Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with experimental and analytical confirmation

Chaos, Solitons & Fractals 75 (2015) 96–110

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with experimental and analytical confirmation A. Arulgnanam a, Awadhesh Prasad b, K. Thamilmaran c,⇑, M. Daniel c a

Department of Physics, St. John’s College, Palayamkottai 627 002, India Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India c Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620 024, India b

a r t i c l e

i n f o

Article history: Received 21 November 2014 Accepted 7 February 2015

a b s t r a c t A new route to strange nonchaotic attractor (SNA), known as multilayered bubble route to SNA, has been identified in a quasiperiodically forced series LCR circuit with a simple nonlinear element. Upon increasing the system control parameter, the stable orbits of the torus become unstable, which induces formation of bubbles in the neighborhood of the resonating region of the torus. We have observed three tori with three smooth branches in the Poincaré map which gradually loose their smoothness and ultimately approach bubble formation, and then approach fractal behavior via SNAs before the onset of chaos. The bubbles gradually enlarge and subsequently another three layers of bubbles are formed as a function of the control parameter. The layers get increasingly wrinkled as a function of the control parameter, resulting in the creation of SNAs which are characterized by Poincaré maps. The multilayered bubble route to SNA is then confirmed by experimental Poincaré maps and explicit analytical solution is developed to further confirm it. Numerically observed bubbling route is characterized qualitatively in terms of phase portraits, power spectrum and further characterized quantitatively, by singular-continuous spectrum analysis, phase sensitivity measure, distribution of finite time Lyapunov exponents, largest Lyapunov exponent and its variance. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Strange nonchaotic attractors (SNAs) that are created when there is quasiperiodic driving are of particular interest among diverse dynamical states encountered in the study of nonlinear dynamical systems [1,2]. SNA is an example of transitional dynamics, occurring between regimes of chaotic strange attractors and quasiperiodic attractors [3]. They posess geometrically complicated ⇑ Corresponding author. E-mail address: [email protected] (K. Thamilmaran). URL: http://www.elsevier.com (K. Thamilmaran). http://dx.doi.org/10.1016/j.chaos.2015.02.006 0960-0779/Ó 2015 Elsevier Ltd. All rights reserved.

structure, by showing fractal nature. But dynamically, they do not show sensitive dependence on initial conditions as seen from negative Lyapunov exponents, that is, they are strange but nonchaotic. Following the pioneering work of Grebogi et al. [4], SNAs have been extensively investigated numerically in dynamical systems, such as biological oscillators [5], driven Duffing type oscillators [6–9] and in certain maps, namely driven velocity-dependent systems [10], two dimensional maps [11], quasiperiodically forced logistic map [12–14], one dimensional cubic map [15– 17], Harper map [18], map representing driven damped superconducting quantum interference device [19–21] and SNAs in HH-neural oscillator [22], in neon glow

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discharge experiment [23] and in quasiperiodically forced, buckled, magnetoelastic ribbon [24]. In some physically relevant situations, the existence of SNAs have also been demonstrated experimentally such as in electronic circuits [25–27], Different routes to SNAs and different scenarios for the formation of SNAs along with their distinct signatures/mechanisms have been summarized in references [27,10]. Recently, three prominent routes, namely HeagyHammel, fractalization and type-II intermittency routes to SNAs have been identified and reported in a quasiperiodically forced negative conductance series LCR circuit with diode as the nonlinear element [26]. In almost all the experimental studies related to nonlinear electronic circuits, either negative impedance converter along with a pn-junction diode or a pair of them or Chua’s diode is used as nonlinear element, and it requires eight to twelve elements to constitute the nonlinear element subcircuit. It will be quite valuable from nonlinear dynamics point of view to construct a nonlinear electronic circuit with simple nonlinear element which exhibits a wide spectrum of strange but nonchaotic dynamical phenomena. The simple nonlinear element used in the present study, consists of one op-amp and three linear resistors in its subcircuit (totally, only four elements) [28]. Using this simple nonlinear element in a forced series LCR circuit, it was recently found that the system showed high complexity in its dynamics, including the standard period-doubling and intermittency routes to chaos. In all the mechanisms of the different routes to SNA, truncation of torus doubling and torus destruction are found to be the common features. Though the above phenomena have been discussed in several papers, it has not been fully understood yet. In the present work, with our simple nonlinear element in a quasiperiodically forced series LCR circuit, we identify a fascinating new route for the formation of SNAs which we term as the multilayered bubble route to SNA. In order to confirm the existence of the multilayered bubble route to SNA in the proposed circuit, a detailed numerical analysis of the state equations in a rescaled form is carried out for suitable values of the parameters. Also, this route is confirmed both by experimental and analytical studies. The paper is organized as follows. In Section 2, we present the realization of the quasiperiodically forced series LCR circuit with the simplified nonlinearity, using sinusoidal forces as the quasiperiodic forcing. The birth of SNA via multilayered bubble route is analyzed in Section 3. In order to support the numerically simulated results, experimental observation of multilayered bubbling route to SNA and their characterization are discussed in Section 4. An explicit analytical solution is also developed to further support the work and discussed in Section 5. The results are summarized and concluded in Section 6.

