# On the universality of the frequency spectrum and band-gap optimization of quasicrystalline-generated structured rods

## Abstract

The dynamical properties of periodic two-component phononic rods, whose elementary cells are generated adopting the Fibonacci substitution rules, are studied through the recently introduced method of the toroidal manifold. The method allows all band gaps and pass bands featuring the frequency spectrum to be represented in a compact form with a frequency-dependent flow line on the surface describing their ordered sequence. The flow lines on the torus can be either closed or open: in the former case, (i) the frequency spectrum is periodic and the elementary cell corresponds to a canonical configuration, (ii) the band gap density depends on the lengths of the two phases; in the latter, the flow lines cover ergodically the torus and the band gap density is independent of those lengths. It is then shown how the proposed compact description of the spectrum can be exploited (i) to find the widest band gap for a given configuration and (ii) to optimize the layout of the elementary cell in order to maximize the low-frequency band gap. The scaling property of the frequency spectrum, that is a distinctive feature of quasicrystalline-generated phononic media, is also confirmed by inspecting band-gap/pass-band regions on the torus for the elementary cells of different Fibonacci orders.

This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

### 1. Introduction

In the last 50 years, the investigation of wave propagation in structured media and their applications in different areas of engineering have attracted significant interest from the scientific community. In this context, the contribution of Prof. Slepyan and his collaborators was essential for understanding and predicting several phenomena, in particular, transition waves in periodic and bistable structures [1–4], interaction between surface modes and fractures [5,6], dissipation and phase transition in lattice materials [7–9] and solitary nonlinear waves [10,11]. These fundamental studies, together with the results obtained by other authors [12–15], have inspired a very active field of research, concerning the design of phononic structures with the aim of achieving and controlling non-standard wave propagation phenomena, such as wave focusing [16], frequency filtering [17], cloaking [18,19] and negative refraction [20,21]. Recently, the intriguing dynamical properties of a class of two-phase periodic structured solids, whose unit cells are generated according to the Fibonacci substitution rule, have been presented [22,23]. This particular family of composite structures belongs to the subset of *quasicrystalline* media [24] and the portion of Floquet–Bloch frequency spectra of its members are characterized by a self-similar pattern which scales according to the factors linked to the Kohmoto's invariant of the family itself [25].

This work provides new insights on the relationship between the geometrical and constitutive properties of the elementary cells and the layout of pass bands/band gaps for the same type of quasicrystalline-generated phononic rods. By considering harmonic axial waves, we show that the corresponding frequency spectrum can be represented on a two-dimensional toroidal manifold similar to that introduced in [26,27] to study Floquet–Bloch waves in periodic laminates. This manifold is universal for all two-phase configurations and the dispersion properties of the concerned rod can be inferred from the features of the frequency-parametrized flow lines lying on the toroidal domain, which is composed of band gap and pass-band regions. We identify a particular subclass of rods whose flow lines on the torus are closed, thus describing a periodicity in the spectrum at an increasing frequency, and show that the subclass coincides with that of the so-called *canonical structures* introduced by Morini and Gei [23]. The local scaling governing the pass band/band gap layout about certain relevant frequencies (i.e. the *canonical frequencies*) is confirmed and highlighted through the analysis of the flow lines on the torus.

The universal representation of the spectrum on the toroidal surface allows us to rigorously estimate the band gap density for rods of any arbitrary Fibonacci elementary cell. We find that for canonical configurations, this quantity varies with the ratio between the lengths of the phases, corresponding to the slope of the flow lines. Conversely, for generic non-canonical rods, the band gap density is independent of the lengths of the cells and is defined by the ratio between the area of the band gap subdomain and the total surface of the torus [28–30]. The provided examples show that this ratio can be easily evaluated numerically.

We further demonstrate how the compact representation of the spectrum on the two-dimensional torus can be exploited to either optimize the design of the elementary cells to achieve the widest low-frequency band gap or to determine rigorously where the maximal band gap is located in the spectrum for a given configuration. In the examples that we report, we have based this investigation on analytical expressions of the boundaries of band gap regions that can be easily obtained for low-order elementary cells. Unlike the standard procedure based on partial evaluation of the spectrum [31–33], the proposed optimization strategy provides exact rigorous results, and it can be easily generalized to Fibonacci cells of higher order.

### 2. Waves in quasicrystalline-generated phononic rods

We introduce a particular class of infinite, one-dimensional, two-component phononic rods consisting of a repeated elementary cell where two distinct elements, say *L* and *S*, are arranged in series according to the Fibonacci sequence [24]. The repetition of such a cell implies periodicity along the axis and then the possibility of applying the Floquet–Bloch technique in order to study harmonic wave propagation. The two-component Fibonacci sequence is based on the following substitution rule [34]

*i*th (

*i*= 0, 1, 2, …) element of the Fibonacci sequence, here denoted by ${\mathcal{F}}_{i}$, obeys the recursive rule ${\mathcal{F}}_{i}={\mathcal{F}}_{i-1}{\mathcal{F}}_{i-2}$, where the initial conditions are ${\mathcal{F}}_{0}=S$ and ${\mathcal{F}}_{1}=L$ (in figure 1, elementary cells designed according to sequences ${\mathcal{F}}_{2}$, ${\mathcal{F}}_{3}$ and ${\mathcal{F}}_{4}$ are displayed).

^{1}The total number of elements of ${\mathcal{F}}_{i}$ corresponds to the Fibonacci number ${\stackrel{~}{n}}_{i}$ given by the recurrence relation ${\stackrel{~}{n}}_{i}={\stackrel{~}{n}}_{i-1}+{\stackrel{~}{n}}_{i-2}$, with

*i*≥2, and ${\stackrel{~}{n}}_{0}={\stackrel{~}{n}}_{1}=1$. The limit of ${\stackrel{~}{n}}_{i+1}/{\stackrel{~}{n}}_{i}$ for $i\to \mathrm{\infty}$ corresponds to the so-called

*golden*mean ratio $(1+\sqrt{5})/2$.

Further in the text, we will refer to those structured rods as Fibonacci structures. According to the general criterion for the classification of the one-dimensional quasiperiodic patterns proposed in [35], these structures are *quasicrystalline*. Quasicrystalline media possess characteristic features that make them an intermediate class between periodic ordered crystals and random media [36,37]. An example of these interesting and intriguing properties is the self-similarity of the frequency spectrum [23]. The focus of this paper is on the analysis of the universal structure of this spectrum and on its application to predict, modulate and optimize the corresponding stop/pass band layout. We will show that the universality of the spectrum is closely related to the properties of the Floquet–Bloch dispersion relation exploited in [22] and summarized in this Section.

