Physics Letters A 376 (2012) 1907–1914

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Physics Letters A

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Torus-doubling process via strange nonchaotic attractors

Takahito Mitsui a,b,∗, Seiji Uenohara c, Takashi Morie c, Yoshihiko Horio d, Kazuyuki Aihara b

a FIRST, Aihara Innovative Mathematical Modelling Project, JST, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japanb Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japanc Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, Kitakyushu 808-0196, Japand Graduate School of Advanced Science and Technology, Tokyo Denki University, 5 Asahi-cho, Adachi-ku, Tokyo 120-8551, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 January 2012Received in revised form 2 April 2012Accepted 3 April 2012Available online 5 April 2012Communicated by C.R. Doering

Keywords:Torus doublingStrange nonchaotic attractors

Torus-doubling bifurcations typically occur only a finite number of times. It has been assumed that torus-doubling bifurcations in quasiperiodically forced systems are interrupted by the appearance of strangenonchaotic attractors (SNAs). In the present Letter, we study a quasiperiodically forced noninvertiblemap and report the occurrence of a torus-doubling process via SNAs. The mechanism of this process isnumerically clarified. Furthermore, this process is experimentally demonstrated in a switched-capacitorintegrated circuit.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Torus-doubling bifurcations have been studied in various dy-namical systems, ranging from finite-dimensional [1–7] to infinite-dimensional systems [8], and also have been observed in severalexperiments on fluid convection [9–11], chemical reactions [12],and various nonlinear circuits [13–15]. Unlike the period-doublingcascades of cycles, torus-doubling bifurcations typically occur onlya finite number of times [3].

In this study, we focus torus-doubling phenomena in quasiperi-odically forced systems, where a torus never becomes resonant.A remarkable aspect of these systems is the appearance of strangenonchaotic attractors (SNAs). An SNA is a geometrically strange at-tractor that is not exponentially sensitive to initial conditions [16](see Refs. [17–19] for reviews). In the literature [3,20], it has beenassumed that torus-doubling bifurcations are interrupted by theappearance of strange nonchaotic attractors. Using the quasiperiod-ically forced logistic map, Kuznetsov et al. [4] have shown that thetorus-doubling bifurcation curve terminates when a torus touchesa line that corresponds to an extremal point of the map. The endpoint of a torus-doubling bifurcation curve is called the torus-doubling terminal (TDT) point. Beyond the TDT point, the torus-doubling bifurcation cannot occur owing to the emergence of su-perstable orbits on the torus, and the torus transits into an SNA ora chaotic attractor as the parameters vary.

* Corresponding author at: FIRST, Aihara Innovative Mathematical ModellingProject, JST, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan.

E-mail address: mitsui@sat.t.u-tokyo.ac.jp (T. Mitsui).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.04.005

In a previous study [21], we implemented an electronic circuitfor a quasiperiodically forced noninvertible map and demonstratedthat the circuit faithfully reproduced various kinds of dynamicalbehavior of the map. In the present Letter, we study the quasiperi-odically forced noninvertible map and report the occurrence of atorus-doubling process via strange nonchaotic attractors. It should bementioned that the torus-doubling process via SNAs has appeareda few times in the literature [22,23] (see Section 4). However, thereis no in-depth analysis of this process. We numerically clarify themechanism of this process and confirm it experimentally.

The Letter is organized as follows. In Section 2, the torus-doubling process via SNAs is numerically studied. In Section 3, thetorus-doubling process via SNAs is demonstrated using the experi-mental circuit. Section 4 concludes the Letter with a summary anda discussion.

2. Numerical study

We consider the following quasiperiodically forced noninvert-ible map

xn+1 = h(xn) + b sin 2πθn,

θn+1 = θn + ω (mod 1) (1)

with

h(x) ≡ kx − 1

1 + e−x/ε+ a,

where x ∈ R, θ ∈ T1 ∼= [0,1), 0 < k < 1, ε > 0, and a ∈ R. Here, θ

is the phase of the quasiperiodic force with an amplitude b and anirrational frequency ω.

