Flat-topped one-dimensional maps have been used in the control of chaos in one-dimensional dynamical systems. In these applications, this mechanism is known as simple limiter control. In this paper, we will consider the introduction of simple limiters u in the tent map, according to a time-dependent scheme defined by a binary sequence s, the iteration pattern. We will define local and Milnor attractors in this non-autonomous context and study the dependence of their existence and coexistence on the value of the limiter u and on the pattern s. Using symbolic dynamics, we will be able to characterize the families of pairs (u,s) for which these attractors exist and coexist, as well as fully describe them. We will observe that this non-autonomous context provides a richness of behaviors that are not possible in the autonomous case.

The behavior of orbits for iterated flat top maps has been widely studied since the dawn of discrete dynamics as a research field. However, little is known about orbit behavior if the map changes along with the iterations. In this work we consider a family of flat top tent maps and investigate in which ways the iteration pattern (symbolic template) can affect the structure of the bifurcation scenarios.

In this work we will consider a family of nonautonomous dynamical systems x(k)(+1) = f(k)(x(k), lambda), lambda is an element of [-1, 1] (N0), generated by a one-parameter family of flat-topped tent maps g(alpha) (x), i.e., f(k)(x, lambda) = g(lambda k) (x) for all k is an element of N-0. We will reinterpret the concept of attractive periodic orbit in this context, through the existence of some periodic, invariant and attractive nonautonomous sets and establish sufficient conditions over the parameter sequences for the existence of such periodic attractors.

In this paper we study a 2-parameter family of 2-periodic nonautonomous systems generated by the alternate iteration of two stunted tent maps. Using symbolic dynamics, renormalization and star product in the nonautonomous setting, we compute the convergence rates of sequences of parameters obtained through consecutive star products/renormalizations, extending in this way Feigenbaum's convergence rates. We also define sequences in the parameter space corresponding to anharmonic period doubling bifurcations and compute their convergence rates. In both cases we show that the convergence rates are independent of the initial point, concluding that the nonautonomous setting has universal properties of the type found by Feigenbaum in families of autonomous systems. (C) 2019 Elsevier B.V. All rights reserved.

In this paper, we will consider a family of 2-periodic nonautonomous dynamical systems, generated by the alternate iteration of two stunted tent maps and study its bifurcation skeleton. We will describe the bifurcation phenomena along and around the bones accomplished with the combinatorial data furnished by the respective symbolic dynamics.

Based on symbolic dynamics of Lorenz maps, we prove that, provided one conjecture due to Morton is true, then Lorenz knots associated to orbits of points in the renormalization intervals of Lorenz maps with reducible kneading invariant of type (X, Y)*S, where the sequences X and Y are Farey neighbors verifying some conditions, are hyperbolic.

We introduce the notions of equilibrium distribution and time of convergence in discrete non-autonomous graphs. Under some conditions we give an estimate to the convergence time to the equilibrium distribution using the second largest eigenvalue of some matrices associated with the system.

In this work, we associate a p-periodic nonautonomous graph to each p-periodic nonautonomous Lorenz system with finite critical orbits. We develop Perron–Frobenius theory for nonautonomous graphs and use it to calculate their entropy. Finally, we prove that the topological entropy of a p-periodic nonautonomous Lorenz system is equal to the entropy of its associated nonautonomous graph.

We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots coded by some of these families of words are hyperbolic, by showing that they are neither satellites nor torus knots and making use of Thurston’s theorem. Infinite families of hyperbolic Lorenz knots are generated in this way, to our knowledge, for the first time. The techniques used can be generalized to study other families of Lorenz knots.

For an interval map, the poles of the Artin–Mazur zeta function provide topological invariants which are closely connected to topological entropy. It is known that for a time-periodic nonautonomous dynamical system F with period p, the p-th power [ζF(z)]p of its zeta function is meromorphic in the unit disk. Unlike in the autonomous case, where the zeta function ζf(z) only has poles in the unit disk, in the p-periodic nonautonomous case [ζF(z)]p may have zeros. In this paper we introduce the concept of spectral invariants of p-periodic nonautonomous discrete dynamical systems and study the role played by the zeros of [ζF(z)]p in this context. As we will see, these zeros play an important role in the spectral classification of these systems.

