TOPOLOGICAL WINDING NUMBERS FOR PERIOD-DOUBLING CASCADES

1996 ◽  
Vol 06 (01) ◽  
pp. 185-187
Author(s):  
CARSTEN KNUDSEN
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Period Doubling ◽  
Accumulation Point ◽  
Winding Number ◽  
Unimodal Maps ◽  
Forced Oscillators ◽  
Essential Properties ◽  
Limiting Value

We define the topological winding number for unimodal maps that share the essential properties of that of winding numbers for forced oscillators exhibiting period-doubling cascades. It is demonstrated how this number can be computed for any of the periodic orbits in the first period-doubling cascade. The limiting winding number at the accumulation point of the first period-doubling cascade is also derived. It is shown that the limiting value for the winding number ω∞ can be computed as the Farey sum of any two neighbouring topological winding numbers in the period-doubling cascade. The derivations are all based on symbolic dynamics and simple combinatorics.

THE WINDING-NUMBER LIMIT OF PERIOD-DOUBLING CASCADES DERIVED AS FAREY-FRACTION

1994 ◽  
Vol 04 (04) ◽  
pp. 999-1002 ◽  
Author(s):  
VOLKER ENGLISCH ◽  
WERNER LAUTERBORN
Keyword(s):  
Period Doubling ◽  
Accumulation Point ◽  
Winding Number ◽  
Farey Fraction

It is shown, that the winding number w∞ at the accumulation point of a period-doubling cascade can be derived as a Farey sum from each two adjacent winding numbers wk and wk+1 of the cascade.

SYMBOLIC DYNAMICS OF UNIMODAL MAPS REVISITED

1989 ◽  
Vol 03 (02) ◽  
pp. 235-246 ◽  
Author(s):  
BAI-LIN HAO ◽  
WEI-MOU ZHENG
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Composition Rule ◽  
Unimodal Maps ◽  
Chaotic Orbits ◽  
Symbolic Sequences ◽  
Solving Equations

Symbolic dynamics of unimodal maps has been recast into a more natural and down-to-numbers way. The median itineraries are built without such artificial constructions as "antiharmonics" and "harmonics" by making use of the newly established periodic window theorem. A generalized composition rule extends the *-composition introduced by Derrida, Gervois and Pomeau. Periodic as well as chaotic orbits are described systematically. The location of all superstable periodic orbits and band-merging points may be calculated by solving equations obtained directly from the corresponding symbolic sequences.

The Dynamics of Symmetric Bimodal Maps

1997 ◽  
Vol 07 (12) ◽  
pp. 2735-2744 ◽  
Author(s):  
Thomas Lofaro
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Period Doubling ◽  
General Setting ◽  
The Family ◽  
Pitchfork Bifurcations ◽  
Parameter Values ◽  
Kneading Theory ◽  
Period Doubling Bifurcations

The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, fα are discussed. We show that for a large class of parameter values the dynamics of fα can be described via an identification with a unimodal map uα. In this parameter regime, a periodic orbit of period 2n + 1 of uα corresponds to a periodic orbit of period 4n + 2 for fα. A periodic orbit of period 2n of uα corresponds to a pair of distinct periodic orbits also of period 2n for fα. In a more general setting we describe the genealogy of periodic orbits in the family fα using symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.

scholarly journals Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

2020 ◽  
Vol 379 (1) ◽  
pp. 103-143
Author(s):  
Oleg Kozlovski ◽  
Sebastian van Strien
Keyword(s):  
Scaling Laws ◽  
Scaling Law ◽  
Period Doubling ◽  
Cantor Sets ◽  
Unimodal Maps ◽  
Renormalization Theory

Abstract We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

FROM SYMBOLIC DYNAMICS TO A DIGITAL APPROACH

2001 ◽  
Vol 11 (06) ◽  
pp. 1683-1694 ◽  
Author(s):  
K. KARAMANOS
Keyword(s):  
Symbolic Dynamics ◽  
Chaotic Attractors ◽  
Unimodal Maps ◽  
Quantitative Manner ◽  
The Asymmetry

We show that the numbers generated by the symbolic dynamics of Feigenbaum attractors are transcendental. Due to the asymmetry of the chaotic attractors of unimodal maps around the maximum in the general case, a standard conjecture, that the occurrence of chaos is related to the transcendence of the number defined by the corresponding symbolic dynamics is reassessed and formulated in a quantitative manner. It is concluded that transcendence may provide an appropriate measure of complexity.

