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).

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.

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.

High-Dimensional Chaos Control Algorithms for Improved Piezoelectric Energy Harvesting Using a Bistable Structure

2015 ◽  
Author(s):  
Daniel Geiyer ◽  
Jeffrey L. Kauffman
Keyword(s):  
Energy Harvesting ◽  
Chaotic Attractor ◽  
Chaos Control ◽  
Invariant Manifolds ◽  
Basin Of Attraction ◽  
High Energy ◽  
High Dimensional ◽  
Piezoelectric Energy Harvesting ◽  
Piezoelectric Energy ◽  
Large Period

Linear cantilevered piezoelectric energy harvesters typically rely on excitation around a resonance frequency for peak operation. Compounding the problem, typical ambient environments either vary dynamically in time or possess energy distributed across a wide spectrum of frequencies. Nonlinear broadband techniques have been implemented with success, but rely on chance that steady-state high energy orbits result as opposed to the low energy or chaotic trajectories that coexist in the basin of attraction. This work aims to implement two high dimensional chaotic controllers for large period orbits located within the chaotic attractor. The first control law is defined using traditional OGY, while the second uses the principles of invariant manifolds and is therefore independent of the system Jacobian. Comparison of the two control methods aims to show that invariant principles are less computationally intensive and result in equivalent stabilized orbits. Furthermore, the only necessary measurement for control design is a single time series representing a state of the system. This article compares two methods of chaos control and their ability to stabilize a large period orbit within the chaotic attractor for improved broadband piezoelectric energy harvesting.

scholarly journals Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields

2011 ◽  
Vol 29 (4) ◽  
pp. 1309-1344 ◽  
Author(s):  
Pablo Aguirre ◽  
◽  
Eusebius J. Doedel ◽  
Bernd Krauskopf ◽  
Hinke M. Osinga ◽  
...  
Keyword(s):  
Vector Fields ◽  
Invariant Manifolds ◽  
Two Dimensional ◽  
Global Bifurcations

scholarly journals THE OSEBERG TRANSITION: VISUALIZATION OF GLOBAL BIFURCATIONS FOR THE KURAMOTO–SIVASHINSKY EQUATION

2001 ◽  
Vol 11 (01) ◽  
pp. 1-18 ◽  
Author(s):  
M. E. JOHNSON ◽  
M. S. JOLLY ◽  
I. G. KEVREKIDIS
Keyword(s):  
Dimensionality Reduction ◽  
Periodic Solutions ◽  
Invariant Manifolds ◽  
Inertial Manifolds ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Approximate Inertial Manifolds ◽  
Numerical Bifurcation ◽  
Sivashinsky Equation

We present and discuss certain global bifurcations involving the interaction of one- and two-dimensional invariant manifolds of steady and periodic solutions of the Kuramoto–Sivashinsky equation. Numerical bifurcation calculations, dimensionality reduction using approximate inertial manifolds/forms, as well as approximation and visualization of invariant manifolds are combined in order to characterize what we term the "Oseberg transition".

scholarly journals GEOMETRY OF HOMOCLINIC CONNECTIONS IN A PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM

2007 ◽  
Vol 17 (04) ◽  
pp. 1151-1169 ◽  
Author(s):  
MARIAN GIDEA ◽  
JOSEP J. MASDEMONT
Keyword(s):  
Symbolic Dynamics ◽  
Body Problem ◽  
Invariant Manifolds ◽  
Libration Point ◽  
Homoclinic Orbits ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lyapunov Orbits ◽  
Homoclinic Connections ◽  
Three Body

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

On the Dynamics Near a Homoclinic Network to a Bifocus: Switching and Horseshoes

2015 ◽  
Vol 25 (11) ◽  
pp. 1530030 ◽  
Author(s):  
Santiago Ibáñez ◽  
Alexandre Rodrigues
Keyword(s):  
Invariant Manifolds ◽  
Switching Behavior ◽  
Nondegeneracy Condition ◽  
Two Dimensional

We study a homoclinic network associated to a nonresonant hyperbolic bifocus. It is proved that on combining rotation with a nondegeneracy condition concerning the intersection of the two-dimensional invariant manifolds of the equilibrium, switching behavior is created: close to the network, there are trajectories that visit the neighborhood of the bifocus following connections in any prescribed order. We discuss the existence of suspended horseshoes which accumulate on the network and the relation between these horseshoes and the switching behavior.

BASIN BIFURCATIONS OF TWO-DIMENSIONAL NONINVERTIBLE MAPS: FRACTALIZATION OF BASINS

1994 ◽  
Vol 04 (02) ◽  
pp. 343-381 ◽  
Author(s):  
C. MIRA ◽  
D. FOURNIER-PRUNARET ◽  
L. GARDINI ◽  
H. KAWAKAMI ◽  
J.C. CATHALA
Keyword(s):  
Phase Plane ◽  
The Other ◽  
Two Dimensional ◽  
Basin Boundary ◽  
Critical Curves ◽  
Noninvertible Maps

Properties of the basins of noninvertible maps of a plane are studied using the method of critical curves. Different kinds of basin bifurcation, some of them leading to basin boundary fractalization are described. More particularly the paper considers the simplest class of maps that of a phase plane which is made up of two regions, one with two preimages, the other with no preimage.

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