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 A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

2005 ◽  
Vol 15 (03) ◽  
pp. 763-791 ◽  
Author(s):  
B. KRAUSKOPF ◽  
H. M. OSINGA ◽  
E. J. DOEDEL ◽  
M. E. HENDERSON ◽  
J. GUCKENHEIMER ◽  
...  
Keyword(s):  
Vector Field ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Lorenz System ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
The Lorenz System ◽  
Global Invariant

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

scholarly journals NONORIENTABLE MANIFOLDS IN THREE-DIMENSIONAL VECTOR FIELDS

2003 ◽  
Vol 13 (03) ◽  
pp. 553-570 ◽  
Author(s):  
HINKE M. OSINGA
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Period Doubling ◽  
Building Blocks ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Three Dimensional Vector Field

It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a Möbius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.

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 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).

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.

Sign in / Sign up

Export Citation Format

Share Document