Research paper
Convergence rates for sequences of bifurcation parameters of nonautonomous dynamical systems generated by flat top tent maps

https://doi.org/10.1016/j.cnsns.2019.105007Get rights and content

Highlights

  • We use symbolic dynamics to study a family of periodic nonautonomous tent systems.

  • Sequences of parameters corresponding to consecutive renormalizations are convergent.

  • These convergence rates only depend on the type of renormalization.

  • We obtain new period doubling sequences corresponding to non renormalizable systems.

  • The previous sequences of parameters converge with rate of order 2.

Abstract

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.

Introduction

The dynamics of discrete nonautonomous dynamical systems is an active field of research in many areas of applications. Periodic nonautonomous systems are a suitable theoretical framework for study systems in fluctuating environments. In particular, they have been used, for example, as models of populations with periodic forcing ([7] and previous papers of the author) or the DNA replication and repair (see [9]).

The interest in studying the dynamics of stunted tent maps goes back to the work of Metropolis et al. ([5]). These maps are good models to study related families of differentiable maps, since they have a simple translation to symbolic dynamics, covering all possible kneading data. Families of flat topped maps have been used to describe limiter control to chaos in electronic circuits and telecommunications (see [12]), in the study of biological populations (see [8]) and in cardiology, to control cardiac arrhythmia ([4]).

In this paper we study a 2-parameter family of 2-periodic nonautonomous systems generated by the alternate iteration of two stunted tent maps. The main results of the paper are contained in Theorems 5 and 6. In Theorem 5, using symbolic dynamics, we compute convergence rates for sequences of bifurcation parameters associated to renormalizable systems, in the spirit of some of those obtained by Feigenbaum ([2]), these rates are universal in the sense that they do not depend on the parameters where the sequence starts. Besides the expected Feigenbaum diagram properties, we have identified sequences of period doubling bifurcations, not associated to renormalizable systems and so not present in the autonomous context. These sequences can be seen as anharmonic period doubling sequences, (for results on this topic see [6] and references therein). The convergence rates of these last sequences are stated in Theorem 6.

The paper is organized as follows: In Section 2 we introduce nonautonomous 2-periodic stunted tent systems and their associated symbolic dynamics. In Section 3 we develop the tools needed in the following sections, namely *-product and renormalization, introduced in the nonautonomous setting in [3]. In Section 4 we compute the convergence rates of a sequence of parameters associated to renormalizable systems and prove its universality, in the sense that they do not depend on the parameters where the sequence starts, going back to the work proposed in [10] for a family of Lorenz maps. Finally, in Section 5 we study sequences of anharmonic period doubling bifurcations with identical convergence rates. The last section is devoted to the proofs of the main results of 3 *-Product and renormalization, 4 Convergence rates to 5.

Section snippets

Family of stunted tent maps

Consider the family of functions fv:[1,1][1,1], with parameter v[1,1], defined byfv(x)={12x,x>1v2v,v12x1v21+2x,x<v12with v[1,1].

The function fv is called stunted tent map and is obtained from the tent map T,T(x)={12x,x01+2x,x<0,taking fv constant in the interval [v12,1v2], (see Fig. 1).

The interval [v12,1v2] is called plateau. The family of stunted tent maps is denoted by T.

Let (vi)i[1,1]N0 be a 2-periodic sequence. The 2-periodic nonautonomous dynamical system defined by (

*-Product and renormalization

In [1] the authors introduced an inner composition law, the *-product, applied to unimodal families. In the autonomous setting it is well known the relation between systems where the kneading data is *-factorizable and renormalizable systems. The relation between the renormalizability of periodic nonautonomous systems of two-piecewise monotone maps and reducibility of its kneading data with respect to the *-product was established in [3].

To improve reading fluency we move all the proofs in this

Convergence rates

Feigenbaum observed that, for the unimodal family fλ(x)=λx(1x), as well as many other families of maps in the interval, the sequence of parameters (λn)n, corresponding to period doubling bifurcations, is convergent and satisfieslimnλnλn1λn+1λn=4.6692This convergence rate is linear in the sense thatlimnλn+1λn(λnλn1)p={0,p<1,p>1.Generally, a convergent sequence un that satisfieslimnlog(un+2un+1un+1un)log(un+1ununun1)=lhas convergence rate of order l in the sense thatlimnun+1

Anharmonic period doubling bifurcations

Traditionally, convergence rates of this type in dynamical systems occur in sequences of renormalizable maps such that, generally, each map is a renormalization of the consecutive one.

In the following we will study some period doubling sequences of 2-systems that do not fit in this pattern. Nevertheless we will prove that the corresponding one dimensional parameter sequences converge with convergence rate of order 2.

Consider X ∈ Σ, a right-sequence, and define the sequence of symbolic sequences

Proofs from Section 3

Proof

Theorem 3

We aim to prove that σ2n(X*Y) ≤ X*Y, for all n such that 2n < |X*Y|. Suppose that ε(X)=1, thus Y^k=Yk, for all k{0,,|Y|2}.

There is 0 ≤ m < |X|/2 and 1 ≤ k ≤ |Y| such that σ2n(X*Y)=X2m+1X|X|1Yk.

If k=|Y| we have σ2n(X*Y)=σ2m(X)<X<X*Y and the result follows. Suppose now that 1 ≤ k < |Y|.

Since X is maximal we have σ2m(X) < X. Hence, there is q|X|2m such thatX2m+1X2m+q1=X1Xq1andε(X1Xq1)X2m+q<ε(X1Xq1)Xq.If 2m+q<|X| then [σ2n(X*Y)]q=X2m+q and we are done. Else, if 2m+q=|X|, then X2m+q=0

Acknowledgements

This work was partially supported by FCT-Portugal, throw project PEst-OE/MAT/UI0117/2014.

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