2. Circuit realization The proposed circuit is a quasiperiodically forced series LCR circuit to which a simple nonlinear element constructed using one op-amp and three linear resistors [28], is connected in parallel as shown in Fig. 1. The circuit contains, a capacitor C, an inductor L, a resistor R, and two sinusoidal

R

iL

L

F1(t)

v

C F2(t)

iN +

iC +

_

_

NR

Fig. 1. Circuit realization of the quasiperiodically forced series LCR circuit with the simplified nonlinear element, N R and two sinusoidal signals F 1 ðtÞ and F 2 ðtÞ. The values of the circuit elements are chosen as C = 10.0 nF, L = 50.2 mH, R = 2.4 kX, and the frequencies of the external quasiperiodic signals are chosen as m1 ¼ 5:658 kHz, m2 ¼ 30:800 kHz, and the amplitude of the external quasiperiodic driving forces is F 2 ¼ 2:650 V and F 1 is chosen as the control parameter.

signals in the form of F 1 sin X1 t and F 2 sin X2 t, where X1 =2pm1 and X2 =2pm2 ; m1 and m2 are the frequencies of the quasiperiodic signal respectively. In the figure, v and iL denote the voltage across the capacitor C and the current flowing through the inductor L, respectively. iC and iN are the current flowing through the capacitor and the nonlinear element respectively. The state equations of the circuit are given by

C

dv ¼ iL  gðv Þ; dt

ð1aÞ

L

diL ¼ RiL  v þ F 1 sin X1 t þ F 2 sin X2 t; dt

ð1bÞ

where

gðv Þ ¼ Gb v þ 0:5ðGa  Gb Þ½jv þ Bp j  jv  Bp j;

ð1cÞ

is the mathematical form of the piecewise linear function representing the characteristic curve as shown in Fig. 2(b), which is obtained from the nonlinear subcircuit shown in Fig. 2(a). F 1 and F 2 are the amplitudes of the two external quasiperiodic forces. The nonlinear element ðN R Þ in the subcircuit has been constructed with one opamp and three linear resistors ðR1 ; R2 ; R3 Þ as shown in Fig. 2(a). It is used to realize the negative slope as Ga and the positive slope as Gb in the characteristic curve shown in Fig. 2(b). It is to be noted that, in the absence of F 2 , the circuit in Fig. 1 is found to exhibit chaos with high complexity, not only through the familiar period-doubling route but also via intermittency route [28]. Therefore, the natural question arises as to what will be the effect on the dynamics of the system, if a second sinusoidal-type external force is connected in series in the circuit. In order to study the dynamics of the circuit numerically, Eq. (1) is converted into a convenient normalized form, by using the following rescaled parameters: x ¼ v =Bp , y ¼ iL =GBp , G ¼ 1=R, x1 ¼ X1 C=G, x2 ¼ X2 C=G, t ¼ C s=G, a ¼ Ga =G, b ¼ Gb =G, b ¼ C=LG2 , f 1 ¼ bF 1 =Bp and f 2 ¼ bF 2 =Bp . The evolution equations so obtained are represented as a set of autonomous equations as follows.

x_ ¼ y  gðxÞ;

ð2aÞ

y_ ¼ by  bx þ f 1 sinðhÞ þ f 2 sinð/Þ;

ð2bÞ

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R1

i(mA)

+ +

+

-

IC741 − R2

Gb D_

-3.8 -Bp

D0

+3.8 +Bp

0

D+ v(V)

Ga

NR R3

(a)

(b)