Let us introduce the mechanical and geometric parameters of elements *L* and *S*. The lengths of the two phases are indicated with *l*_{L} and *l*_{S}, while *A*_{X}, *E*_{X} and ϱ_{X} (*X*∈{*L*, *S*}) denote cross-section area of each bar, Young's modulus and mass density per unit volume of the two adopted materials. For both elements, we define the displacement function and the axial force along the rod as *u*(*z*) and *N*(*z*) = *EAu*′(*z*), respectively, where *z* is the coordinate describing the longitudinal axis. The governing equation of harmonic waves in each section assumes the form

_{X}/

*E*

_{X}corresponds to the reciprocal of the square of the speed of propagation of longitudinal waves in material

*X*. The solution of (2.2) is given by

*C*

^{X}

_{1}and

*C*

^{X}

_{2}are integration constants, to be determined by the boundary conditions.

To obtain the dispersion diagram of the periodic rod, displacement *u*_{r} and axial force *N*_{r} at the right-hand boundary of the elementary cell have to be given in terms of those at the left-hand boundary, namely *u*_{l} and *N*_{l} (figure 1), as

*U*_{j}= [

*u*

_{j}

*N*

_{j}]

^{T}(

*j*=

*r*,

*l*) and

*T*_{i}is a transfer (or transmission) matrix of the cell ${\mathcal{F}}_{i}$. This matrix is the result of the product ${\mathit{T}}_{i}=\prod _{p=1}^{{\stackrel{~}{n}}_{i}}{\mathit{T}}^{X},$ where

*T*^{X}(

*X*∈{

*L*,

*S*}) is the transfer matrix, which relates quantities across a single element, given by

Transfer matrices *T*_{i} have some important properties that can be exploited: (i) they are unimodular, i.e. $det{\mathit{T}}_{i}=1$, and (ii) follow the recursion rule

*T*_{0}=

*T*^{S}and

*T*_{1}=

*T*^{L}.

The Floquet–Bloch theorem implies that ${\mathit{U}}_{\mathrm{r}}=\mathrm{exp}(\text{i}k{L}_{i}){\mathit{U}}_{\mathrm{l}}$, where *L*_{i} is the total length of the fundamental cell ${\mathcal{F}}_{i}$ and the imaginary unit appearing in the argument of the exponential function should not be confused with the index *i*. By combining this condition with (2.4), we obtain the dispersion equation

Equivalently, we can study the dispersion properties of these structures by evaluating the eigenvalues of the transfer matrix. As *T*_{i} is unimodular, it turns out that the characteristic equation of the waveguide is given by

*e*

^{ikLi}=

*λ*in equation (2.8) and multiplying it by e

^{−ikLi}, the condition e

^{ikLi}+ e

^{−ikLi}−

*T*_{i}= 0 is achieved, leading to

*η*

_{i}=

*T*_{i}/2.

By observing equation (2.9), we can easily deduce that all the information concerning harmonic axial wave propagation in a Fibonacci structure is contained in the half trace *η*_{i} of the corresponding transfer matrix. Waves propagate when |*η*_{i}| < 1 ($k{L}_{i}\in \mathbb{R}\mathrm{\setminus}\{x:x=h\pi ,\hspace{0.17em}h\in \mathbb{Z}\}$), band gaps correspond to the ranges of frequencies where |*η*_{i}| > 1 (*k* is a complex number with a non-vanishing imaginary part), whereas |*η*_{i}| = 1 characterizes standing waves ($k{L}_{i}\in \{x:x=h\pi ,h\in \mathbb{Z}\}$).

We note that both the transfer matrix (2.5) and the dispersion relation (2.9) possess a form identical to that derived in [38,39] and used in [26,27,40] to study antiplane shear waves in periodic two-phase, multiphase and quasicrystalline laminates, respectively. Further in the paper, we will exploit this mathematical analogy generalizing the approach proposed in [26] to study the universal structure of the frequency spectrum of Fibonacci phononic rods.

### 3. Universal structure of the frequency spectrum

The analysis of the universal structure of the frequency spectrum will take advantage of the introduction of the following variables [26,28–30]:

*T*_{i}, together with the relationship (2.6), implies the following recursive rule for the half trace

*η*

_{i+1}[23]:

The quantity

*L*and

*S*, and it depends on their constitutive parameters but not on lengths of the single elements

*L*and

*S*. When

*γ*= 1 there is no contrast between phases and the waveguide behaves as a homogeneous one. Expressions (3.3) show that for any given value of

*γ*,

*η*

_{0},

*η*

_{1}and

*η*

_{2}are 2

*π*-periodic functions of

*ζ*

_{S}and

*ζ*

_{L}. The generic half trace

*η*

_{i}can be derived by means of successive iterations of the recursive formula (3.2) by assuming (3.3) as initial conditions. Therefore, at any order

*i*,

*η*

_{i}is also a 2

*π*-periodic function, separately, of

*ζ*

_{S}and

*ζ*

_{L}as it is defined through sums and products of functions with the same period. This implies that we can consider the half trace

*η*

_{i}as a function of a two-dimensional torus of edge length 2

*π*, whose toroidal and poloidal coordinates are

*ζ*

_{S}and

*ζ*

_{L}, respectively. This function is independent of the lengths of the two phases

*L*and

*S*. The toroidal domain is composed of two complementary subspaces that are associated with the two inequalities introduced earlier in the discussion after equation (2.9), namely |

*η*

_{i}(

*ζ*

_{S},

*ζ*

_{L})| < 1 identifies a pass-band subdomain, whereas |

*η*

_{i}(

*ζ*

_{S},

*ζ*

_{L})| > 1 corresponds to a band-gap one. The two regions might not be simply connected and the collection of lines of separation between the two subdomains, in which |

*η*

_{i}(

*ζ*

_{S},

*ζ*

_{L})| = 1, denotes a standing wave solution. The measures of the two regions are univocally determined by the value of the parameter

*γ*.

A sketch of the toroidal domains for cells ${\mathcal{F}}_{2}$ and ${\mathcal{F}}_{3}$ is displayed in figure 2*a*,*b* where the set of physical properties tabled in table 1 have been assumed (for that choice, *γ* ≈ 2.125). In both plots, the pink zone corresponds to the pass-band region, whereas the band-gap one is painted in grey.

