Homoclinic Orbits, Subharmonics and Global Bifurcations in Forced Oscillations

1981 ◽  
Author(s):  
Bernie D. Greenspan ◽  
Philip J. Holmes
Keyword(s):  
Homoclinic Orbits ◽  
Forced Oscillations ◽  
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 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).

Study of the dynamics of ТПА 350 automatic mill working stand elements

2020 ◽  
Vol 76 (8) ◽  
pp. 830-840
Author(s):  
S. R. Rakhmanov ◽  
V. V. Povorotnii
Keyword(s):  
Mechanical System ◽  
Dynamic Model ◽  
Maximum Amplitude ◽  
Calculation Scheme ◽  
Dynamic Loads ◽  
Forced Oscillations ◽  
Mass Model ◽  
Automatic Mill ◽  
Hollow Billet ◽  
Oscillation Movement

To form a necessary geometry of a hollow billet to be rolled at a pipe rolling line, stable dynamics of the base equipment of the automatic mill working stand has a practical meaning. Among the forces, acting on its parts and elements, significant by value short-time dynamic loads are the least studied phenomena. These dynamic loads arise during transient interaction of the hollow billet, rollers, mandrel and other mill parts at the forced grip of the hollow billet. Basing of the calculation scheme and dynamic model of the mechanical system of the ТПА 350 automatic mill working stand was accomplished. A mathematical model of dynamics of the system “hollow billet (pipe) – working stand” within accepted calculation scheme and dynamic model of the mechanical system elaborated. Influence of technological load of the rolled hollow billet variation in time was accounted, as well as variation of the mechanical system mass, and rigidity of the ТПА 350 automatic mill working stand. Differential equations of oscillation movement for four-mass model of forked sub-systems of the automatic mill working stand were made up, results of their digital calculation quoted. Dynamic displacement of the stand elements in the inter-roller gap obtained, which enabled to estimate the results of amplitude and frequency characteristics of the branches of the mill rollers setting. It was defined by calculation, that the maximum amplitude of the forced oscillations of elements of the ТПА 350 automatic mill working stand within the inter-roller gap does not exceed 2 mm. It is much higher than the accepted value of adjusting parameters of the deformation center of the ТПА 350 automatic mill. A scheme of comprehensive modernization of the rollers setting in the ТПА 350 automatic mill working stand was proposed. It was shown, that increase of rigidity of rollers setting in the ТПА 350 automatic mill working stand enables to stabilize the amplitude of forced oscillations of the working stand elements within the inter-rollers gap and considerably decrease the induced nonuniform hollow billet wall thickness and increase quality of the rolled pipes at ТПА 350.

CALCULATION OSCILLATIONS OF VARIOUS ELEMENTS OF THE ELASTIC SYSTEM OF THE CENTER-FREE GRINDING MACHINE SASL 5AD

2020 ◽  
pp. 160-165
Author(s):  
Джугурян Т.Г. ◽  
Марчук В.І. ◽  
Марчук І. В.
Keyword(s):  
Horizontal Plane ◽  
Natural Frequencies ◽  
Elastic System ◽  
Experimental Studies ◽  
Piezoelectric Sensors ◽  
Forced Oscillations ◽  
Method Of Calculation ◽  
Vibration Resistance ◽  
Grinding Machine ◽  
Experimental Researches

During the design of operations of centerless intermittent grinding of surfaces there is a need to identify the natural frequencies of oscillations of the elements of the technological system of grinding. The method of calculation of rigidity, vibration resistance and forced oscillations of the elements of the circular grinding machine is offered in the article. Carrying out of experimental researches of rigidity of elastic system of the SASL 5AD grinding machine. We conducted preliminary experimental studies to measure the oscillations of various elements of the elastic system of the SASL 5AD grinding machine in the horizontal plane by piezoelectric sensors during grinding with continuous and discontinuous circles with different geometric parameters.

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