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Fig. 1. Function h(x) for k = 0.1, ε = 0.02, and a = 0.5. The arrows indicate thepoints of zero derivative.

First, let us consider the unforced case with b = 0. Conse-quently, map (1) reduces to a one-dimensional (1D) map xn+1 =h(xn), proposed as a model of chaotic dynamics of a biologicalneuron [24]. As shown in Fig. 1, the function h(x) has two pointsof zero derivative at x = x± = −ε ln[(1 − 2kε ∓ √

1 − 4kε )/(2kε)],where x− < 0 < x+ . These points correspond to the lines ofzero derivative {(x, θ) ∈ R × T

1|x = x±}. Depending on the pa-rameters, the 1D map exhibits rich, nonlinear phenomena suchas period-doubling cascades to chaos, intermittency, and dev-il’s staircases. If the 1D map xn+1 = h(xn) has an invariant setdenoted by Λ, the direct product Λ × T

1 is an invariant setof map (1). For example, if Λ is a fixed point, map (1) ex-hibits a quasiperiodic motion on an invariant circle. This invari-ant circle is regarded as the cross-section of a two-frequencytorus in three-dimensional phase space. Conventionally, this cir-cle is called a “torus”. Note that the period doubling of a cyclein the 1D map implies the doubling of a torus in map (1) forb = 0.

Our interests lie in torus-doubling phenomena under quasiperi-odic forcing with b �= 0. Accordingly, we fix ε = 0.02 and ω =(√

5 − 1)/2, the golden mean. Owing to the condition 0 < k < 1,the typical state points converge to an attractor. We characterizeattractors in terms of both the Lyapunov exponent in the direc-tion of x, given by λ = limn→∞(1/n) ln |∂xn/∂x0|, and the phasesensitivity exponent ν , defined as a power-law exponent of thephase sensitivity function Γ (n) ≡ minx0,θ0 [max1� j�n |∂x j/∂θ0|] ∼ nν

(n → ∞) [25]. The exponent ν gives the measure of the sensi-tivity of a nonchaotic attractor with respect to an infinitesimaldifference in phase θ . Then, a quasiperiodic attractor of smooth in-variant curves (torus) is characterized by λ < 0, ν = 0; an SNA, byλ < 0, ν > 0; and a chaotic attractor, by λ > 0.

2.1. The case of small k

Here, we consider a case where k has a small value, i.e., k =10−4; in this case, the torus-doubling process via SNAs is demon-strated clearly. Fig. 2(a) is the phase diagram in the a–b plane fork = 10−4, and the boxed region is magnified in Fig. 2(b). We con-sider only the case a � 1/2 owing to the symmetry with respectto a = 1/2. The SNA region is shown in gray. The other regions arelabeled by a pair of symbols 1T (attracting single torus), 2T (at-tracting doubled torus), α, and β . In regions with α, an attractingtorus does not intersect with the lines of zero derivative in thephase space, while in regions with β , it does. The boundary of re-gions with α and β is shown by a blue dotted line. For small b, lessthan bc ≈ 0.0290504, a torus-doubling bifurcation occurs when theparameters (a,b) cross the red solid line. In this bifurcation, an at-tracting doubled torus and an unstable single torus emerge from aparent single torus, as shown in Figs. 3(a) and (b).

