Power spectrum analysis and missing level statistics of microwave graphs with violated time reversal invariance
Malgorzata Bialous, Vitalii Yunko, Szymon Bauch, Michal Lawniczak, Barbara Dietz, Leszek Sirko
PPower spectrum analysis and missing level statistics of microwave graphs withviolated time reversal invariance
Ma(cid:32)lgorzata Bia(cid:32)lous, Vitalii Yunko, Szymon Bauch, Micha(cid:32)l (cid:32)Lawniczak, Barbara Dietz, ∗ and Leszek Sirko † Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668 Warszawa, Poland (Dated: March 2, 2018)We present experimental studies of the power spectrum and other fluctuation properties in thespectra of microwave networks simulating chaotic quantum graphs with violated time reversal in-variance. On the basis of our data sets we demonstrate that the power spectrum in combinationwith other long-range and also short-range spectral fluctuations provides a powerful tool for theidentification of the symmetries and the determination of the fraction of missing levels. Such a pro-cedure is indispensable for the evaluation of the fluctuation properties in the spectra of real physicalsystems like, e.g., nuclei or molecules, where one has to deal with the problem of missing levels.
PACS numbers: 05.40.-a,05.45.Mt,05.45.Tp,03.65.Sq
Introduction. — In the last decades the concept ofquantum chaos, that is, the understanding of the fea-tures of the classical dynamics in terms of the spectralproperties of the corresponding quantum system, like nu-clei, atoms, molecules, quantum wires and dots or othercomplex systems [1–3], has been elaborated extensively.It has been established by now that the spectral proper-ties of generic quantum systems with classically regulardynamics agree with those of Poissonian random num-bers [4] while they coincide with those of the eigenvaluesof random matrices [6] from the Gaussian orthogonal en-semble (GOE) and the Gaussian unitary ensemble (GUE)for classically chaotic systems with and without time-reversal ( T ) invariance [5], respectively, in accordancewith the Bohigas-Giannoni-Schmit (BGS) conjecture [7].A multitude of studies with focus on problems from thefield of quantum chaos have been performed by now theo-retically and numerically. However, there are nongenericfeatures in the spectra of real physical systems that arenot yet fully understood. Such problems are best tackledexperimentally with the help of model systems like mi-crowave billiards [8, 9] and microwave graphs [10, 11]. Inthe experiments with microwave billiards the analogy be-tween the scalar Helmholtz equation and the Schr¨odingerequation of the corresponding quantum billiard is ex-ploited. Microwave graphs [10, 11] simulate the spectralproperties of quantum graphs [12–14], networks of one-dimensional wires joined at vertices. They provide anextremely rich system for the experimental and the theo-retical study of quantum systems, that exhibit a chaoticdynamics in the classical limit.The idea of quantum graphs was introduced by Li-nus Pauling to model organic molecules [15] and theyare also used to simulate, e.g., quantum wires [16],optical waveguides [17] and mesoscopic quantum sys-tems [18, 19]. The validity of the BGS conjecturewas proven rigourously for graphs with incommensurablebond lengths in Refs. [20, 21]. Accordingly, the fluctua-tion properties in the spectra of classically chaotic quan-tum graphs with and without T invariance are expected to coincide with those of random matrices from the GOEand the GUE, respectively. This was confirmed experi-mentally [10, 11] for the nearest-neighbor spacing distri-bution using microwave networks [22–26].The statistical analysis of the spectral properties ofa quantum system and the comparison with the con-ventional GOE or GUE results requires complete se-quences of eigenvalues belonging to the same symmetryclass [7, 27]. Accordingly, the experimental determina-tion of the chaoticity of a system on the basis of thespectral fluctuation properties might be far from simple,since several effects, like, e.g., nongeneric contributions asin the case of the stadium billiard [28], the existence oftiny islands of regular dynamics in the chaotic sea [29],mixed symmetries or incomplete spectra may result indeviations from the random-matrix theory (RMT) pre-dictions.We are not aware of experimental studies including theanalysis of long-range spectral fluctuations in incompletespectra of chaotic systems with violated T invariance,which, as outlined below, is essential to be able to obtainconclusive results on the spectral properties. Our objec-tive is to fill this gap. T violation was tested experimen-tally, e.g., in nuclear spectra and in compound-nucleusreactions [30, 31] and in electron transport through quan-tum dots, where T violation is induced by a magneticfield [32]. Furthermore, T violation in scattering sys-tems was studied thoroughly