We derive an algorithm to determine recursively the lap number (minimal number of monotone pieces) of the iterates of unimodal maps of an interval with free end-points. The algorithm is obtained by the sign analysis of the itineraries of the critical point and of the boundary points of the interval map. We apply this algorithm to the estimation of the growth number and the topological entropy of maps with direct and reverse bifurcations.
In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.
The work described in this paper can be interpreted as an application of the order patterns of symbolic dynamics when dealing with unimodal maps. Specifically, it is shown how Gray codes can be used to estimate the probability distribution functions (PDFs) of the order patterns of parametric unimodal maps. Furthermore, these PDFs depend on the value of the parameter, what eventually provides a handle to estimate the parameter value from symbolic sequences (in form of Gray codes), even when the critical point depends on the parameter.
In 2010, Weixiao Shen pointed out to us that the proof of Theorem 3.2 of our 2002 paper in Ann ENS was flawed, and he kindly provided an argument to fix this proof. (We do not make claims on the lower bounds for the stationary density anymore.) In 2017, Wael Bahsoun, Christopher Bose, and Marks Ruziboev pointed out to us that the proof of Corollary 5.2 in the same paper is flawed. We explain how to recover Corollary 5.2 up to replacing the set $M'_q$ on p. 93 by a slightly smaller set (this does not affect the rest of the paper).
We consider C^2 families t->f_t of C^4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure m_t of f_t, as a function of t on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set D of CE parameters, and, for each s in D, a subset D0 of D of polynomially recurrent parameters containing s as a Lebesgue density point, and constants C>1, G >4, so that, for every 1/2-Holder function A (of 1/2-Holder norm |A|) and all t in D0, |\int A dm_t -\int A dm_s| < C |A| |t-s|^{1…
Ge, Rusjan, and Zweifel (J. Stat. Phys. 59, 1265 (1990)) introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A non-universal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. There are an infinite number of exceptional sequences, however, which lead to superexponential scaling. The ori…
The escape rate of a stochastic dynamical system can be found as an expansion in powers of the noise strength. In previous work the coefficients of such an expansion for a one-dimensional map were fitted to a general form containing a few parameters. These parameters were found to be related to the fractal structure of the repeller of the system. The parameter alpha, the "noise dimension", remains to be interpreted. This report presents new data for alpha showing that the relation to the dimensions is more complicated than predicted in earlier work and oscillates as a function of the map param…
We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f with unique acim mu, a perturbation X, and an observable phi. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functi…
We uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps characterized each by a Lyapunov exponent that diverges to minus infinity. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the preimages of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sens…
We examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity $z>1$. First we determine analytically the sensitivity to initial conditions $ξ_{t}$. Then we consider a renormalization group (RG) operation on the partition function $Z$ of the multifractal attractor that eliminates one half of the multifractal points each time it is applied. Invariance of $Z$ fixes a length-scale transformation factor $2^{-η}$ in terms of the generalized dimensions $D_β$. There exists a gap $Δη$ in the values of $η$ equal to $λ_{q}=1/(1-q)=D_{\inft…
We investigate the effects of linear and optimal nonlinear control in the simple case of one-dimensional unimodal maps. We show that linear feedback relates unimodal maps to invertible ``Henon-like'' maps. This observation should be useful in relating the considerable bodies of knowledge which exist for the two types of systems. In the case of the optimal nonlinear feedback scheme of de Sousa Vieira and Lichtenberg we investigate the relationship between controlled and uncontrolled maps, particularly the preservation of the period doubling route to chaos.
We explore the order in chaos by studying the geometric structure of time series that can be deduced from a visibility mapping. This visibility mapping associates to each point $(t,x(t))$ of the time series to its horizontal visibility basin, i.e. the number of points of the time series that can be seen horizontally from each respective point. We apply this study to the class of unimodal maps that give rise to equivalent bifurcation diagrams. We use the classical logistic map to illustrate the main results of this paper: there are visibility patterns in each cascade of the bifurcation diagram,…
The synchronization behavior of delay coupled chaotic smooth unimodal maps over a ring network with stochastic switching of links at every time step is reported in this paper. It is observed that spatiotemporal synchronization never appears for nearest neighbor connections; however, stochastic switching of connections with homogeneous delay $(τ)$ is capable of synchronizing the network to homogeneous steady state or periodic orbit or synchronized chaotically oscillating state depending on the delay parameter, stochasticity parameter and map parameters. Linear stability analysis of the synchron…
Recently, Buzzi showed in the compact case that the entropy map $f\rightarrow$ $h_{top}(f)$ is lower semi-continuous for all piecewise affine surface homeomorphisms. We prove that topological entropy for the Lozi maps can jump from zero to a value above 0.1203 as one crosses a particular parameter and hence it is not upper semi-continuous in general. Moreover, our results can be extended to a small neighborhood of this parameter showing the jump in the entropy occurs along a line segment in the parameter space.
We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations.
We recall that, at both the intermittency transitions and at the Feigenbaum attractor in unimodal maps of non-linearity of order $ζ>1$, the dynamics rigorously obeys the Tsallis statistics. We account for the $q$-indices and the generalized Lyapunov coefficients $λ_{q}$ that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the edge of chaos with the appearance of a special value for the entropic index $q$. The physical area of the Tsallis statistics is further probed by considering the dynamics near …
We present a method for computing the topological entropy of one-dimensional maps. As an approximation scheme, the algorithm converges rapidly and provides both upper and lower bounds.
The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this letter, two complementary proofs are given that the foliation of the space by invariant curves implies that the algebraic entropy is zero.