The torus-doubling bifurcation can be described in terms of ra-tional approximations for the irrational frequency ω [4] (see also[26] and references therein). The rational approximations of thegolden mean ω are given by the ratios of Fibonacci numbers, ωl =Fl−1/Fl (l = 1,2, . . .), where F0 = 0, F1 = 1, and Fl+1 = Fl + Fl−1.The lth approximation to map (1) is given by the periodicallyforced map:

xn+1 = h(xn) + b sin 2πθn,

θn+1 = θn + ωl (mod 1). (2)

Then, the union of all attracting sets of map (2) for different ini-tial phases θ0 ∈ [0,1/Fl) results in the lth approximation to theoriginal attractor in the quasiperiodically forced map (1). For suf-ficiently large l, a single torus may be approximated by the unionof all stable period-Fl cycles of map (2). The torus-doubling pointis approximated by the period-doubling points of these cycles. Be-cause these period-doubling points depend on the initial phase θ0,the torus-doubling bifurcation is defined if this phase dependencedisappears as l → ∞. Fig. 3(d) shows the Lyapunov exponent λ ofeach stable cycle in map (2) with l = 7 as a function of the initialphase θ0, where the union of all the cycles approximates the singletorus of Fig. 3(a). Fig. 3(e) shows the same result but for l = 10. Wecan see that the phase dependence disappears as l increases. Bydefinition, the Lyapunov exponent λ is zero at the doubling point[see Fig. 3(c)].

The torus-doubling bifurcation is possible only if the torusis away from the lines of zero derivative, x = x± [4]. As theparameters are increased along the torus-doubling bifurcationcurve, the deformation of the torus becomes larger. Eventually,the torus touches one of the lines of zero derivative; for instance,

Fig. 2. (a) Phase diagram in the a–b plane for k = 10−4. The boxed region is magnified in Fig. 2(b). The SNA region is shaded in gray. The other regions are labeled by a pairof symbols 1T (attracting single torus), 2T (attracting doubled torus), α, and β . In regions with α, an attracting torus does not intersect with the lines of zero derivative inthe phase space, whereas in regions with β , it does. The boundary of regions with α and β is shown by a blue dotted line. The torus-doubling bifurcation occurs on the redsolid line that terminates at the TDT point marked by a cross (+). (For interpretation of the references to color in this figure legend, the reader is referred to the web versionof this Letter.)

T. Mitsui et al. / Physics Letters A 376 (2012) 1907–1914 1909

Fig. 3. Torus-doubling bifurcation for b = 0.025. (a) Attracting single torus at a = −0.0377. (b) Attracting doubled torus (red solid line) and unstable parent torus (blue dashedline) at a = −0.0375. A line of zero derivative, x = x− ≈ −0.2624471875, is shown by the gray dotted line in (a) and (b). (c) Lyapunov exponents λ of attracting torus (λ < 0)and repelling torus (λ > 0) as functions of a. The torus-doubling bifurcation occurs at a ≈ −0.037643, where λ = 0. (d) Lyapunov exponent λ of each stable cycle in map (2)as a function of initial phase θ0. l = 7 (F7 = 13) and a = −0.0377. (e) Same as (d) but for l = 10 (F10 = 55). (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this Letter.)

Fig. 4. Torus-doubling process via SNAs for b = 0.03. (a) Attracting single torus at a = −0.0275. (b) SNA at a = −0.0273975. (c) Attracting doubled torus (red solid line) andunstable parent torus (blue dashed line) at a = −0.0273970. Insets show an enlarged part of each attractor. A line of zero derivative, x = x− ≈ −0.2624471870, is shown bythe gray dotted line in (a), (b), and (c). (d) Lyapunov exponents λ of stable torus (λ < 0) and unstable torus (λ > 0) as functions of a. (e) Lyapunov exponent λ of each stablecycle in map (2) as a function of initial phase θ0. l = 7 (F7 = 13) and a = −0.0275. (f) Same as (e) but for l = 10 (F10 = 55). The arrows indicate the positions of superstableorbits. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

x = x− ≈ −0.2624471875 for a < 0.5. The torus-doubling bifur-cation curve terminates when this tangency occurs at (a,b) =(ac,bc) ≈ (−0.029397840,0.0290504) [the point marked with across in Figs. 2(a) and (b)].