in experiments with mi-crowave billiards [33, 34]. The effects of T violation onthe spectral properties of the eigenvalues of closed quan-tum systems have also been investigated in such sys-tems [35–37]. However, it is difficult if not impossibleto obtain complete T violation in microwave billiards,whereas its achievement is straightforward in microwavenetworks [22–26].In this Letter we will develop a procedure to obtaininformation on the chaoticity and T symmetry of a clas-sical system from the spectral properties of the corre-sponding quantum system in the presence of missing lev-els. Incomplete spectra are actually a problem one has a r X i v : . [ n li n . C D ] S e p to cope with in real physical systems like, e.g., nuclei andmolecules [38–41], so such a procedure is a requisite fortheir analysis [42, 43]. It is applied to the spectra of ir-regular, fully connected microwave networks simulatingquantum graphs with violated T invariance. The impactof missing levels on the spectral fluctuation propertiesis particularly large for long-range spectral fluctuations.It was demonstrated numerically in Ref. [44] that thepower spectrum [45, 46] is a powerful statistical mea-sure to discriminate between deviations caused by miss-ing levels and by the mixing of symmetries. Additionalevidence for these effects may be obtained on the basis ofcommonly used statistical measures for short- and long-range spectral fluctuations [43]. Accordingly, in orderto unambiguously identify the symmetry of the systemand the fraction of missing levels, we considered all thesestatistical measures. Experimental setup. — We simulate quantum graphsexperimentally by using a network of coaxial microwavecables, that are coupled by junctions at the vertices. Aphotograph of one example is shown in Fig. 1. The mi-crowave networks comprised 6 junctions, that were allconnected with each other by coaxial cables, in order tosimulate a fully connected quantum graph. The coax-ial cables (SMA-RG402) consist of an inner conductor ofradius r = 0 .
05 cm, which was surrounded by a con-centric conductor of inner radius r = 0 .
15 cm. Thespace between them was filled with Teflon. Measure-ments yielded a dielectric constant ε (cid:39) .
06. Below thecut-off frequency of the TE mode ν c (cid:39) cπ ( r + r ) √ ε =33 .
26 GHz [47, 48] only the fundamental TEM mode canpropagate inside a coaxial cable. Note, that not the ge-ometric lengths L i of the coaxial cables, but the opti-cal lengths L opti = L i √ ε yield the lengths of the bondsin the corresponding quantum graph. The analogy be-tween a quantum graph and a microwave network withthe same topology relies on the formal equivalence of thewave equations governing the wave function ψ ij ( x ) of aparticle moving in the bond connecting vertices i and j of a quantum graph and the potential difference U ij ( x )between the inner and the outer conductors in the corre-sponding coaxial cable. In the first case, the equation isgiven by the one-dimensional Schr¨odinger equation withNeumann boundary conditions at the vertices connectingthe different bonds. In the second case, it coincides withthe Telegraph equation, again with Neumann boundaryconditions at the junctions connecting the coaxial cables.The T violation was induced with five Anritsu PE8403microwave circulators with low insertion loss which op-erate in the frequency range from 7 −
14 GHz. Theseare non-reciprocal three-port passive devices. A waveentering the circulator through port 1, 2 or 3 exits atport 2, 3, or 1, respectively, as illustrated schematicallyin the right-upper inset of Fig. 1. The scattering matrix
FIG. 1: Photograph of one realization of a microwave net-work. An ensemble of 30 different networks was created bychanging the lengths of four bonds using the phase shiftersvisible at the bottom of the graph. Time-reversal invariancewas induced by five microwave circulators. One is shown en-larged in the right-upper inset to illustrate their functionality.For the measurement of the scattering matrix, the vector net-work analyzer (VNA) was coupled to the network via a HP85133-616 flexible microwave cable; see lower inset. element S was measured using an Agilent E8364B mi-crowave vector network analyzer (VNA), connected to asix-arm vertex of the network via a HP 85133-616 flexiblemicrowave cable; see lower inset in Fig. 1. Figure 2 showsa part of one measured reflection spectrum. Due to theunavoidable absorption in the walls of the cables used asbonds it exhibits weakly overlapping resonances, of whichthe positions yield the eigenvalues of the correspondingquantum graph. Accordingly, their determination wasa non-trivial task. We compared the measured reflec-tion spectra of an ensemble of 30 different realizationsof graphs with the same total optical length L (cid:39) . L i (cid:39) −
65 cm with phase shifters (see Fig. 1) insteps of ± .
