Spectral properties of the period-doubling operator
Victor Varin
备注：
This article was declined by "Nonlinearity" on the ground of "Conflict of Interests", since it criticizes implicitly some members of the editorial board
We compute the spectrum of the Feigenbaum period-doubling operator in the space of bounded analytical functions in an ellipse. The spectral properties of the period-doubling operator in this space are not the same as in the space of even analytical functions. In particular, it was found that the dimension of the unstable manifold is not one (Feigenbaum's conjecture), but three. We analyze several articles devoted to this problem and compare different approaches and algorithms.
We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a ``Perron-Frobenius type operator'', to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point.
The period-doubling sequence is one of the most well-known aperiodic $0$-$1$ sequences. In this paper, a complete description of its symbolic recurrence plot is given, and formulas for asymptotic values of basic recurrence quantifiers are derived.
Explicit formulas for complexity and unique invariant measure of the period-doubling subshift can be derived from those for the Thue-Morse subshift, obtained by Brlek, De Luca and Varricchio, and Dekking. In this note we give direct proofs based on combinatorial properties of the period-doubling sequence. We also derive explicit formulas for correlation integral and other recurrence characteristics of the period-doubling subshift. As a corollary we obtain that the determinism of this subshift converges to 1 as the distance threshold approaches 0.
We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that, with sufficient non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics are completely c…
The Henon family has been shown to have period-doubling cascades. We show here that the same occurs for a much larger class: Large perturbations do not destroy cascades. Furthermore, we can classify the period of a cascade in terms of the set of orbits it contains, and count the number of cascades of each period. This class of families extends a general theory explaining why cascades occur.
Based on numerical results of a Silnikov equation, three period-doubling cascades, corresponding respectively to three different characters of the rotation number of a limit closed orbit, are studied, and the Feigenbaum constant is used successfully in the estimation of the critical parameter values for the period-doubling bifurcation. The conceptions of separaror and pseudo-attractor are also introduced in the discussion part of this paper.
A general procedure which defines a partial ordering of cyclic permutations induced by continuous maps is known for constructing immediate successors to a cycle. We expound on this procedure in terms of labelled digraphs and characteristic polynomials then apply this study to period doubling, the most common route to chaos for a nonlinear dynamical system.
We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point, i.e., the combinatorial description of the boundary of chaos coincides with the topological description. We also show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable.
We consider the conservative Hénon family at the period-doubling bifurcation of its fixed point and demonstrate that the separatrices of the fixed saddle point nearing the bifurcation split exponentially: given that $λ_+$ is the smaller of the eigenvalues of the saddle point, the angle between the separatrices along the homoclinic orbit satisfies $$\sin α= O(e^{-{π^2 \over \log |λ_+|}})+ O\left( e^{-2 (1-κ) {π^2 \over \log |λ_+|}} \right),$$ for any positive $κ<1$.
The Feigenbaum-Cvitanovic equation -lambda * g(x)= g(g(lambda * x)) is solved over the interval 0 <= x <= 1 with a Chebyshev series representation of g(x). Accurate expansion coefficients are tabulated for solutions g(x) = 1+O(x^z) with even exponents from z=2 up to z=14.
We consider the infinite one-sided sequence generated by the period-doubling substitution $σ(a,b)=(ab,aa)$, denoted by $\mathbb{D}$. Since $\mathbb{D}$ is uniformly recurrent, each factor $ω$ appears infinite many times in the sequence, which is arranged as $ω_p$ $(p\ge 1)$. Let $r_p(ω)$ be the $p$-th return word over $ω$. The main result is: for each factor $ω$, the sequence $\{r_p(ω)\}_{p\geq1}$ is $Θ_1$ or $Θ_2$, which are substitutive sequences and determined completely in this paper.