For b � bc , the single torus transits to a doubled torus througha narrow parameter region of SNAs. The torus-doubling processvia SNAs involves two different transitions. The first transition isthe gradual fractalization of torus [27,28]. In this transition, a torus

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Fig. 5. (a) Phase diagram in the a–b plane for k = 0.1. The boxed region is magnified in Fig. 5(b). The SNA and the chaotic attractor regions are shown in gray and black,respectively. The other regions are labeled using symbols 1T (attracting single torus), 2T (attracting doubled torus), and 4T (attracting quadrupled torus). The first torus-doubling bifurcation occurs on the red solid curve that terminates at the point marked by a cross (+); the second torus-doubling bifurcation occurs along the green dashedcurve. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

becomes increasingly wrinkled and finally changes into an SNAwithout apparent mediation of any nearby unstable invariant set.Figs. 4(a) and (b) show a wrinkled single torus at a = −0.02750and an SNA at a = −0.0273975, respectively, for b = 0.03. Thefractalization point is estimated at a ≈ −0.027403 for b = 0.03 bycomputing the phase sensitivity exponent ν . The second transi-tion is the reverse of the Heagy–Hammel transition. In the Heagy–Hammel transition, an attracting doubled torus collides with anunstable parent torus in a dense set of points; consequently, anSNA emerges [20]. Fig. 4(c) shows an attracting doubled torus andan unstable parent torus at a = −0.0273970, just before the col-lision at a ≈ −0.2739726 for b = 0.03. The Lyapunov exponent λ

remains negative in the torus-doubling process via SNAs, as shownin Fig. 4(d).

In the torus-doubling process via SNAs, the attracting torus(single or doubled) intersects with one of the lines of zero deriva-tive, x = x± [see Figs. 4(a) and (c)]. Thus, the attracting torussupports superstable orbits and cannot be marginally (un)stable, atleast uniformly in θ . In terms of approximating map (2), the Lya-punov exponent λ of each stable cycle depends on its initial phaseθ0 owing to the presence of superstable orbits even if l increases[see Fig. 4(e) for l = 7 and Fig. 4(f) for l = 10]. Therefore, in thiscase, the torus cannot undergo the torus-doubling bifurcation.

2.2. The case of relatively large k

As k increases, the dynamical behavior in the vicinity of theTDT point becomes more diversified. Let us consider the casek = 0.1, where the 1D map xn+1 = h(xn) can undergo the sec-ond period-doubling bifurcation. Fig. 5(a) shows the phase diagramfor k = 0.1, and the boxed region is magnified in Fig. 5(b). Here,chaotic attractors and quadrupled tori also appear. We can seethat chaotic attractors, SNAs, and single and doubled tori coexistin the vicinity of the TDT point [4]. However, we emphasize thatthe torus-doubling process via SNAs can occur for large b.

3. Experimental study

The torus-doubling bifurcation and the torus-doubling pro-cess via SNAs can be demonstrated experimentally. We per-formed experiments on the switched-capacitor integrated circuitof the chaotic neuron proposed in our previous studies [29,21].A schematic diagram of the circuit is shown in Fig. 6(a). The cir-cuit was designed to implement the chaotic neuron model [24]under quasiperiodic forcing as follows:

yn+1 = kyn + α f (yn + d cos 2πθn) + c ≡ g(yn, θn),

θn+1 = θn + ω (mod 1), (3)

Fig. 6. (a) Schematic diagram of switched-capacitor integrated circuit used for im-plementing Eq. (3) with C y = Cξ = Ce A = 12.7 pF, Ck = 11.1 pF, Cα = 6.4 pF,Cu = 2.5 pF, and V B = −1.65 V. The circuit operates according to a nonoverlap-ping three-phase clock, which consists of signals φB , φC , and φD . (b) Measuredcharacteristic of nonlinear function f (y).