112 cm, thus yielding slightly differing posi-tions of the resonances. An estimate using Weyl’s law forquantum graphs [13] indicated that approximately 4 %of the eigenvalues were missing.
Fluctuations in the experimental spectra. — For theanalysis of the spectral properties of the microwave net-works, first, the resonance frequencies need to be rescaledin order to eliminate system specific properties like thetotal length L of the graph. This is done with the helpof Weyl’s law, which states, that the resonance density ρ ( ν ) = L / (2 π ) is uniform. Accordingly, the rescaledeigenvalues are determined from the resonance frequen-cies as (cid:15) i = ν i L / (2 π ), with the frequencies sorted suchthat ν i ≤ ν i +1 . FIG. 2: A reflection spectrum starting at 7 GHz. The cir-culators operate only above that frequency. The resonancesobviously overlap. This makes the determination of the res-onance frequencies very difficult and thus calls for a reliabletheoretical description which accounts for missing levels.
A commonly used measure for short-range spectralfluctuations is the nearest-neighbor spacing distribution,that is, the distribution of the spacings between adja-cent eigenvalues, s i = (cid:15) i +1 − (cid:15) i . For long-range spectralfluctuations these are the variance Σ ( L ) of the num-ber of eigenvalues in an interval L and the stiffness ofthe spectrum ∆ ( L ), given by the least-squares devia-tion of the integrated resonance density of the eigenval-ues from the straight line best fitting it in the interval L [6]. The histogram and the circles in Fig. 3 show thenearest-neighbor spacing distribution P ( s ) in (a), its in-tegral I ( s ) = (cid:82) s d s (cid:48) P ( s (cid:48) ) in (b), the number varianceΣ in (c) and the stiffness ∆ in (d). The experimen-tal curves were generated by computing the averages ofthe statistical measures obtained for each of the 30 mi-crowave networks. Here, for each of them 250 resonancefrequencies could be identified. While the short-rangespectral fluctuations in (a) and (b) seem to coincide wellwith those of the eigenvalues of random matrices fromthe GUE (full black lines), this is not the case for thelong-range spectral fluctuations in (c) and (d).Another statistical measure for long-range spectralfluctuations is the power spectrum of the deviation ofthe q th nearest-neighbor spacing from its mean value q , δ q = (cid:15) q +1 − (cid:15) − q . It is given in terms of the Fourierspectrum from ’time’ q to k , S ( k ) = | ˜ δ k | , with˜ δ k = 1 √ N N − (cid:88) q =0 δ q exp (cid:18) − πikqN (cid:19) (1)when considering a sequence of N levels. The power spec-trum has not established itself widely, even though, as wewill demonstrate in this Letter, it provides a particularlyuseful statistical measure, especially in the presence ofmissing levels. It was shown in Refs. [45, 46], that for FIG. 3: (Color online) Spectral properties of the rescaled res-onance frequencies. Panels (a)-(d) show the nearest-neighborspacing distribution P ( s ) (histogram), its integral I ( s ) (cir-cles), the number variance Σ (circles) and the stiffness ∆ statistic (circles), respectively. The experimental results arecompared to those of the eigenvalues of random matricesfrom the GUE curves (black full lines) and the correspond-ing missing-level statistics (red [gray] dashed lines) with ϕ =0 . k/N (cid:28) k = k/N exhibits a power law depen-dence (cid:104) S (˜ k ) (cid:105) ∝ (˜ k ) − α . Here, for regular systems α = 2and for chaotic ones α = 1 independently of whether T invariance is preserved or not. The power spectrumand this power law behavior was studied numerically inRef. [49–52], experimentally in a microwave billiard withclassically chaotic dynamics in Ref. [53] and for a sin-gular rectangular microwave billiard in Ref. [54]. Re-cently, it was successfully applied to the measured molec-ular resonances in Er and
Er [39]. These systemspreserve T invariance, whereas for the case of violated T invariance in the presence of missing levels there wasa lack of experimental studies. This was the motivationfor the experiments presented in this Letter.In Fig. 4 the experimental power spectrum (circles) iscompared to that for the eigenvalues of random matricesfrom the GUE (black full line). Both curves are plot-ted versus ˜ k . We observe that, firstly, both curves startto deviate from each other below log ˜ k (cid:46) − .