Fig. 2. (a) Realization of the nonlinear element (N R ). R1 ¼ 1:990k; R2 ¼ 1:981k and R3 ¼ 1:989k are three linear resistors with one op-amp l741C operating in dual voltage power supply of 9 V and (b) its ðv  iÞ characteristic curve with negative slope Ga ¼ 0:56 mS, positive slope Gb ¼ 2:5 mS and BP ¼ 3:8 V is the break point in the characteristic curve at which the negative conductance changes into positive conductance.

h_ ¼ x1 ;

ð2cÞ

/_ ¼ x2 ;

ð2dÞ

and the nonlinear function gðxÞ is given by

gðxÞ ¼ bx þ 0:5ða  bÞ½jx þ 1:0j  jx  1:0j;

ð2eÞ

where overdot stands for differentiation with respect to s. Basically, the piecewise linear nonlinear element in this quasiperiodically forced series LCR circuit is a non-smooth system having two nonintersecting discontinuous boundP P aries at 1 fx ¼ 1g and 2 fx ¼ 1g and degree of smoothness equal to two across each of them. However, the occurrence of chaotic (or) strange nonchaotic dynamics is to be attributed to a sequence of smooth bifurcations only, and discontinuity induced bifurcations, if any, are not found to change the nature of the attractor. Nevertheless, Eqs. (2) are then numerically integrated using Runge– Kutta fourth order routine with a step size of 0.0001, until a discontinuity is detected. When the trajectory crosses a discontinuous boundary, the state variables are reset properly using the Zero Time Discontinuous Mapping (ZDM), as outlined in [29]. This is done at both the discontinuities present in the system, so as to ensure a correct picture of the global dynamics. Now, the dynamics of the system depends upon the rescaled parameters a; b; b; x1 ; x2 ; f 1 and f 2 . Various interesting dynamical transitions from torus to SNA and subsequently to chaotic attractor occur on increasing the value of the amplitude of one of the sinusoidal forces f 1 for fixed value of other system parameters. 3. Multilayered bubbling route to SNA – A numerical study Strange nonchaotic attractors are created through the new novel route, namely, multilayered bubbling route, in our circuit system for the fixed values of a ¼ 1:344, b = 6.000, b ¼ 1:147, x1 ¼ 0:853, x2 ¼ 4:645, f 2 ¼ 0:800 and by choosing the amplitude of the other driving force f 1 as the control parameter. The above route is confirmed through both qualitative and quantitative measures. Qualitatively SNAs are confirmed through Poincaré surface of section by distinguishing between torus and SNAs

geometrically. Quantitative confirmation is provided using the following four different measures: (i) the largest Lyapunov exponent and its variance to distinguish between torus to SNA and SNA to chaos, (ii) singularcontinuous spectrum analysis to confirm the presence of SNAs by analyzing whether the system undergoes regular or irregular motion or the state in between them, (iii) separation of nearby points between two trajectories, and (iv) distribution of finite time Lyapunov exponents, which is used to confirm SNAs and their mechanism and the phase sensitivity measure, which is the specific quantity for the characterization of the quasiperiodic system. 3.1. Poincaré surface of section In order to elucidate the emergence of multilayered bubbling route to SNA in the present system, the Poincaré surface of section in the ð/  xÞ plane is plotted by varying the control parameter, f 1 . On increasing the amplitude f 1 from 0, in the range f 1  (0, 0.6727) 1-torus appears. On further increasing the amplitude, in the range f 1  (0.6728, 0.7578) 2-tori has been observed and 3-tori is observed in the range f 1  (0.7579, 0.8402). In the Poincaré surface section plot, in the above ranges, one smooth branch is observed for 1-torus, two and three smooth branches are observed for 2-tori and 3-tori in their ð/  xÞ plane respectively. In Fig. 3(a) with three smooth branches representing 3-tori is shown and the orbit is observed to be completely stable. Bubbles started appearing in the strands of period 3 tori at f 1 ¼ 0:8403, and continue to appear in the range f 1  (0.8403, 0.8654). On increasing the amplitude f 1 further the size of the bubbles increases and then forms into multilayered bubbles with single, two and three layers in the amplitude ranges f 1  (0.8403, 0.8406), f 1  (0.8407, 0.8412) and f 1  (0.8413, 0.8421) respectively as shown in Figs. 3(b–d). In order to get a clear insight, insets are also presented within the respective figures. If the amplitude is increased further, the layers started disappearing at f 1 ¼ 0:8424 as shown in Fig. 3(e) and become a single bubble at f 1 ¼ 0:8458 as shown in Fig. 3(f). On further increase of the amplitude, in the range f 1  (0.8458, 0.8470), the bubble continues to be single. The mechanism for this route is that, for a