where y is the internal state of the neuron model, and θ isthe phase of the quasiperiodic forcing with amplitude d and ir-rational frequency ω. The damping parameter k and the scalingparameter α are determined by capacitor ratios as k = 1 − Ck/C y

and α = Cα/C y , respectively. The bias parameter c is given byc = (Cc/C y)V B , where V B is the voltage bias. The function f (·)is a monotonically decreasing continuous function determined bythe physical properties of MOSFETs in the circuit. Fig. 6(b) showsthe measured characteristic of f (·). The circuit operates according

T. Mitsui et al. / Physics Letters A 376 (2012) 1907–1914 1911

Fig. 7. Torus-doubling bifurcation for c = −0.62 V (i.e. Cc = 4.8 pF). (a) Single torus at d = 0.09 V. (b) Torus at d = 0.07 V near doubling point. (c) Doubled torus at d = 0.05 V.The dotted curves show the curve of zero derivative, y = −0.645 − d cos 2πθ . Panels (d), (e), and (f) show the ACFs C(n) corresponding to the attractors in (a), (b), and (c),respectively.

to a nonoverlapping three-phase clock, which consists of signalsφB , φC , and φD . At each iteration, a discrete-time quasiperiodicsignal, d cos 2πθn , is input from an analogue input/output board(National Instruments PXI-6289). Throughout the measurements,we set k = 1.6/12.7, α = 6.4/12.7, and ω = (

√5 − 1)/2. The bias c

and the amplitude d are the control parameters. For further detailson the circuit implementation, see [29].

Map (3) can be transformed into the following form yn+1 =kyn + α f (yn) + c + d cos 2πθn , θn+1 = θn + ω (mod 1), by lineartransformations y + d cos 2πθ → y and d

√1 − 2k cos 2πω + k2 →

d and by a shift in the initial phase θ0 + const. → θ0. Conse-quently, map (3) is qualitatively the same with map (1). Notethat the function g(y, θ) is noninvertible with respect to y owingto the nonlinear characteristic of f (·). As discussed in the previ-ous section, the type of torus doubling is affected by the curvesof zero derivative of function g(y, θ), which are approximated byy = −0.64 − d cos 2πθ and y = 0.62 − d cos 2πθ , respectively.

3.1. Experimental method for characterizing attractors

We use the following measures to distinguish between the dif-ferent types of attractors observed. The normalized autocorrela-tion function (ACF) of variable y is given by C(n) = 〈 yt yt+n〉/〈 y2

t 〉,where yn = yn −〈yt〉 and 〈yt〉 denotes the long-time average of yt

with respect to time t . If the time series yn is quasiperiodic, theACF C(n) is a quasiperiodic function of time n [30]. Accordingly,the ACF C(n) takes values close to 1 in a quasiperiodic manner; ifthe frequency ω is well approximated by a rational number p/q,then C(q) ≈ C(0) ≡ 1. For the time series yn corresponding to anSNA, typically, the ACF C(n) neither comes close to 1 nor decaysto 0 [30,31].

In several experimental studies [32–34], the information di-mension D1 has been used to distinguish between SNAs andchaotic attractors because the standard time-series methods usedfor estimating Lyapunov exponents do not give reliable results forSNAs [35]. The information dimension of an attractor is definedas D1 = limr→0 I(r)/ log2(1/r), where r is the edge length of theboxes covering the attractor. Here, the information I(r) is given by

I(r) = −∑N(r)i=1 pi log2 pi , where pi is the frequency at which a typ-

ical trajectory visits the ith box and N(r) is the minimum numberof boxes required to cover the attractor. For map (3), the informa-tion dimension D1 cannot be less than 1 due to the quasiperiodicmotion of θ . Furthermore, it has been proven that the informationdimension D1 is bounded by the Lyapunov dimension DL fromabove [36]. For SNAs with one zero and one negative Lyapunovexponent, DL = 1 and therefore D1 = 1 [37–39]. This predictioncoincides with the Kaplan–Yorke conjecture such that D1 = DL fortypical attractors [40]. On the other hand, for chaotic attractorswith one zero and one positive Lyapunov exponent, DL = 2 andwe estimate D1 = 2 according to the Kaplan–Yorke conjecture [32,34].