5. Sec-ondly, the experimental (cid:104) S (˜ k ) (cid:105) does not exhibit a clearpower law behavior for small ˜ k . These deviations, andalso those observed for the long-range spectral fluctua-tions in Fig. 3 (c) and (d) cannot result from a mixingof symmetries [29, 49–52], since the short-range spectralfluctuations are well described by GUE statistics. How-ever, similar to Ref. [39], they can be attributed to thesmall fraction of missing levels, as demonstrated in thesequel. Missing level statistics. — As stated above, the com-pleteness of energy spectra is a rather rare situation in ex-
FIG. 4: (Color online) The average power spectrum. Theexperimental results (black circles) are compared to that forthe eigenvalues of random matrices from the GUE (black fullline) and the corresponding missing-level statistics (red [gray]dashed line). The fraction of the observed levels was unam-biguously determined to ϕ = 0 . ± . perimental investigations [40–42]. The problem of miss-ing levels can be circumvented in open systems, like mi-crowave billiards or microwave networks, where scatter-ing matrix elements are available. Their fluctuation prop-erties provide measures for the chaoticity, e.g., in termsof their correlation functions [33, 34] or the enhancementfactor [11, 55, 56]. For closed systems, analytical expres-sions were derived for incomplete spectra based on RMTin Ref. [43]. The nearest-neighbor spacing distribution isexpressed in terms of the ( n +1)st nearest-neighbor spac-ing distribution P ( n, s ), with P (0 , s ) = P ( s ). It is wellapproximated by P ( n, s ) (cid:39) γs µ e − κ s , where for n = 0 , µ = 1 , µ = 2 , γ and κ are obtained from the normalizationof P ( n, s ) to unity and the scaling of s to average spacingunity, respectively. If the fraction of detected eigenvalues ϕ is close to unity, the nearest-neighbor spacing distribu-tion accounting for missing levels, is given by p ( s ) (cid:39) P (cid:18) sϕ (cid:19) + (1 − ϕ ) P (cid:18) , sϕ (cid:19) + .... (2)Similarly, the number variance Σ and the stiffness ∆ may be expressed in terms of those for complete spectra( ϕ = 1), σ ( L ) = (1 − ϕ ) L + ϕ Σ (cid:18) Lϕ (cid:19) (3)and δ ( L ) = (1 − ϕ ) L
15 + ϕ ∆ (cid:18) Lϕ (cid:19) . (4)In Fig. 3 the functions Eq. (2)-(4) are plotted for ϕ =0 .
965 as red [gray] dashed lines. The agreement with the corresponding experimental results is remarkable. Likethe experimental nearest-neighbor spacing distributionthe curve obtained from Eq. (2) is close to that of theeigenvalues of random matrices from the GUE. This fea-ture enabled the assignment of the GUE as the RMTmodel applicable to the experimental data. In order tocorroborate that the deviations from GUE observed inFigs. 3 and 4 indeed are solely due to missing levels weanalysed power spectra. An analytical expression wasderived for the power spectrum of incomplete spectra inRef [44], (cid:104) s (˜ k ) (cid:105) = ϕ π K (cid:16) ϕ ˜ k (cid:17) − k + K (cid:16) ϕ (cid:16) − ˜ k (cid:17)(cid:17) − − ˜ k ) + 14 sin ( π ˜ k ) − ϕ , (5)which for ϕ = 1 yields that for complete spectra. Here,0 ≤ ˜ k ≤ K ( τ ) is the spectral form factor, whichequals K ( τ ) = τ for the GUE.This analytical result is shown as red dashed curve inFig. 4. The fraction of observed levels, actually, was de-termined to ϕ = 0 . ± .
005 from the power spectrum,which depends particularly sensitively on the value of ϕ .This is illustrated in Fig. 5, where we compare its asymp-totic behavior to experimental results. Here, the fraction ϕ was varied by randomly eliminating resonance frequen-cies. The power spectra for different values of ϕ lie closeto one another. However, they still are clearly distin-guishable. To illustrate this, each curve was shifted byunity with respect to its lower neighbor in Fig. 5. Evenfor the case of only 70 % of observed levels, we find goodagreement between the analytical result Eq. (5) and theexperimental one. Conclusions. — We present first experimental studiesof the fluctuation properties in incomplete spectra ofmicrowave networks simulating chaotic quantum graphswith broken time reversal symmetry. The experimen-tal results are in good agreement with the analytical ex-pressions for missing level statistics Eqs. (2)-(4) derivedin Ref. [43] and Eq. (5) for the power spectrum givenin Ref. [44]. All these expressions explicitly take intoaccount the fraction of observed levels ϕ , however, thepower spectrum is particularly sensitive to it. Therefore,we used it to determine the fraction of observed levels, ϕ = 0 . ± . ϕ for ϕ (cid:38) .