E_{S} = E_{L} = 3.3 GPa |
ϱ_{S} = ϱ_{L} = 1140 kg m^{−3} |
A_{L} = 4A_{S} = 1.963 × 10^{−3} m^{2} |
l_{L} = 0.07 m |

Equation (2.9) shows that |*η*_{i}(*ζ*_{S}, *ζ*_{L})| is invariant under the transformation

*π*-periodic torus. The latter can be conveniently represented through the so-called

*square identification*[41], in which the curved domain is flattened and transformed to a square whose edges are still described by coordinates

*ζ*

_{S}and

*ζ*

_{L}, both ranging now between 0 and

*π*. In the new square representation, the band-gap subdomain (|

*η*

_{i}(

*ζ*

_{S},

*ζ*

_{L})| > 1) is denoted by ${\mathbb{D}}_{i}(\gamma )$. In the following, the square equivalent

*π*-periodic torus with the domain ${\mathbb{D}}_{i}(\gamma )$ will be indicated with ${\mathbb{T}}_{i}$. At times, we will also refer to it as the ‘reduced torus’ for the cell ${\mathcal{F}}_{i}$.

In figure 2*c*,*d*, the reduced tori ${\mathbb{T}}_{2}$ and ${\mathbb{T}}_{3}$ are reported. The light blue, light red and light brown regions in both plots denote the subdomains ${\mathbb{D}}_{2}(\gamma )$ and ${\mathbb{D}}_{3}(\gamma )$ determined for *γ* ≈ 8.031, 2.125 and 1.170, respectively. In particular, the light red ones are the representation of the band gap domains depicted in grey on the original 2*π*-periodic tori reported just above in the same figure 2*a*,*b*, respectively.

The spectrum for a Fibonacci rod of any arbitrary order can, therefore, be studied by analysing the dynamic flow parametrized ** ζ**(

*ω*) = (

*ζ*

_{S}(

*ω*),

*ζ*

_{L}(

*ω*)) on the corresponding reduced torus, where the frequency

*ω*plays the role of a time-like parameter. This flow is the image on ${\mathbb{T}}_{i}$ of the trajectories described by the angles

*ζ*

_{S}and

*ζ*

_{L}on the original torus. Two examples of the latter are the blue lines reported in figure 2

*a*,

*b*. In order to represent these flow lines on ${\mathbb{T}}_{i}$, we interpret expression (3.1) as the equation of a rectilinear trajectory lying on the square. Now, for any arbitrary Fibonacci cell ${\mathcal{F}}_{i}$ for which a specific indication for lengths

*l*

_{L}and

*l*

_{S}is provided, we can depict the trajectory (3.1) on ${\mathbb{T}}_{i}$ as those illustrated for ${\mathcal{F}}_{2}$ and ${\mathcal{F}}_{3}$ in the two plots of figure 2

*c*,

*d*. For this purpose, if we consider values of the frequency such that $\sqrt{{\varrho}_{X}/{E}_{X}}\omega {l}_{X}>\pi $, by recalling the invariance of ${\mathbb{T}}_{i}$ and of its subdomain ${\mathbb{D}}_{i}(\gamma )$ with respect to transformations (3.5), expression (3.1) can be written in the transformed form as

*c*,

*d*, and the flow

**(**

*ζ**ω*) can be expressed as

*c*,

*d*are the images of the flow lines illustrated in figure 2

*a*,

*b*, respectively. By examining these lines, we can easily observe that they trace a closed trajectory on the torus. In the next section, the class of structures, whose spectra are described by this particular type of flow lines, is defined and characterized in detail.

The values of *ω*, for which the lines of the flow (3.7) intersect the boundary of the subdomain ${\mathbb{D}}_{i}(\gamma )$, coincide with the extremes of the band gaps. These intersections are highlighted with red points in figure 2*c*,*d* for waveguides generated by ${\mathcal{F}}_{2}$ and ${\mathcal{F}}_{3}$ for *γ* ≈ 2.125. The same band gaps are illustrated in the classical dispersion diagrams of figure 2*e*,*f* .

A parametric equation for the flow lines on ${\mathbb{T}}_{i}$ is easily derived from equation (3.6)

*c*,

*d*). In particular, the segment emerging from the origin for

*ω*= 0

^{+}(i.e.

*m*=

*n*= 0) has equation

*ζ*

_{L}(

*ω*) =

*βζ*

_{S}(

*ω*). In the next section, we discuss how rational and irrational values of ratio (3.10) are associated with Fibonacci rods possessing periodic and non-periodic spectra, respectively, corresponding to closed and open trajectories on the 2

*π*-periodic torus, respectively. Both these two different behaviours are studied by analysing the flow lines on ${\mathbb{T}}_{i}$. Relevant indications concerning the band gap density and the different properties of rods with periodic and non-periodic spectra are obtained by using this universal approach.

### 4. Analysis of the flow lines on the reduced torus

Let us analyse the different types of trajectories (lines) that can describe the flow ** ζ**(

*ω*) on the torus. The condition for closed periodic lines is the existence of a frequency interval

*Ω*such that [41]

We can deduce from expression (4.3) that the trajectories on the torus are periodic if the ratio *β* is a rational number. This condition is exactly the same as that introduced in [23] and necessary to realize Fibonacci structures with a periodic spectrum, which are called in that article *canonical structures*. Therefore, canonical configurations correspond to closed flow trajectories on the torus. Considering the original 2*π*-periodic torus, these are closed helicoidal orbits on the surface as those reported in figure 2*a*,*b*. The two whole numbers *j* and *q* represent the number of cycles, namely 2*π* rotations, about the toroidal and poloidal axes. As an example, both blue trajectories of figure 2*a*,*b* correspond to *j* = 1 and *q* = 2 and then to *β* = 2. On ${\mathbb{T}}_{i}$, the closed flow lines associated with canonical structures become a *finite* number of parallel segments. The periodicity of the dispersion diagram is verified in figure 2*e*,*f* where the band gap limits already highlighted in the companion graphs plotted above (i.e. (c) and (d), respectively) are marked with red points.