3.2. Experimental results

We observed attractors for several values of c and d, as shownin Figs. 7 and 8. The ACF C(n) and the information dimension D1were calculated from a time series of 105 data points.

The torus-doubling bifurcation is observed for c = −0.62 V (i.e.Cc = 4.8 pF) when the amplitude d decreases. Figs. 7(a), (b), and(c) show a single torus at d = 0.09 V, a torus at d = 0.07 V nearthe doubling point, and a doubled torus at d = 0.05 V, respectively.As shown in Figs. 7(d)–(f), the corresponding ACFs C(n) take valuesclose to one. Thus, all the attractors in this bifurcation route areconsidered to be quasiperiodic tori. The ACF in Fig. 7(e) does notattain values close to 1 for large n. This is because the torus nearthe doubling point has a weak stability in the direction of y and isnot robust against noise in the circuit.

On the other hand, the torus-doubling process via SNAs is ob-served for c = −0.26 V (i.e. Cc = 2.0 pF). Figs. 8(a), (b), and (c)show a single torus at d = 0.370 V, a wrinkled single torus atd = 0.340 V, and an SNA at d = 0.319 V, respectively. The sin-gle torus becomes increasingly wrinkled and appears to changeinto the SNA without any crisis-like event. Therefore, this tran-sition is considered to be the gradual fractalization of the torus[27]. Figs. 8(d), (e), and (f) show an SNA at d = 0.316 V, a dou-bled torus at d = 0.309 V, and a doubled torus at d = 0.300 V,

1912 T. Mitsui et al. / Physics Letters A 376 (2012) 1907–1914

Fig. 8. Torus-doubling process via SNAs for c = −0.26 V (i.e. Cc = 2.0 pF). (a) Single torus at d = 0.370 V. (b) Single torus at d = 0.340 V. (c) SNA at d = 0.319 V. (d) SNA atd = 0.316 V. (e) Doubled torus at d = 0.309 V. (f) Doubled torus at d = 0.300 V. The dotted curves show the line of zero derivative, y = −0.645 − d cos 2πθ . Panels from (g)to (l) show the ACFs C(n) corresponding to the attractors in (a) to (f), respectively.

respectively. In Fig. 8(e), it is observed that the two curves of thedoubled torus are about to merge nonuniformly. Thus, this transi-tion is considered as the reverse of the Heagy–Hammel transition.The ACF C(n) takes values close to 1 for the tori but does not at-tain values close to 1 for the SNAs, as shown in Figs. 8(g)–(l). Let usintroduce a maximum value of ACF, Cmax = max1�n�N C(n), whereN � 1. Fig. 9 shows a plot of Cmax vs. d, where N = 2000. Thetwo smooth curves that constitute a doubled torus are “disjoint”.In other words, the second-iterate map has two attractors, each ofwhich is a single invariant curve. We can identify a transition re-gion denoted by the label “I” in Fig. 9, where the “two curves” arenot disjoint but the dynamics is roughly quasiperiodic in that Cmaxis almost unity.

Fig. 9 also shows a plot of information dimension D1 vs. d. Thedetails of numerical estimates for D1 are given in Appendix A.

The information dimension D1 is close to 1 for all the attrac-tors observed in the torus-doubling process via SNAs. In partic-ular, D1 � 1.08 for the SNA in Fig. 8(c), and D1 � 1.06 for theSNA in Fig. 8(d). Even in the largest case, D1 � 1.16 for an SNAat d = 0.314 V. Due to the effect of noise, the information di-mension D1 of the attractors becomes slightly larger than 1, thetheoretical value for SNAs, but D1 is well below 2, the theoret-ical value for chaotic attractors. Thus, the geometrically strangeattractors observed in this process are considered to be non-chaotic.