We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and its reducibility. This reveals that the parameter values for which the invariant curve doubles its period are contained in regions of the parameter space where the invariant curve is reducible. Then we present two additional studies to explain this fact. In first place we consider the images and th…
In this paper we show that the invariant Cantor set of period doubling type of any infinitely renormalizable area-preserving map in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point is always contained in a Lipschitz curve but never contained in a smooth curve. This extends previous results by de Carvalho, Lyubich and Martens about strongly dissipative maps of the plane close to unimodal maps to the area-preserving setting. The method used for constructing the Lipschitz curve is very similar to the method used in the dissipative case but proving the nonexistence of…
The appearance of numerous period-doubling cascades is among the most prominent features of {\bf parametrized maps}, that is, smooth one-parameter families of maps $F:R \times {\mathfrak M} \to {\mathfrak M}$, where ${\mathfrak M}$ is a smooth locally compact manifold without boundary, typically $R^N$. Each cascade has infinitely many period-doubling bifurcations, and it is typical to observe -- such as in all the examples we investigate here -- that whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic $F$. We illustrate this …
In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\{ T_t(z)=i t\tan z\}_{0< t\leq π}$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as $t$ increases we find single instances of "period quadrupling", "period splitting" and standard "period doubling"; there follows a general pattern of "period merging" where two attracting cycles of period $2^n$ "merge" into one attracting cycle of period $2^{n+1}$, and "cycle doubling"…
The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser in the nineteen-seventieth to study the asymptotic small scale geometry of the attractor of one-dimensional systems which are at the transition from simple to chaotic dynamics. This geometry turns out to not depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point which is also hyperbolic among generic smooth enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicit…
In this study, replication of a period-doubling cascade in coupled systems with delay is rigorously proved under certain assumptions, which guarantee the existence of bounded solutions and replication of sensitivity. A novel definition for replication of sensitivity is utilized, in which the proximity of solutions is considered in an interval instead of a single point. Examples with simulations supporting the theoretical results concerning sensitivity and period-doubling cascade are provided.
We consider the infinite one-sided sequence over alphabet $\{a,b\}$ generated by the period-doubling substitution $σ(a)=ab$ and $σ(b)=aa$, denoted by $\mathbb{D}$. Let $r_p(ω)$ be the $p$-th return word of factor $ω$. The main result of this paper is twofold. (1) For any factor $ω$ in $\mathbb{D}$, the return word sequence $\{r_p(ω)\}_{p\geq1}$ is $Θ_1$ or $Θ_2$. Both of them are substitutive sequences and determined completely in this paper. (2) For any factor $ω$ in $Θ_1$ (resp. $Θ_2$), the return word sequence $\{r_p(ω)\}_{p\geq1}$ is still $Θ_1$ or $Θ_2$. We call it the reflexivity propert…
In the present study, we investigate the dynamics of impulsive differential equations driven by a chaotic system. We rigorously prove that, likewise the drive, the response impulsive system is also chaotic. Our results are based on the presence of sensitivity and infinitely many unstable periodic motions. The theoretical results are supported by simulations.