In figures 3, 4 and 5, examples of periodic flow lines for canonical structures generated by the repetition of cells ${\mathcal{F}}_{2}$, ${\mathcal{F}}_{3}$ and ${\mathcal{F}}_{4}$ are reported. For the calculations, we considered two phases *S* and *L* of the same material (*E*_{S} = *E*_{L} and ϱ_{S} = ϱ_{L}, see table 1) so that parameters *γ* and *β* become

^{2}depend only on ratio

*A*

_{S}/

*A*

_{L}, while the direction of flow is defined by

*l*

_{S}/

*l*

_{L}. Moreover, according to the classification provided in [23], the analysed rods belong to the second family of canonical configurations.

In the plots on the left-hand side of each of figures 3–5, diagrams are presented of the half traces *η*_{2}, *η*_{3}, *η*_{4} reported as functions of *ω* for an interval of frequencies which coincides with the half period of the spectrum. We use coloured dots to earmark the extremes of the intervals where |*η*_{2}|, |*η*_{3}|, |*η*_{4}| > 1, defining the band gaps. The flow lines on ${\mathbb{T}}_{2}$, ${\mathbb{T}}_{3}$ and ${\mathbb{T}}_{4}$ are reported on the right-hand side of each figure. Their intersection with the boundaries of ${\mathbb{D}}_{2}$, ${\mathbb{D}}_{3}$ and ${\mathbb{D}}_{4}$, which identify the extreme of the band gaps, are indicated with the same coloured dots. We used the same colour cod in both diagrams of traces and ${\mathbb{T}}_{i}$ in order to associate the corresponding band gap in the two different representations. We note that the flow diagrams in ${\mathbb{T}}_{i}$ highlight all the band gaps contained in the half period of the canonical structures, and then the successive band gaps can be visualized using the same finite number of segments on ${\mathbb{T}}_{i}$ and applying the transformation (3.6). Therefore, for canonical structures generated by any arbitrary cell ${\mathcal{F}}_{i}$, the band gap density *φ*_{i} is given by the ratio between the measure of the intersections between the flow lines and the subdomain ${\mathbb{D}}_{i}$, and the total length of the flow lines. The latter is given by the sum of all the parallel segments reported in figures 3–5 and corresponding to $\sqrt{{j}^{2}+{q}^{2}}\pi $. This ratio depends on both the area of ${\mathbb{D}}_{i}$ and the direction of the flow lines, and then on both *γ* and *β* parameters.

The values of the band gap density for three different examples of canonical structures with elementary cells from ${\mathcal{F}}_{2}$ to ${\mathcal{F}}_{8}$ are reported in figure 6. We assumed the same constitutive properties used for the results shown in figures 3–5 (table 1) and three different ratios *l*_{S}/*l*_{L} which, in this particular case, correspond to three values of *β*, namely 1, 2 and 7/2 (see equation (4.4)_{2}). According to the definition provided in [23], those three ratios are associated with canonical structures which belong to the first, the second and the third family. The three families are distinguished by different stop and pass band layouts, but they all possess periodic spectra with properties depending exclusively on *β*. Figure 6 shows that the value of the band gap density is different for cells of the same order *i*, but with distinct values of the parameter *β*. This confirms, as we have already mentioned, that the band gap density of canonical rods depends on the ratio *l*_{S}/*l*_{L}. As a consequence, if we assume given constitutive properties of the phases *S* and *L* (i.e. *E*_{S}, *E*_{L}, ϱ_{S} and ϱ_{L}) and given cross-sections *A*_{L} and *A*_{S}, and then we determine univocally the domain ${\mathbb{T}}_{i}$ and the area of the subspace ${\mathbb{D}}_{i}$, we can modulate the band gap density by simply varying the ratio *l*_{S}/*l*_{L}. Indeed, by changing this parameter, we assign a different direction to the flow lines on the torus or equivalently to the slope of the segments on the square identification of ${\mathbb{T}}_{i}$, determining the band gap intervals which coincide with intersections of the flow trajectories with the subdomain ${\mathbb{D}}_{i}$.

By observing figure 6, we note that, for all the three types of canonical rods here analysed, the band gap density increases with the index *i* following a logarithmic trend. This is in agreement with the results presented in [42,43] for electronic and optic systems subjected to quasiperiodic Fibonacci potentials.

In addition to the canonical ones, we can define a different class of waveguides whose ratio *β* is irrational. In this case, the spectrum is not periodic and the corresponding flow lines are open and cover *ergodically* the whole torus with uniform measure [44]. In this situation, it is commonly said that the orbits are *dense* on the torus [45]. Consequently, the flow trajectories on ${\mathbb{T}}_{i}$ consist of an *infinite* number of parallel segments which, in turn, cover ergodically the whole square domain. Therefore, the band gap density is given by the ratio between the area of the subdomain ${\mathbb{D}}_{i}$ and the area *π*^{2} of the square. Since the measure of ${\mathbb{D}}_{i}$ is determined only by the parameter *γ*, which is independent of the ratio *l*_{S}/*l*_{L}, for non-canonical rods the band gap density does not depend on that ratio.

The fundamental differences between the flow lines of a canonical waveguide and those of non-canonical one are pointed out in figure 7. Figure 7*a*,*b* display the variation of the half trace *η*_{2} with the frequency and the trajectories on the reduced torus ${\mathbb{T}}_{2}$ for a canonical structure with parameters listed in table 1 and *l*_{S}/*l*_{L} = 1/2, the same considered in figure 3. The variation of *η*_{2} is plotted for a frequency range equal to its period (0 < *ω*≲305 krad s^{−1}). The corresponding extremes of band gaps both in the half-trace diagram and in ${\mathbb{T}}_{2}$ are marked using points with the same colours. As anticipated, due to the periodicity of the flow lines, all band gaps and pass bands in the frequency spectrum can be represented through the two parallel segments reported in figure 7*b*. Indeed, by observing this figure, the first and the third band gap, whose extremes are denoted by red and green points, respectively, overlap as well as the second and the fourth ones whose extremes are marked with magenta and yellow points, respectively.