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A. Arulgnanam et al. / Chaos, Solitons & Fractals 75 (2015) 96–110 0.9

(b)

0.5

(a)

0.5

0.1

0.1

x

x

-0.3 -0.3

-0.7

-0.7 -1.1

-1.1

-1.5 0

1

2

3

φ

4

5

6

0

(c)

0.5

1

2

3

φ

4

5

(d)

0.5

-0.3

-0.3

x

0.1

x

0.1

6

-0.7

-0.7

-1.1

-1.1

0

1

2

3

φ

4

5

6

0

(e)

0.5

1

2

3

φ

4

5

(f)

0.5

0.1

-0.3

-0.3

x

x

0.1

6

-0.7

-0.7

-1.1

-1.1

0

1

2

3

φ

4

5

6

0

(g)

0.9

1

2

3

φ

4

5

(h)

0.9

0.1

0.1

x

0.5

x

0.5

6

-0.3

-0.3

-0.7

-0.7

-1.1

-1.1

0

1

2

3

φ

4

5

6

0

1

2

3

φ

4

5

6

Fig. 3. The transition from 3-tori to SNA: projection of numerically simulated Poincaré surface of section plots of Eq. (2) with / modulo 2p in the ð/  xÞ plane. (a) period 3-tori when f 1 ¼ 0:7609, (b) single layered bubble when f 1 ¼ 0:8406, (c) two layered bubble when f 1 ¼ 0:8412, (d) three layered bubble when f 1 ¼ 0:8415, (e) disappearance of multilayer when f 1 ¼ 0:8427, (f) single bubble when f 1 ¼ 0:8458, (g) SNA when f 1 ¼ 0:8527 and (h) chaotic attractor when f 1 ¼ 0:9119.

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particular choice of the system parameter values, the system exhibits period-3 tori. By slowly increasing the control parameter, the system exhibits swollen shape type bifurcation (bubble) similar to that of Sekikawa et al. [30]. We call this swollen shape as period one bubble or single layer bubble. Interestingly, upon increasing the control parameter little further, second layer of period two bubble is formed. This happens until the formation of third layer of period three bubble is formed. On increasing the control parameter continuously further, reverse bifurcation takes place, that is, the period three layer bubble vanishes followed by the disappearance of a period two layer bubble. Once it reaches the single layer period one bubble, the bubble starts wrinkling and the torus gets destructed for further increase of control parameter. The destructed part of the bubble finally gives birth to SNA. But the remaining part of the torus other than bubble region is unaffected in the range of the amplitude f 1  (0.8471, 0.8530) as shown in Fig. 3(g). On further increase of the control parameter, the torus gets extremely wrinkled and the chaotic state appears at f 1 ¼ 0:9119 as shown in Fig. 3(h). This phenomenon is named as the multilayered bubbling transition to SNA, while bubbles appear in the neighborhood regions resonating torus, SNAs are formed due to wrinkling of bubbles. This multilayered bubbling route is observed in a rather narrow range of frequency, that is, x1  (0.8510, 0.8530) as a function of amplitude of the sinusoidal forcing

3.2. Largest Lyapunov exponent and its variance The first quantitative measure that we use to characterize the transition region from torus to SNA and from SNA to

(a)

-2 -7

(b)

-2

f1 = 0.7609 3-tori

Amplitude

Amplitude

in the range f 1  (0.7579, 0.8530). It is to be noted that, this route is significantly different from the well known fractalization route, where the entire strands of the n-period torus continuously deform and get extremely wrinkled as a function of the control parameter. The formation of SNAs through multilayered bubbling route has been identified for the first time to the best of our knowledge. For further qualitative analysis, the power spectrum of the above multilayered bubble route is presented in Fig. 4(a–d). Since, there is not much variation in the power spectra of 3-tori and multilayered layered bubbles, we present here only the power spectrum of 3-tori, 3-layered bubbles, SNAs and chaos. From Fig. 4(a), it has been observed that, a discrete set of frequencies appeared in the power spectrum corresponding to 3-tori indicating its quasiperiodic nature. For the bubbles as given in Fig. 4(b), it is seen that there is not much variation in the amplitudes of the discrete frequencies. But, for the case of SNA (see Fig. 4(c)), harmonics also appears along with the discrete set of frequencies indicating that they are neither quasiperiodic nor chaotic. Broad band nature is observed in the power spectrum thus indicating the chaotic behavior as shown in Fig. 4(d).