It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a "universal" area-preserving map $F_*$ -- a map with orbits of all binary periods $2^k, k \in \fN$. In this paper, we consider maps in some neighbourhood of $F_*$ and study their dynamics. We first demonstrate that the map $F_*$ admits a "bi-infinite heteroclinic tangle": a sequence of periodic points $\{z_k\}$, $k \in \fZ$, |z_k| \converg…
It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\fR}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a "universal" area-preserving map $F_*$ -- a map with orbits of all binary periods $2^k, k \in \fN$. In this paper, we consider {\it infinitely renormalizable} maps -- maps on the renormalization stable manifold in some neighborhood of $F_*$ -- and study their dynamics. For all such infinitely renormalizable maps in a neighborhood of the fi…
We explore fundamental questions about the renormalization group through a detailed re-examination of Feigenbaum's period doubling route to chaos. In the space of one-humped maps, the renormalization group characterizes the behavior near any critical point by the behavior near the fixed point. We show that this fixed point is far from unique, and characterize a submanifold of fixed points of alternative RG transformations. We build on this framework to systematically distinguish and analyze the allowed singular and `gauge' (analytic and redundant) corrections to scaling, explaining numerical r…
We propose an extension of the one dimensional (doubling) renormalization operator to the case of maps on the cylinder. The kind of maps considered are commonly referred as quasi-periodic forced one dimensional maps. We prove that the fixed point of the one dimensional renormalization operator extends to a fixed point of the quasi-periodic forced renormalization operator. We also prove that the operator is differentiable around the fixed point and we study its derivative. Then we consider a two parametric family of quasi-periodically forced maps which is a unimodal one dimensional map with a f…
Unlike classical bifurcations, border-collision bifurcations occur when, for example, a fixed point of a continuous, piecewise $\mathcal{C}^{1}$ map crosses a boundary in state space. Although classical bifurcations have been much studied, border-collision bifurcations are not well understood. This paper considers a particular class of border-collision bifurcations, i.e., border-collision period-doubling bifurcations. We apply a subharmonic perturbation to the bifurcation parameter, which is also known as alternate pacing, and we investigate the response under such pacing near the original bif…
It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $\field{R}^2$. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date. We argue that the period doubling renormalization fixed p…
The period doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated. The Jacobian Rigidity Conjecture says that the period doubling Cantor sets of two-dimensional Henon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, e.g. the one-dimensional case. The other extreme case is when the maps preserve area, e.g. the average Jacobian is one. Indeed, the period doubling Cantor set of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalizati…
Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asymptotics near the origin, thus providing a family of extremal examples of singular measures, including the Thue--Morse measure.
Spiral wave patterns observed in models of cardiac arrhythmias and chemical oscillations develop alternans and stationary line defects, which can both be thought of as period-doubling instabilities. These instabilities are observed on bounded domains, and may be caused by the spiral core, far-field asymptotics, or boundary conditions. Here, we introduce a methodology to disentangle the impacts of each region on the instabilities by analyzing spectral properties of spiral waves and boundary sinks on bounded domains with appropriate boundary conditions. We apply our techniques to spirals formed …
In this paper, we show the following: the Hausdorff dimension of the spectrum of period-doubling Hamiltonian is bigger than $\log α/\log 4$, where $α$ is the Golden number; there exists a dense uncountable subset of the spectrum such that for each energy in this set, the related trace orbit is unbounded, which is in contrast with a recent result of Carvalho (Nonlinearity 33, 2020); we give a complete characterization for the structure of gaps and the gap labelling of the spectrum. All of these results are consequences of an intrinsic coding of the spectrum we construct in this paper.
We consider a family of strongly-asymmetric unimodal maps $\{f_t\}_{t\in [0,1]}$ of the form $f_t=t\cdot f$ where $f\colon [0,1]\to [0,1]$ is unimodal, $f(0)=f(1)=0$, $f(c)=1$ is of the form and $$f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)& \mbox{ for }x<c, \\ 1-K_+|x-c|^β+ o(|x-c|^β) &\mbox{ for }x>c, \end{array}\right. $$ where we assume that $β>1$. We show that such a family contains a Feigenbaum-Coullet-Tresser $2^\infty$ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $2^\infty$ map turn out to be super-exponential and…
Predicting the onset of period-doubling bifurcations via dominant eigenvalue extracted from autocorrelation
Faculty of Science, Kunming University of Science and Technology（昆明理工大学理学院）；School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Center for Mathematical Science, Huazhong University of Science and Technology（华中科技大学数学中心）；Steklov-Wuhan Institute for Mathematical Exploration, Huazhong University of Science and Technology（华中科技大学Stecklov武汉数学探索中心）；Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Systems, Great Bay University（大湾区大学广东省数学与神经动力系统重点实验室）；Department of Mathematics and Department of Physics, Great Bay University（大湾区大学数学系和物理系）