The three pairs of figures 7*c*,*d*, 7*e*,*f* , 7*g*,*h* illustrate the diagrams of the half trace *η*_{2} and the flow lines on ${\mathbb{T}}_{2}$ for cells ${\mathcal{F}}_{2}$ with ${l}_{S}/{l}_{L}=1/2+\sqrt{1/500}$, ${l}_{S}/{l}_{L}=1/2+3\sqrt{1/500}$ and ${l}_{S}/{l}_{L}=1/2+10\sqrt{1/500}$, respectively. We assumed three different perturbations of the length ratio in order to have three irrational values of *β* and then three examples of non-canonical configurations. Their spectra are studied in the same range of frequencies of the canonical waveguide in figure 7*a*,*b*. We observe that for the three irrational ratios the half trace *η*_{2} is no longer periodic, and the number of band gaps in the same frequency range increases with respect to the canonical case. Due to the lack of periodicity, band gaps are characterized by widths and relative distances that are all different from each other. This implies that the representation of each of them on ${\mathbb{T}}_{2}$ is associated with a different parallel segment, as shown in figure 7*d*,*f* ,*h*. These segments are the images on the reduced torus of the three flow lines, which, in this case, are infinite. At an increase of the frequency range for the half traces in figure 7*c*,*e*,*g*, more and more segments are needed in order to depict the set of band gaps on the right-hand counterparts (figure 7*d*,*f* ,*h*, respectively), up to cover the whole domain of ${\mathbb{T}}_{2}$. Therefore, for all the three non-canonical rods analysed, it is confirmed that the band gap density *φ*_{i} is given by the ratio between the area of ${\mathbb{D}}_{2}$ and *π*^{2}. In general,

*γ*and is

*independent*of

*l*

_{S}/

*l*

_{L}.

We can now generalize the analysis provided for waveguides generated by ${\mathcal{F}}_{2}$ to any arbitrary Fibonacci cell ${\mathcal{F}}_{i}$. In analogy with the previous examples, we consider two phases with the same properties (table 1) and ${l}_{S}/{l}_{L}=\sqrt{3/10}$ and ${l}_{S}/{l}_{L}=\sqrt{1/2}$, corresponding to $\beta =\sqrt{10/3}$ and $\beta =\sqrt{2}$. We solve numerically the dispersion relation (2.9) over increasing intervals of frequencies, and at each iteration we estimate the ratio between the total length of the band gaps and the whole length of the frequency range. Calculations are carried out for structures designed according to cells ${\mathcal{F}}_{2}$, ${\mathcal{F}}_{3}$, ${\mathcal{F}}_{4}$ and ${\mathcal{F}}_{5}$; the results are shown in figures 8 and 9. Red lines with circle marks and blue lines with square marks map the band gap density for ${l}_{S}/{l}_{L}=\sqrt{3/10}$ and ${l}_{S}/{l}_{L}=\sqrt{1/2}$, respectively. For both cases, and in each panel, we note the convergence of the data to the black horizontal line that corresponds to *φ*_{i} in (4.5). These ratios can be estimated numerically or analytically for cell ${\mathcal{F}}_{2}$ (see explicit expression derived in [26]), and in this case they are 0.5090 for ${\mathcal{F}}_{2}$, 0.5098 for ${\mathcal{F}}_{3}$, 0.5938 for ${\mathcal{F}}_{3}$ and 0.6334 for ${\mathcal{F}}_{5}$. The convergence observed for all panels in figures 8 and 9 demonstrates that for non-canonical structures the band gap density at a given value of *γ* is *independent* of the lengths of the phases *S* and *L*. Therefore, we can state that the band gap density is a *universal* property of classes of non-canonical waveguides characterized by a prescribed *γ* and an elementary cell ${\mathcal{F}}_{i}$. This is in agreement with the results reached in [26], where it is shown that for irrational values of a parameter analogous to our *β* the band gap density of two phase laminates is independent of the thicknesses of the layers.

### 5. Band gap optimization using universality properties

The compact representation of the frequency spectrum on ${\mathbb{T}}_{i}$ is now used to formulate rigorously and solve two types of optimization problems in periodic quasicrystalline-generated rods. We focus on the case of ${\mathcal{F}}_{3}$ for which analytical representations of the boundaries of the band gaps are available, but the same approach can be easily applied to higher-order cells with the aid of implicit expressions similar to those obtained in [27].

The band gap subdomain ${\mathbb{D}}_{3}$ is composed of two identical regions for any values of the parameter *γ*: one, namely ${\mathbb{D}}_{3}^{-}$, lies on the portion of ${\mathbb{T}}_{3}$ delimited by the intervals 0 ≤ *ζ*_{S} ≤ *π* and 0 ≤ *ζ*_{L} ≤ *π*/2; the other, ${\mathbb{D}}_{3}^{+}$, occupies the portion delimited by the intervals 0 ≤ *ζ*_{S} ≤ *π* and *π*/2 ≤ *ζ*_{L} ≤ *π* (figures 2*d* and 4*b*). The former is considered for the maximization of gap width, but the same methodology can be applied to ${\mathbb{D}}_{3}^{+}$. All points of the boundary of ${\mathbb{D}}_{3}^{-}$ satisfy the condition *η*_{3}(*ζ*_{S}, *ζ*_{L}) = − 1 and define the curves ${\mathcal{C}}_{l}^{-}$ and ${\mathcal{C}}_{u}^{-}$ whose analytical expressions are

*ω*

^{B}−

*ω*

^{A}} is related to the length of the associated interval along the flow line, whose endpoints

*A*(

*ζ*

^{A}

_{S},

*ζ*

^{A}

_{L}) and

*B*(

*ζ*

^{B}

_{S},

*ζ*

^{B}

_{L}) lie on ${\mathcal{C}}_{u}^{-}$ and ${\mathcal{C}}_{l}^{-}$, respectively, through the relationship

*A*and

*B*belong to the flow lines on ${\mathbb{T}}_{3}$, their coordinates satisfy the relationships

*ζ*

^{•}

_{X}=

*ζ*

_{X}(

*ω*

^{•}). Equations (5.2) and (5.3), together with expressions (5.1) for the curves ${\mathcal{C}}_{l}^{-}$ and ${\mathcal{C}}_{u}^{-}$, enable us to maximize the width of {

*ω*

^{B}−

*ω*

^{A}} through the flow lines defined on the basis of the physical and geometrical properties of the elementary cells.