9. Long-range spectral fluc-tuations were then used to confirm this assignment. Theexcellent agreement between the experimental and theanalytical results, demonstrated in Fig. 5 for a range of0 . ≤ ϕ ≤ . T invariance. FIG. 5: Illustration of the sensitivity of the average powerspectrum to changes in the fraction of observed levels ϕ . Thesymbols show results which were generated from the experi-mental data by randomly eliminating resonance frequencies.Triangles-right show the data from Fig. 4 with ϕ = 0 . ϕ = 0 .
9, triangles-up to ϕ = 0 . ϕ = 0 .
8, triangles-down to ϕ = 0 .
75 and squares to ϕ = 0 .
7. In order to allow a distinct demonstration of theexcellent agreement between the theoretical and the experi-mental results, all except the curve for ϕ = 0 .
965 were shiftedwith respect to their lower neighbor by unity.
This work was partially supported by the Min-istry of Science and Higher Education grant UMO-2013/09/D/ST2/03727 and the EAgLE project (FP7-REGPOT-2013-1, Project Number: 316014). ∗ Electronic address: [email protected] † Electronic address: [email protected][1] J. M. G. G´omez, K. Kar, V. K. B. Kota, R. A. Molina, A.Rela˜no, and J. Retamosa, Phys. Rep. , 103 (2011).[2] H. A. Weidenm¨uller and G. E. Mitchell, Rev. Mod. Phys. , 539 (2009).[3] F. Haake, Quantum Signatures of Chaos (Springer-Verlag, Heidelberg, 2001).[4] M.V. Berry and M. Tabor, Proc. R. Soc. A , 375(1977).[5] The Gaussian ensembles are ensembles of random matri-ces, of which the entries are Gaussian distributed withzero mean. Random matrices from the GOE and theGUE are real symmetric and hermitian, respectively.[6] M. L. Mehta,
Random Matrices (Academic Press, Lon-don, 1990).[7] O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev.Lett. , 1 (1984).[8] H.-J. St¨ockmann, Quantum Chaos: An Introduction (Cambridge University Press,Cambridge,2000)[9] B. Dietz and A. Richter, CHAOS , 097601 (2015).[10] O. Hul, S. Bauch, P. Pako´nski, N. Savytskyy, K.˙Zyczkowski, and L. Sirko, Phys. Rev. E , 056205(2004).[11] M. (cid:32)Lawniczak, S. Bauch, O. Hul, and L. Sirko, Phys. Rev. E , 046204 (2010).[12] T. Kottos, U. Smilansky, Phys. Rev. Lett. , 4794(1997).[13] T. Kottos, U. Smilansky, Ann. Phys. , 76 (1999).[14] P. Pako´nski, K. ˙Zyczkowski, M. Ku´s J. Phys. A , 673 (1936).[16] J. A. Sanchez-Gil, V. Freilikher, I. Yurkevich, and A. A.Maradudin, Phys. Rev. Lett. , 948 (1998).[17] R. Mittra, S. W. Lee, Analytical Techniques in the Theoryof Guided Waves (Macmillan, NY, 1971).[18] D. Kowal, U. Sivan, O. Entin-Wohlman, and Y. Imry,Phys. Rev. B , 9009 (1990).[19] Y. Imry, Introduction to Mesoscopic Systems (Oxford,NY, 1996).[20] S. Gnutzmann and A. Altland, Phys. Rev. Lett. ,194101 (2004).[21] Z. Pluhaˇr and H. A. Weidenm¨uller, Phys. Rev. Lett. ,144102 (2014).[22] M. (cid:32)Lawniczak, O. Hul, S. Bauch, P. Seba, and L. Sirko,Phys. Rev. E , 056210 (2008).[23] M. (cid:32)Lawniczak, S. Bauch, O. Hul, and L. Sirko, Phys. Scr. T143 , 014014 (2011).[24] O. Hul, M. (cid:32)Lawniczak, S. Bauch, A. Sawicki, M. Ku´s, L.Sirko, Phys. Rev. Lett , 040402 (2012).