Chaotic scattering on a double well: Periodic orbits, symbolic dynamics, and scaling

1993 ◽  
Vol 3 (4) ◽  
pp. 475-485 ◽  
Author(s):  
Vincent Daniels ◽  
Michel Vallières ◽  
Jian‐Min Yuan
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Scattering

scholarly journals Chaotic period doubling

2009 ◽  
Vol 29 (2) ◽  
pp. 381-418 ◽  
Author(s):  
V. V. M. S. CHANDRAMOULI ◽  
M. MARTENS ◽  
W. DE MELO ◽  
C. P. TRESSER
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Chaotic Dynamics ◽  
A Priori ◽  
Period Doubling ◽  
Small Scale ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Renormalization Operator ◽  
Renormalization Theory

AbstractThe period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space ofC2+αunimodal maps, forα>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space ofC2unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to geta prioribounds. In this smoother class, calledC2+∣⋅∣, the failure of hyperbolicity is tamer than inC2. Things get much worse with just a bit less smoothness thanC2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space ofC1+Lipunimodal maps has infinite topological entropy.

GLOBAL BEHAVIOR OF THE CHARGED ISOSCELES THREE-BODY PROBLEM

1994 ◽  
Vol 04 (04) ◽  
pp. 865-884 ◽  
Author(s):  
PAU ATELA ◽  
ROBERT I. McLACHLAN
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Body Problem ◽  
Kepler Problem ◽  
Global Bifurcation ◽  
Three Body Problem ◽  
Anisotropic Kepler Problem ◽  
The One ◽  
Three Body ◽  
Limiting Value

We study the global bifurcation diagram of the two-parameter family of ODE’s that govern the charged isosceles three-body problem. (The classic isosceles three-body problem and the anisotropic Kepler problem (two bodies) are included in the same family.) There are two major sources of periodic orbits. On the one hand the “Kepler” orbit, a stable orbit exhibiting the generic bifurcations as the multiplier crosses rational values. This orbit turns out to be the continuation of the classical circular Kepler orbit. On the other extreme we have the collision-ejection orbit which exhibits an “infinite-furcation.” Up to a limiting value of the parameter we have finitely many periodic orbits (for each fixed numerator in the rotation number), passed this value there is a sudden birth of an infinite number of them. We find that these two bifurcations are remarkably connected forming the main “skeleton” of the global bifurcation diagram. We conjecture that this type of global connection must be present in related problems such as the classic isosceles three-body problem and the anisotropic Kepler problem.

scholarly journals Parabolic limits of renormalization

2000 ◽  
Vol 20 (1) ◽  
pp. 173-229 ◽  
Author(s):  
BENJAMIN HINKLE
Keyword(s):  
Fixed Point ◽  
Period Doubling ◽  
Unit Interval ◽  
Local Analysis ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Combinatorial Description ◽  
Combinatorial Rigidity

A unimodal map $f:[0,1] \to [0,1]$ is renormalizable if there is a sub-interval $I \subset [0,1]$ and an $n > 1$ such that $f^n|_I$ is unimodal. The renormalization of $f$ is $f^n|_I$ rescaled to the unit interval.We extend the well-known classification of limits of renormalization of unimodal maps with bounded combinatorics to a classification of the limits of renormalization of unimodal maps with essentially bounded combinatorics. Together with results of Lyubich on the limits of renormalization with essentially unbounded combinatorics, this completes the combinatorial description of limits of renormalization. The techniques are based on the towers of McMullen and on the local analysis around perturbed parabolic points. We define a parabolic tower to be a sequence of unimodal maps related by renormalization or parabolic renormalization. We state and prove the combinatorial rigidity of bi-infinite parabolic towers with complex bounds and essentially bounded combinatorics, which implies the main theorem.As an example we construct a natural unbounded analogue of the period-doubling fixed point of renormalization, called the essentially period-tripling fixed point.

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