#### (a) Identification of the widest band gap for a prescribed structure

We first consider a given cell ${\mathcal{F}}_{3}$ with prescribed physical and geometrical properties. Our purpose is to determine the interval {*ω*^{B} − *ω*^{A}} defining the widest band gap in the frequency spectrum of the structure. As *β* = (*ζ*^{B}_{L} − *ζ*^{A}_{L})/(*ζ*^{B}_{S} − *ζ*^{A}_{S}), expression (5.2) can be written in the following form:

*ν*

_{S}and

*β*are known and the goal is achieved by finding the value of the translation coefficient

*α*associated with the largest Δ

*ζ*

_{S}. By imposing that both the points

*A*and

*B*lie on the flow line (5.3) and that $A\in {\mathcal{C}}_{l}^{-}$ and $B\in {\mathcal{C}}_{u}^{-}$, the following equations for the coordinates

*ζ*

^{B}

_{S}and

*ζ*

^{A}

_{S}are established

*ζ*

_{S}=

*ζ*

^{B}

_{S}−

*ζ*

^{A}

_{S}becomes

*α*between (5.5) and (5.6), it turns out that

*ζ*

^{A}

_{S}and

*ζ*

^{B}

_{S}that maximize the quantity (5.7) and are also a solution of equation (5.8). Then, the corresponding

*α*can be evaluated by means of (5.5) and (5.6). The problem can be solved graphically for any cell ${\mathcal{F}}_{3}$ through the two diagrams reported in figure 10. For the calculations, we considered a non-canonical configuration with the parameters listed in table 1 and ${l}_{S}/{l}_{L}=1/2+3\sqrt{1/500}$.

The contour plot in figure 10*a* shows the variation of the function (5.7) on the whole two-dimensional domain 0 ≤ {*ζ*^{A}_{S}, *ζ*^{B}_{S}} ≤ *π*, while the red line reported in the same figure is determined by the values of *ζ*^{A}_{S} and *ζ*^{B}_{S} satisfying equation (5.8). Point *P*, whose coordinates are the solution to (5.8) and maximize Δ*ζ*_{S}, is denoted by the yellow dot. It corresponds to the intersection between the red line and the blue curve, defined in this case through the equation Δ*ζ*_{S} = 0.589. We note that this point also coincides with the intersection between the curve (5.8) and the line *ζ*^{B}_{S} = *π* − *ζ*^{A}_{S}. Consequently, the coordinates *ζ*^{A}_{S} and *ζ*^{B}_{S} can be derived as the solution of the system

_{2}into (5.9)

_{1}, we obtain

*ζ*

^{A}

_{S}= 1.278 and

*ζ*

^{B}

_{S}= 1.863. Using these values in equation (5.5) (or (5.6)),

*α*= 1.692 is determined. Remembering that in this case ${l}_{S}/{l}_{L}=1/2+3\sqrt{1/500}=1/\beta $, equation (5.3) provide

*ζ*

^{A}

_{L}= 0.324 and

*ζ*

^{B}

_{L}= 1.246.

We determined the translation coefficient of the flow segment corresponding to the widest band gap among all those detected in the spectrum of the structure, as well as the coordinates on ${\mathbb{T}}_{3}$ of the points *A*(*ζ*^{A}_{S}, *ζ*^{A}_{L}) and *B*(*ζ*^{B}_{S}, *ζ*^{B}_{L}), associated with *ω*^{A} and *ω*^{B}. *A* and *B* are denoted by red dots in figure 11*b*, and the width *ω*^{B} − *ω*^{A} = 22.066 krad s^{−1} can be calculated through equation (5.4). On the basis of the definition (3.6), *ω*^{A} and *ω*^{B} are given by

*n*and

*m*are two whole numbers satisfying condition (3.9).

The invariance of ${\mathbb{T}}_{3}$ and ${\mathbb{D}}_{3}$ with respect to the transformations (3.5), together with the conditions $A\in {\mathcal{C}}_{l}^{-}$ and $B\in {\mathcal{C}}_{l}^{+}$, provides the following system of implicit equations

*n*and

*m*corresponding to the extremes

*ω*

^{A}and

*ω*

^{B}of the maximal band gap are given by a pair of integer solutions of system (5.13) that satisfies the relationship (3.9). They can be found through a diagram like the one reported in figure 10

*b*, where the solutions of equation (5.13)

_{1}and (5.13)

_{2}correspond to the red and blue contours, respectively, and the black line is defined by equation (3.9). The green dot denotes the intersection of the three curves at

*n*= 6 and

*m*= 10, which are the required numbers in this case. By substituting them together with the previously calculated

*ζ*

^{A}

_{S},

*ζ*

^{B}

_{S},

*ζ*

^{A}

_{L},

*ζ*

^{B}

_{L}and the physical properties of the cell in expressions (5.11) and (5.12), we finally determine

*ω*

^{A}= 759.69 krad s

^{−1}and

*ω*

^{B}= 781.76 krad s

^{−1}. These extremal values are highlighted using the red dots in the diagram of the half trace

*η*

_{3}reported in figure 11

*a*.

The illustrated method can be easily applied to cells of higher order through the general approach developed in [27], where analytical expressions for the boundaries of the band gap subregions of periodic laminates with an arbitrary number of phases are derived. This original procedure provides several fundamental advantages with respect to the standard optimization methods based on the numerical evaluation of the frequency spectrum. This is obvious especially in the case of non-canonical structures as this is the case where the spectrum is not periodic, and then, in principle, calculations over an infinite frequency domain should be performed to determine the widest band gap. Since, in practice, calculations must be truncated, such an approach yields only an approximate solution. Moreover, there is currently any rigorous way to predict how the considered truncated subdomain allows an accurate estimation compared to the real infinite case. Contrarily, through the formulation over the torus ${\mathbb{T}}_{3}$, the problem is solved in closed form, without any approximation, avoiding the numerical calculations required by the evaluation of large portions of the frequency spectrum. It is also worth remarking that the solutions *m* and *n* can be relatively high, at a frequency for which, due to the effects of lateral inertia, the simple one-dimensional axial model might be no longer valid.

#### (b) Optimization of the lowest band gap through variation of the geometrical properties

The second example of optimization, which can be formulated rigorously and solved by exploiting the representation of the spectrum on ${\mathbb{T}}_{3}$, is here illustrated. Let us consider a cell ${\mathcal{F}}_{3}$ with parameters listed in table 1, such that *γ* ≈ 2.125 and the slope of the flow lines becomes *β* = *l*_{L}/*l*_{S}. Our aim is now to find the value of *β* that maximize the lowest band gap of the spectrum.

