Structure change of stable and unstable manifolds in two-dimensional maps: Period-doubling bifurcation

1992 ◽  
Vol 2 (2) ◽  
pp. 139-146 ◽  
Author(s):  
Yoshihiro Yamaguchi ◽  
Kiyotaka Tanikawa
Keyword(s):  
Period Doubling ◽  
Structure Change ◽  
Period Doubling Bifurcation ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds

PERIOD-DOUBLING GEOMETRY OF THE BELOUSOV-ZHABOTINSKY REACTION DYNAMICS

1996 ◽  
Vol 06 (07) ◽  
pp. 1267-1279 ◽  
Author(s):  
JICHANG WANG ◽  
F. HYNNE ◽  
P. GRAAE SØRENSEN
Keyword(s):  
Stable Manifold ◽  
Period Doubling ◽  
Quantitative Information ◽  
Stable And Unstable Manifolds ◽  
Bz Reaction ◽  
Positive Orthant ◽  
Model Independent ◽  
Unstable Manifolds ◽  
Belousov Zhabotinsky Reaction ◽  
Concentration Space

The intricate geometry of stable and unstable manifolds of a saddle cycle arising from a super-critical period-doubling bifurcation is explored experimentally for the Belousov-Zhabotinsky (BZ) reaction in a CSTR (Continuous flow Stirred Tank Reactor). We find clear experimental evidence that the stable manifold winds round the stable period-doubled orbit arising at the bifurcation. The stable manifold is probed experimentally by perturbations from the period-doubled limit cycle. The method provides a model-independent experimental test for species essential for the complexity, as well as quantitative information about the geometry of the limit cycles, the associated manifolds, and their embedding in the concentration space. The results are supported by simulations of the experiments with a four-dimensional model of the BZ reaction. Here stable manifold has several branches showing some tendency of curling. However the branches may end on the boundary of the positive orthant of the concentration space.

Bifurcations of one- and two-dimensional maps

1984 ◽  
Vol 311 (1515) ◽  
pp. 43-102 ◽  
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Jacobian Determinant ◽  
Two Dimensional ◽  
Hénon Map ◽  
Stable And Unstable Manifolds ◽  
Henon Map ◽  
One Dimensional ◽  
The Family ◽  
Area Preserving

We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C 3 map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y 2 or (in different coordinates) f(y) = Ay(1 —y), in which case F^ e is the Henon map. The maps F e have constant Jacobian determinant e and, as e -> 0, collapse to the family f^. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F/ ue , for small e . Moreover, we are able to extend these results to the area preserving family F/u. 1 , thereby obtaining (partial) bifurcation sets in the (/u, e)-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii’s theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the ( /u, e ) -parameter plane between e = 0 and e = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F /u.e and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.

scholarly journals COMPUTING TWO-DIMENSIONAL GLOBAL INVARIANT MANIFOLDS IN SLOW–FAST SYSTEMS

2007 ◽  
Vol 17 (03) ◽  
pp. 805-822 ◽  
Author(s):  
J. P. ENGLAND ◽  
B. KRAUSKOPF ◽  
H. M. OSINGA
Keyword(s):  
Invariant Manifolds ◽  
Multiple Time Scales ◽  
Test Case ◽  
Multiple Time ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Boundary Problems ◽  
Case Examples ◽  
Unstable Manifolds ◽  
Global Invariant

We present the GLOBALIZEBVP algorithm for the computation of two-dimensional stable and unstable manifolds of a vector field. Specifically, we use the collocation routines of AUTO to solve boundary problems that are used during the computation to find the next approximate geodesic level set on the manifold. The resulting implementation is numerically very stable and well suited for systems with multiple time scales. This is illustrated with the test-case examples of the Lorenz and Chua systems, and with a slow–fast model of a somatotroph cell.

Chaotic Mixing of Highly Viscous Liquids With Rectangular or Elliptical Rotors

2005 ◽  
Author(s):  
M. Erol Ulucakli
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Fixed Points ◽  
Stokes Flow ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Rectangular Cell ◽  
Viscous Liquids ◽  
Unstable Manifolds ◽  
Time Periodic

The objective of this research is to experimentally investigate various mixing regions in a two-dimensional Stokes flow driven by a rectangular or elliptical rotor. Flow occurs in a rectangular cell filled with a very viscous fluid. The Reynolds number based on rotor size is in the order of 0.5. The flow is time-periodic and can be analyzed, both theoretically and experimentally, by considering the Poincare map that maps the position of a fluid particle to its position one period later. The mixing regions of the flow are determined, theoretically, by the fixed points of this map, either hyperbolic or degenerate, and their stable and unstable manifolds. Experimentally, the mixing regions are visualized by releasing a blob of a passive dye at one of these fixed points: as the flow evolves, the blob stretches to form a streak line that lies on the unstable manifold of the fixed point.

A NEW ALGORITHM FOR COMPUTING ONE-DIMENSIONAL STABLE AND UNSTABLE MANIFOLDS OF MAPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250018 ◽  
Author(s):  
HUIMIN LI ◽  
YANGYU FAN ◽  
JING ZHANG
Keyword(s):  
Bisection Method ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Numerical Examples ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Correction Scheme ◽  
Accuracy Criterion ◽  
Tangent Component ◽  
Unstable Manifolds

A new algorithm is presented to compute one-dimensional stable and unstable manifolds of fixed points for both two-dimensional and higher dimensional diffeomorphism maps. When computing the stable manifold, the algorithm does not require the explicit expression of the inverse map. The global manifold is grown from a local manifold and one point is added at each step. The new point is located with a "prediction and correction" scheme, which avoids searching the computed part of the manifold with a bisection method and accelerates the searching process. By using the fact that the Jacobian transports derivatives along the orbit of the manifold, the tangent component of the manifold is determined and a new accuracy criterion is proposed to check whether the new point that defines the manifold is acceptable. The performance of the algorithm is demonstrated with several numerical examples.

Tangencies between stable and unstable manifolds

1992 ◽  
Vol 121 (1-2) ◽  
pp. 73-90 ◽  
Author(s):  
Flaviano Battelli ◽  
Kenneth J. Palmer
Keyword(s):  
Perturbation Theory ◽  
Homoclinic Orbit ◽  
Second Order ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Four Dimensions ◽  
Unstable Manifolds ◽  
Order Contact

SynopsisIn this paper perturbation theory is used to construct systems in four dimensions having two dimensional stable and unstable manifolds which touch along a homoclinic orbit but only with a second order contact.

PERTURBING TWO-DIMENSIONAL MAPS HAVING CRITICAL HOMOCLINIC ORBITS

1999 ◽  
Vol 09 (06) ◽  
pp. 1189-1195 ◽  
Author(s):  
FLAVIANO BATTELLI ◽  
CLAUDIO LAZZARI
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Homoclinic Orbits ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Hyperbolic Fixed Point ◽  
Unstable Manifolds ◽  
Planar Maps ◽  
Necessary And Sufficient ◽  
Critical Lines

Discrete planar maps such as xn+1=f(xn)+μ g(xn, μ), xn∈ℝ2, n∈ℤ, μ∈ℝ, are studied under the assumption that the unperturbed map xn+1=f(xn) has a critical homoclinic orbit to a hyperbolic fixed point. We give either necessary and sufficient conditions for a bifurcation from zero to two homoclinic orbits as the small parameter μ crosses zero. These conditions are stated in terms of geometrical objects such as critical lines, stable and unstable manifolds.

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