scholarly journals Universal Features of Tangent Bifurcation

1985 ◽  
Vol 38 (1) ◽  
pp. 1 ◽  
Author(s):  
R Delbourgo ◽  
BG Kenny
Keyword(s):  
Renormalization Group ◽  
Limit Cycles ◽  
Period Doubling ◽  
Accumulation Point ◽  
Universal Functions ◽  
One Dimensional ◽  
Chaotic Region ◽  
Renormalization Group Equations ◽  
Tangent Bifurcation ◽  
One Dimensional Maps

We exhibit certain universal characteristics of limit cycles pertaining to one-dimensional maps in the 'chaotic' region beyond the point of accumulation connected with period doubling. Universal, Feigenbaum-type numbers emerge for different sequences, such as triplication. More significantly we have established the existence of different classes of universal functions which satisfy the same renormalization group equations, with the same parameters, as the appropriate accumulation point is reached.

ROUTE TO CHAOS VIA INVERSE CASCADE OF CONTINUOUS BIFURCATIONS

1995 ◽  
Vol 05 (01) ◽  
pp. 123-132 ◽  
Author(s):  
M. GUTMAN ◽  
V. GONTAR
Keyword(s):  
Period Doubling ◽  
Scaling Behavior ◽  
One Dimensional ◽  
Inverse Cascade ◽  
Discontinuous Bifurcation ◽  
Universal Properties ◽  
Route To Chaos ◽  
Piecewise Continuous ◽  
One Dimensional Maps

A route to chaos via an inverse cascade of continuous bifurcations that arithmetically reduce the period of successive orbits has been obtained for piecewise continuous one-dimensional maps. We have studied the mechanism of these bifurcations and established that their scaling behavior is governed by constants with new universal properties. The possibility of obtaining discontinuous bifurcation from any selected orbit of a cascade of period-doubling to any orbit of inverse cascade has been demonstrated.

scholarly journals Type-II intermittency in a class of two coupled one-dimensional maps

2000 ◽  
Vol 5 (3) ◽  
pp. 233-245 ◽  
Author(s):  
J. Laugesen ◽  
E. Mosekilde ◽  
T. Bountis ◽  
S. P. Kuznetsov
Keyword(s):  
Period Doubling ◽  
Coupled System ◽  
Two Dimensions ◽  
Type Ii ◽  
One Dimensional ◽  
System A ◽  
Subcritical Hopf Bifurcation ◽  
Theoretical Predictions ◽  
The Individual ◽  
One Dimensional Maps

The paper shows how intermittency behavior of type-II can arise from the coupling of two one-dimensional maps, each exhibiting type-III intermittency. This change in dynamics occurs through the replacement of a subcritical period-doubling bifurcation in the individual map by a subcritical Hopf bifurcation in the coupled system. A variety of different parameter combinations are considered, and the statistics for the distribution of laminar phases is worked out. The results comply well with theoretical predictions. Provided that the reinjection process is reasonably uniform in two dimensions, the transition to type-II intermittency leads directly to higher order chaos. Hence, this transition represents a universal route to hyperchaos.

scholarly journals METRIC PROPERTIES OF CHAOTIC REGION IN ONE-DIMENSIONAL MAPS

1984 ◽  
Vol 33 (3) ◽  
pp. 341
Author(s):  
WANG YOU-QIN ◽  
CHEN SHI-GANG
Keyword(s):  
One Dimensional ◽  
Metric Properties ◽  
Chaotic Region ◽  
One Dimensional Maps
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