This one, i.e. {*ω*^{B} − *ω*^{A}}, is detected by the intersection between the region ${\mathbb{D}}_{3}^{-}$ and the flow segment starting from the origin of the plane *Oζ*_{S}*ζ*_{L}. Similarly to the case studied in §a, $A({\zeta}_{S}^{A},{\zeta}_{L}^{A})\in {\mathcal{C}}_{l}^{-}$ and $B({\zeta}_{S}^{B},{\zeta}_{L}^{B})\in {\mathcal{C}}_{l}^{+}$, and

*α*= 0 in this problem. Equation (5.14)

_{1}, together with the condition $A\in {\mathcal{C}}_{l}^{-}$, provides the following expression for

*β*:

_{2}and imposing $B\in {\mathcal{C}}_{u}^{-}$, we obtain

*l*

_{S}, and then

*ν*

_{S}, is known, the expression for the width of the band gap (5.2) can be written in the normalized form

*β*is given by (5.15),

*ζ*

^{A}

_{L}and

*ζ*

^{A}

_{L}can be expressed as functions of

*ζ*

^{A}

_{S}and

*ζ*

^{A}

_{S}using (5.1). We now have to determine the values of

*ζ*

^{A}

_{S}and

*ζ*

^{B}

_{S}that maximize $\mathrm{\Delta}\overline{\omega}$ and are solution of equation (5.16). The problem is solved graphically using the diagram reported in figure 12

*a*. The contour plot herein shows the variation of $\mathrm{\Delta}\overline{\omega}$ on the whole two-dimensional domain 0 ≤ {

*ζ*

^{A}

_{S},

*ζ*

^{B}

_{S}} ≤

*π*, while the red line reported in the same figure is the plot of equation (5.16). Point

*Q*, whose coordinates are the solution of (5.16) and maximize $\mathrm{\Delta}\overline{\omega}$, is denoted by the yellow dot. It corresponds to the intersection between the red line and the blue contour, the latter defined through equation $\mathrm{\Delta}\overline{\omega}=1.46$. For the set of constitutive and geometrical parameters here considered, we have

*ζ*

^{A}

_{S}= 1.215 and

*ζ*

^{B}

_{S}= 2.675. By employing these values in equation (5.1), we get

*ζ*

^{A}

_{L}= 0.345 and

*ζ*

^{B}

_{L}= 0.769, and then, eventually,

*β*= (

*ζ*

^{B}

_{L}−

*ζ*

^{A}

_{L})/(

*ζ*

^{B}

_{S}−

*ζ*

^{A}

_{S}) = 0.284. Therefore, this solution provides the slope of the flow segment corresponding to the widest lowest band gap, and its extreme

*A*(

*ζ*

^{A}

_{S},

*ζ*

^{A}

_{L}) and

*B*(

*ζ*

^{A}

_{S},

*ζ*

^{A}

_{L}) on the reduced torus ${\mathbb{T}}_{3}$ are marked with the red dots in figure 12

*b*. This result is valid for any given value of

*l*

_{L}≠0 which is assumed to be known for the calculations, and then the optimization procedure does not depend separately on lengths

*l*

_{S}and

*l*

_{L}, but only on their ratio

*β*.

The illustrated method provides an exact solution to the problem of the maximization of the lowest band gap, which is of practical importance in several operative scenarios involving different types of phononic structures (see, e.g. [31–33]). The formulation over the reduced torus can be easily extended to the case of an arbitrary cell ${\mathcal{F}}_{i}$ and represents a promising alternative to the direct approach based on partial evaluation of the frequency spectrum evaluation for all possible ratios *l*_{S}/*l*_{L}.

### 6. Scaling of the band gaps observed on the reduced torus

The universal representation of the spectrum on the reduced torus ${\mathbb{T}}_{i}$ can be exploited to check the local scaling occurring between band gaps at determined frequencies, as shown earlier in [22,23,40] for different types of quasicrystalline phononic structures. Following their approach, let us identify with *R*_{i} = (*x*_{i}, *y*_{i}, *z*_{i}) a point whose coordinates correspond to *x*_{i} = *η*_{i+2}, *y*_{i} = *η*_{i+1} and *z*_{i} = *η*_{i}. On the basis of the recursive relation (3.2), the change of point *R*_{i} to *R*_{i+1} can be described as the evolution of the nonlinear discrete map

*i*. It is worth noting that

*J*=

*I*/4, where

*I*is the Kohmoto's invariant defined in [23].

For any given value of the frequency, and then of the flux variables *ζ*_{S} and *ζ*_{L}, equation (6.2) defines a manifold whose equation in the continuous three-dimensional space *Oxyz* is *x*^{2} + *y*^{2} + *z*^{2} − 2*xyz* − 1 = *J*(*ω*), the so-called Kohmoto's surface [25]. The points obtained by iterating map (6.1) are all confined on this surface and describe an open, *discrete* trajectory. Each Kohmoto's surface possesses six saddle points, say ± *P*_{k} (*k* = 1, 2, 3), whose coordinates are $\pm {P}_{1}=(\pm 2\sqrt{1+J(\omega )},0,0)$, $\pm {P}_{2}=(0,\pm 2\sqrt{1+J(\omega )},0)$, $\pm {P}_{3}=(0,0,\pm 2\sqrt{1+J(\omega )})$. They are connected through a closed (periodic) orbit generated by the six-cycle transformation obtained by applying six times map (6.1), in other words, ${\mathcal{T}}^{6}({P}_{k})={P}_{k}$. Moreover, it can be also verified that ${\mathcal{T}}^{3}({P}_{k})=-{P}_{k}$. The frequencies *ω*_{c} at which a generic *R*_{i} coincides with one of these saddle points are called canonical frequencies and are exactly midway of the semi-period of the spectrum of canonical structures [23]. For instance, in the cases addressed in figures 3–5 and 7, the period is approximately 305 krad s^{−1} and canonical frequencies are approximately 305/4 krad s^{−1} and (3/4) 305 krad s^{−1}.