Lorenz knots and links have been an area of research for over thirty years. Their study combines several fields of mathematics, including topology, geometry, and dynamical systems. The introduction of the Lorenz template permitted a dimension reduction, the definition of the one-dimensional Lorenz map, and the use of symbolic dynamics to code its orbits. Meanwhile, Thurston’s geometrization theorem implies that all Lorenz knots can be classified as torus knots, satellites, or hyperbolic knots. Based on a list of hyperbolic knots presented by Birman and Kofman, we define three families of Lorenz knots, compute their hyperbolic volume using the topology and geometry software SnapPy and use the results to present conjectures concerning their hyperbolicity.

We apply kneading theory to describe the knots and links generated by the iteration of renormalizable nonautonomous dynamical systems with reducible kneading invariants, in terms of the links corresponding to each factor. As a consequence we obtain explicit formulas for the genus for this kind of knots and links.

The purpose of this paper was to introduce the symbolic formalism based on kneading theory, which allows us to study the renormalization of non-autonomous periodic dynamical systems.

We define nonautonomous graphs as a class of dynamic graphs in discrete time whose time-dependence consists in connecting or disconnecting edges. We study periodic paths in these graphs, and the associated zeta functions. Based on the analytic properties of these zeta functions we obtain explicit formulae for the number of n-periodic paths, as the sum of the nth powers of some specific algebraic numbers.

We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant $(K_f^-,K_f^+)=(X,Y)*(S,W)$, in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from the ones corresponding to the factors of the $*$-product. Using this result we obtain explicit formulas for the genus and the braid index of this renormalizable Lorenz knots and links. Then we obtain explicit formulas for sequences of these invariants, associated to sequences of renormalizable Lorenz maps with kneading invariant $(X,Y)*(S,W)^{*n}$, concluding that both grow exponentially. This is specially relevant, since it is known that topological entropy is constant on the archipelagoes of renormalization.

We present some recent results concerning the structure of renormalizable Lorenz links. Then we use these results to derive formulae for the computation of knot and link invariants. Finally we analyze the complexity of the algorithms obtained and compare it with the complexity of the algorithms derived from the definitions, obtaining a reduction from exponential complexity, in the classic algorithms, to linear in our algorithms.

We describe the Williams zeta functions and the twist zeta functions of sub-Lorenz templates generated by renormalizable Lorenz maps, in terms of the corresponding zeta-functions of the sub-Lorenz templates generated by the renormalized map and by the map that determines the renormalization type.

Given two different representations of a Lorenz link, we compare how they affect the computation of the multivariable Alexander polynomial. We also compare the Alexander polynomial with the trip number and genus. Our experimental results lead us to conjecture that, for Lorenz knots, the Alexander polynomial is an equivalent invariant to the pair (trip number, genus). Finally, we give a counterexample in the case of Lorenz links.

Bursting behavior is ubiquitous in physical and biological systems, specially in neural cells where it plays an important role in information processing. This activity refers to a complex oscillation characterized by a slow alternation between spiking behavior and quiescence. In this paper, the interesting phenomena which transpire when two cells are coupled together, is studied in terms of symbolic dynamics. More specifically, we characterize the topological entropy of a map used to examine the role of coupling on identical bursters. The strength of coupling leads to the introduction of a second topological invariant that allows us to distinguish isentropic dynamics. We illustrate the significant effect of the strength parameter on the topological invariants with several numerical results.

One of the interesting complex behaviors in many cell membranes is bursting, in which a rapid oscillatory state alternates with phases of relative quiescence. Although there is an elegant interpretation of many experimental results in terms of nonlinear dynamical systems, the dynamics of bursting models is not completely described. In the present paper, we study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics on the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.