Bifurcations ◽  
Chaotic Motions ◽  
Axially Moving

An analysis on the chaotic dynamics of a six-dimensional nonlinear system which represents the averaged equation of an axially moving viscoelastic belt is given in this paper for the first time. We combine the theory of normal form and the global perturbation method to investigate the global bifurcations and chaotic dynamics of the axially moving viscoelastic belt. Firstly, the theory of normal form is used to reduce six-dimensional averaged equation to the simpler normal form. Then, the global perturbation method is employed to analyze the global bifurcations and chaotic dynamics of six-dimensional nonlinear system. The analysis results indicate that there exist the homoclinic bifurcations and the single-pulse in six-dimensional averaged equation. Finally, numerical simulations are also used to investigate the nonlinear dynamic characteristics of the axially moving viscoelastic belt. The results of numerical simulations demonstrate that there exist the chaotic motions and the jumping orbits of the axially moving viscoelastic belt.

GLOBAL BIFURCATIONS AND CHAOTIC DYNAMICS IN SUSPENDED CABLES

2009 ◽  
Vol 19 (11) ◽  
pp. 3753-3776 ◽  
Author(s):  
HONGKUI CHEN ◽  
ZHAOHUA ZHANG ◽  
JILONG WANG ◽  
QINGYU XU
Keyword(s):  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Global Bifurcation ◽  
Homoclinic Orbits ◽  
Invariant Sets ◽  
Global Bifurcations ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Suspended Cables

The global bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The governing equations are obtained to describe the nonlinear transverse vibrations of suspended cables. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degrees-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary and principal parametric resonance of suspended cables is considered. With the method of multiple scales, parametrically and externally excited system is transformed to the averaged equation, based on which, the recently developed global bifurcation method is employed to detect the presence of orbits which are homoclinic to certain invariant sets for the resonant case. The analysis of the global bifurcations indicates that there exist the generalized Šhilnikov type multipulse homoclinic orbits in the averaged equation of suspended cables. The results obtained here mean that chaotic motions can occur in suspended cables. Numerical simulations also verify the analytical predictions. It is found, according to the results of numerical simulations, that the Šhilnikov type multipulse homoclinic orbits exist in the nonlinear motion of the cables.

scholarly journals Homoclinic and heteroclinic bifurcations in a two-dimensional endomorphism

2003 ◽  
Vol 16 (2) ◽  
pp. 273-283
Author(s):  
Ilham Djellit ◽  
Mohamed Ferchichi
Keyword(s):  
Saddle Point ◽  
Chaotic Attractor ◽  
Invariant Manifolds ◽  
Basin Of Attraction ◽  
Homoclinic Orbits ◽  
Specific Information ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Homoclinic Bifurcations ◽  
Noninvertible Maps

Our study concerns global bifurcations occurring in noninvertible maps, it consists to show that there exists a link between contact bifurcations of a chaotic attractor and homoclinic bifurcations of a saddle point or a saddle cycle being on the boundary of the chaotic attractor. We provide specific information about the intricate dynamics near such points. We study particularly a two-dimensional endomorphism of (Z\ - Z$ - Z\) type. We will show that points of contact, between boundary of the attractor and its basin of attraction, converge toward the saddle point or the saddle cycle. These points of contact are also points of intersection between the stable and unstable invariant manifolds. This gives rise to the birth of homoclinic orbits (homoclinic bifurcations).

Pyrolysis of methane in a single pulse shock tube

1972 ◽  
Vol 22 (3) ◽  
pp. 303-317 ◽  
Author(s):  
D. H. Napier ◽  
N. Subrahmanyam
Keyword(s):  
Shock Tube ◽  
Single Pulse
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