Note that the curves of zero derivative are away from the torusin the torus-doubling bifurcation, as shown in Fig. 7, but not soin the torus-doubling process via SNAs, as shown in Fig. 8. Theseobservations are consistent with the numerical results presentedin Section 2.

T. Mitsui et al. / Physics Letters A 376 (2012) 1907–1914 1913

Fig. 9. Maximal value of ACF, Cmax (red circle), and information dimension D1 (bluediamond) as functions of d in the torus-doubling process via SNAs for c = −0.62 V(i.e. Cc = 2.0 pF). The label “I” denotes a transition region between the SNA and thedoubled torus. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this Letter.)

4. Conclusion

In this study, we investigated the torus-doubling bifurcationand the torus-doubling process via SNAs using a quasiperiodicallyforced noninvertible map. The torus-doubling process via SNAsconsists of two different transitions: the gradual fractalization of atorus and the reverse of the Heagy–Hammel transition. We also ex-perimentally demonstrated both types of torus-doubling phenom-ena in a switched-capacitor integrated circuit, which implementsthe quasiperiodically forced noninvertible map.

Some transitions shown in the literature appear to be thetorus-doubling process via SNAs. Venkatesan et al. showed a torus-doubling process through a narrow parameter region of SNAs fora quasiperiodically forced Duffing oscillator (see route “GF2” inFig. 1(c) of [22]). Agrawal et al. showed a similar torus-doublingprocess in a coupled Rössler system under quasiperiodic forcing(see the region in the vicinity of (ε, g) ∼ (0.3,0.1) in Fig. 2 of[23]). As these examples show, the torus-doubling process via SNAsoccurs also in quasiperiodically forced invertible systems, whereinthe termination of the torus-doubling bifurcation curves is due tothe loss of smooth dependence of the Lyapunov vectors on thephase variable of the torus [5]. Although the termination mecha-nism of the torus-doubling bifurcation curves is different betweennoninvertible and invertible systems, the phase dependence of theorbital asymptotic stability inhibits the torus-doubling bifurcationin both the systems.

In computational neuroscience, there is increasing interest inneural information processing of multi-frequency signals: the pitchperception of complex sounds in auditory systems [41,42], theamplitude-modulated signal processing by class II neurons [43],the vibrational resonance in single and coupled neurons [44,45],and so on. Recently, Lim et al. have shown the appearance ofSNAs in a quasiperiodically forced Hodgkin–Huxley neuron [46],in a quasiperiodically forced Morris–Lecar neuron [47], and in aquasiperiodically forced Hindmarsh–Rose neuron [48], using real-istic parameter values. They propose that strange nonchaotic firingpatterns may become a dynamical origin of complex physiologicalrhythms. We hope that our study will stimulate further investiga-tions for complex neuronal dynamics.

Acknowledgements

We thank Dr. M.U. Kobayashi, Dr. H. Takahashi and Dr. Biswamb-har Rakshit for their valuable comments. This research is sup-ported by the Aihara Project, the FIRST program from JSPS, ini-tiated by CSTP.

Fig. 10. Plot of information I(r) vs. log2(1/r) for the single torus in Fig. 8(a) atd = 0.370 V, the SNA in Fig. 8(d) at d = 0.316 V, and the doubled torus in Fig. 8(f)at d = 0.300 V. The straight lines are the results of least squares fitting to the datawithin the range of 2−5.5 < r < 2−3.

Appendix A. Numerical estimates of information dimension D1

The information dimension D1 was estimated for all the attrac-tors discussed in Section 3. Fig. 10 shows plots of I(r) vs. log2(1/r)for the single torus in Fig. 8(a) at d = 0.370 V, the SNA in Fig. 8(d)at d = 0.316 V, and the doubled torus in Fig. 8(f) at d = 0.300 V.The straight lines are the results of least squares fitting to the datawithin a range of edge lengths 2−5.5 < r < 2−3. In the fine scale,r � 2−5.5, the slope slightly increases due to the effect of noise inthe circuit.

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