[25] M. (cid:32)Lawniczak, A. Sawicki, S. Bauch, M. Ku´s, andL. Sirko, Phys. Rev E , 022925 (2014).[27] O. Bohigas, R. U. Haq, and A. Pandey, in Nuclear Datafor Science and Technology , ed. by K. H. B¨ockhoff (Rei-del, Dordrecht, 1983).[28] M. Sieber, U. Smilansky, S. C. Creagh, and R. G. Little-john, J. Phys. A , 6217 (1993).[29] B. Dietz, T. Guhr, B. Gutkin, M. Miski-Oglu, and A.Richter, Phys. Rev. E , 022903 (2014).[30] J. B. French, V. K. B. Kota, A. Pandey, and S. Tomsovic,Phys. Rev. Lett. , 2313 (1985).[31] C. E. Mitchell, A. Richter, H. A. Weidenm¨uller, Rev.Mod. Phys. , 2845 (2010).[32] Z. Pluhaˇr, H. A. Weidenm¨uller, J. A. Zuk, C. H.Lewenkopf, and F. J. Wegner, Ann. Phys. , 1 (1995).[33] B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu,A. Richter, F. Sch¨afer, J. Verbaarschot, and H. A. Wei-denm¨uller, Phys. Rev. Lett. , 064101 (2009).[34] B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A.Richter, F. Sch¨afer and H. A. Weidenm¨uller, Phys. Rev.E , 036205 (2010).[35] P. So, S. M. Anlage, E. Ott, and R. N. Oerter, Phys. Rev.Lett. , 2662 (1995).[36] U. Stoffregen, J. Stein, H.-J. St¨ckmann, M. Ku´s, and F.Haake, Phys. Rev. Lett. , 2666 (1995).[37] J. S. A. Bridgewater, A. Gokirmak, and S. M. Anlage,Phys. Rev. Lett. , 2890 (1998).[38] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. Bohn, C.Makrides, A. Petrov, and S. Kotochigova, Nature (Lon-don) , 475 (2014).[39] J. Mur-Petit and R. A. Molina, Phys. Rev. E , 042906(2015).[40] H. I. Liou, H. S. Camarda, and F. Rahn, Phys. Rev. C , 131 (1972).[41] T. Zimmermann, H. K¨oppel, L. S. Cederbaum, G. Per-sch, and W. Demtr¨oder, Phys. Rev. Lett. , 3 (1988). [42] U. Agvaanluvsan, G. E. Mitchell, J. F. Shriner Jr., M.Pato, Phys. Rev. C , 064608 (2003).[43] O. Bohigas and M. P. Pato, Phys. Lett. B , 171(2004).[44] R.A. Molina, J. Retamosa, L. Mu˜noz, A. Rela˜no, and E.Faleiro, Phys. Lett. B , 25 (2007).[45] A. Rela˜no, J.M.G. G´omez, R. A. Molina, J. Retamosa,and E. Faleiro, Phys. Rev. Lett. , 244102 (2002).[46] E. Faleiro, J. M. G. G´omez, R. A. Molina, L. Mu˜noz, A.Rela˜no, and J. Retamosa, Phys. Rev. Lett. , 244101(2004).[47] D. S. Jones, Theory of Electromagnetism (PergamonPress, Oxford, 1964), p. 254.[48] N. Savytskyy, A. Kohler, S. Bauch, R. Bl¨umel, and L.Sirko, Phys. Rev. E , 036211 (2001).[49] J. M. G. G´omez, A. Rela˜no, J. Retamosa, E. Faleiro, L.Salasnich, M. Vraniˇcar, and M. Robnik, Phys. Rev. Lett. , 084101 (2005).[50] L. Salasnich, Phys. Rev. E , 047202 (2005).[51] M. S. Santhanam and J. N. Bandyopadhyay, Phys. Rev.Lett. , 114101 (2005).[52] A. Rela˜no, Phys. Rev. Lett. , 224101 (2008).[53] E. Faleiro, U. Kuhl, R.A. Molina, L. Mu˜noz, A. Rela˜no,and J. Retamosa, Phys. Lett. A , 251 (2006).[54] M. Bia(cid:32)lous, V. Yunko, M. (cid:32)Lawniczak, S. Bauch, B. Dietz,and L. Sirko, in preparation.[55] M. (cid:32)Lawniczak, S. Bauch, and L. Sirko, in Handbook ofApplications of Chaos Theory , eds. Christos Skiadas andCharilaos Skiadas (CRC Press, Boca Raton, USA, 2016),p. 559.[56] M. (cid:32)Lawniczak, M. Bia(cid:32)lous, V. Yunko, S. Bauch, and L.Sirko, Phys. Rev. E91