In the neighbourhood of *ω*_{c}, the corresponding point *R*_{i} locates in the vicinity of a saddle point, therefore the *discrete* trajectory traced by transformation of the point *R*_{i} itself at an increasing index on the Kohmoto's surface is then studied as a small perturbation of the periodic orbit with map (6.1) linearized about the six saddle points. The derived linearized transformation has an eigenvalue that is equal to one and an additional pair of them given by

*κ*

^{+}(

*ω*) governs the local scaling occurring between localized ranges of the spectrum of cell ${\mathcal{F}}_{i}$ and that of ${\mathcal{F}}_{i+6}$, while $\lambda \approx \sqrt{{\kappa}^{+}}$ is the scaling factor between ${\mathcal{F}}_{i}$ and ${\mathcal{F}}_{i+3}$. In particular, across a canonical frequency, the width of a band gap in the diagram of cell ${\mathcal{F}}_{i+6}$ centred at frequency

*ω*

_{c}, say {

*ω*

^{V}

_{i+6}−

*ω*

^{U}

_{i+6}}, is related to that of {

*ω*

^{V}

_{i}−

*ω*

^{U}

_{i}} in the diagram of cell ${\mathcal{F}}_{i}$ centred at the same frequency by the following scaling law:

*κ*=

*κ*

^{+}(

*ω*

_{c}). Similarly, the following relationship can be established between the widths of {

*ω*

^{V}

_{i+3}−

*ω*

^{U}

_{i+3}} and {

*ω*

^{V}

_{i}−

*ω*

^{U}

_{i}}:

As an example, let us consider Fibonacci canonical cells ${\mathcal{F}}_{i}$ whose parameters are those in table 1 (*γ* ≈ 2.125) and *l*_{S}/*l*_{L} = 1 (*β* = 1). For this class of structures, local scaling governed by (6.4) and (6.5) at *ω*_{c} = 37.596 krad s^{−1} is analysed. The numerical results are illustrated in figure 13, where the band gap associated with ${\mathcal{F}}_{5}$ is compared with that corresponding to ${\mathcal{F}}_{8}$ using close-up views of both the diagrams *η*_{5} and *η*_{8} (respectively, (*a*) and (*c*)) and the flow lines on the reduced tori ${\mathbb{T}}_{5}$ and ${\mathbb{T}}_{8}$ (respectively, (*b*) and (*d*)). The canonical frequency *ω*_{c} is indicated with green points and the magenta dot-dashed vertical lines on the left-hand sides, while the extremes of the band gaps *U* and *V* are denoted by the red points on the right. We note also in this case the perfect correspondence between the band gaps detected through the trace diagrams and the intersections of the flow lines with the subdomains ${\mathbb{D}}_{5}$ and ${\mathbb{D}}_{8}$. Concerning the band gap reported in figure 13*a*,*b*, numerical calculations yield *ω*^{V}_{5} − *ω*^{U}_{5} = 3.407 krad s^{−1} and *λ* = 18.12. By using the relationship (6.5), the value *ω*^{V}_{8} − *ω*^{U}_{8} ≈ (*ω*^{V}_{5} − *ω*^{U}_{5})/*λ* = 0.188 krad s^{−1} is obtained, which is in very good agreement with the value provided by direct estimation of the band gap highlighted in figure 13*c*,*d* (i.e. *ω*^{V}_{8} − *ω*^{U}_{8} = 0.186 krad s^{−1}). We record the same scaling behaviour by comparing *ω*^{V}_{5} − *ω*^{U}_{5} with *ω*^{V}_{11} − *ω*^{U}_{11} centred at the same *ω*_{c}. In this case, the scaling factor is *κ* = *κ*^{+}(*ω*_{c}) = 328.25, the actual range *ω*^{V}_{11} − *ω*^{U}_{11} measures 0.0103 krad s^{−1}, whereas relationship (6.4) provides *ω*^{V}_{11} − *ω*^{U}_{11} ≈ (*ω*^{V}_{5} − *ω*^{U}_{5})/*κ* = 0.0104 krad s^{−1}.

The proposed example demonstrates how, in addition to the standard representation of the dispersion diagram, the typical scaling properties can be also pinpointed and estimated through the universal representation of the torus.

### 7. Concluding remarks

The characteristic features of the frequency spectrum for elastic waves propagating in a two-phase periodic medium can be revealed through its universal representation on a two-dimensional toroidal surface composed of pass band and band gap subregions. Frequency-dependent flow lines belonging to the surface can be defined for each configuration of the waveguide. In this paper, we exploited this possibility to investigate axial waves for a class of periodic rods whose elementary cell is generated through the Fibonacci substitution rule, an example of quasicrystalline sequence.

First, we have established the mechanical and geometrical conditions for which an elementary cell of the Fibonacci sequence may display closed flow lines on the torus, a circumstance that corresponds to the periodicity of the frequency spectrum and of the layout of pass bands and band gaps. We concluded that the required combination of parameters corresponds to that leading to the concept of *canonical* structures introduced by Morini & Gei [23]. For these types of arrangements, it turned out that the band gap density depends on the lengths of the two phases. Conversely, for non-canonical rods, the flow lines cover ergodically the torus and their band gap density is independent of the lengths of the constituents.

Second, we addressed analytically two illustrative band gap optimization problems, based on element ${\mathcal{F}}_{3}$ of the Fibonacci sequence. Analytical expressions of the boundaries of band gap regions on the torus were exploited, on the one hand, to guide the design of the elementary cell to achieve the widest low-frequency band gap, on the other, to detect the maximal band gap in the spectrum for a given configuration. Thanks to the availability of the expressions of the boundaries of the band-gap regions on the torus, the proposed optimization technique is considerably more robust in comparison with the standard procedure based on the partial evaluation of the frequency spectrum [31–33], which necessarily relies on numerical algorithms.

In the final section, the local scaling governing the spectrum of quasicrystalline-generated phononic rods about certain relevant frequencies, as revealed in [23], was investigated and confirmed through the analysis of the flow lines on the torus.

The presented approach, based on the representation of pass band and band gap sub-regions on the toroidal manifold, can be easily extended to study other wave phenomena governed by an equation similar to (2.2) in different periodic systems, i.e. prestressed laminates, photonic crystals and composite nanostructures. Moreover, through the definition of an appropriate set of invariants that fully characterize the pass band/band gap layout, similar universality properties can be detected in spectra associated with different types of equations, such as, for example, those related to flexural systems [46,47], thin soft dielectric films [48] and plane strain laminates [49].

### Data accessibility

This article has no data.

### Author's contributions

L.M. and Z.G.T. carried out the numerical calculations. L.M. and M.G. performed the data analysis. L.M., G.S. and M.G. drafted the manuscript. All authors read and approved the manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Funding

L.M. and M.G. were funded by the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie Actions COFUND grant SIRCIW, agreement no. 663830. G.S. was supported from ISF (grant no. 1912/15) and BSF (grant no. 2014358).

## Acknowledgments

The authors gratefully acknowledge Dr Ram Band, Department of Mathematics, Technion-Israel Institute of Technology, for their insightful discussions on the research topic of the paper.

## Footnotes

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