-12 -17

f1 = 0.8415 3-layered bubble

-7 -12 -17

-22

-22 0

0.2

0.4

0.6

0.8

1

0

0.2

Frequency (c)

0.8

1

(d)

-2

f1 = 0.8527 SNA

Amplitude

Amplitude

0.6

Frequency

-2 -7

0.4

-12 -17

f1 = 0.9119 Chaos

-7 -12 -17

-22

-22 0

0.2

0.4

0.6

Frequency

0.8

1

0

0.2

0.4

0.6

0.8

1

Frequency

Fig. 4. Power spectrum of (a) 3-tori for f 1 ¼ 0:7609, (b) three layered bubble for f 1 ¼ 0:8415, (c) strange nonchaotic attractor for f 1 ¼ 0:8527 and (d) chaotic attractor for f 1 ¼ 0:9119.

A. Arulgnanam et al. / Chaos, Solitons & Fractals 75 (2015) 96–110

chaos, is obtained by calculating the largest Lyapunov exponent K and its variance r. The variance, r of the largest asymptotic Lyapunov exponent, K from finite time Lyapunov exponent ð kÞ of length M, is defined as M X k ¼ 1 ki ; M i¼1

r¼

K 1X 2 ðK  ki Þ ; K i¼1

ð3aÞ

ð3bÞ

where K is the total number of finite time Lyapunov exponent ( k) that has been used to calculate the variance of the largest Lyapunov exponent and ki , is the instantaneous Lyapunov exponent at every iteration i. The largest Lyapunov exponent and its variance are calculated in the range f 1  (0.8304, 0.9511). The entire region is divided into three segments. The first segment corresponds to f 1  (0.8304, 0.8757), where the multilayered bubble is formed. A plot of the largest Lyapunov exponent and its variance corresponding to this segment are given in Fig. 5(a) and (b) respectively. When the control parameter is changed

101

at a particular frequency, if the amplitude crosses the critic cal value of f 1 ¼ 0:8400, the system changes to exhibit c torus for ðf 1 < f 1 Þ. In the range 0:8400 < f 1 < 0:8470, multilayered bubbles appear and for further increase of f 1 in the range 0:8473 < f 1 < 0:8533 SNAs appear. In this range, the largest Lyapunov exponent is negative and the corresponding variance is higher, which indicates that the region of transition is from torus to SNA. This is also evident from the smooth and irregular variation respectively in both the largest Lyapunov exponent and in the variance spectrum. This strange nonchaotic dynamic region becomes a chaotic dynamic region beyond the value of f 1 ¼ 0:8527, after the largest Lyapunov exponents crosses zero axis. The other two segments f 1  (0.8908, 0.9119) and f 1  (0.9330, 0.9511) show the usual route to SNA and hence the details are not presented here. 3.3. Singular-continuous spectrum analysis Further, in order to quantitatively confirm that the dynamics of the system is nonchaotic and strange, we use specific measures that have been suggested earlier, in particular, the singular-continuous spectrum analysis [31]. In general, power spectrum of dissipative dynamical system can be either discrete, or continuous, or a combination of both. Discrete spectrum is usually generated by regular motion such as periodic or quasiperiodic motion, where as continuous spectrum corresponds to irregular motion such as chaotic or random motion. A singularcontinuous spectrum is an intermediate stage between discrete and continuous spectra [8]. Using this property, we can identify whether the multilayered bubble indeed leads to strange nonchaotic dynamics or not. To confirm this, we compute the partial Fourier sum [15,16] as

Yða; NÞ ¼

N X fyk gexpð2pikaÞ;

ð4Þ

k¼1

where a is proportional to the irrational driving frequency

X1 and fyk g is the time series of the variable y of length N. When N is regarded as time, jYða; NÞj2 grows with time N as [31] 2 jYða; NÞj  N l ;

ð5Þ

where l is a scaling exponent. The time evolution of Yða; NÞ can be represented by an orbit, or a walker in the complex plane ðRe½Yða; NÞ; Im½Yða; NÞÞ and for a singularcontinuous spectrum (when the dynamics is strange), it implies that the walk in the plane (Re½Y; Im½Y) will be a

Fig. 5. Different regions of multilayered bubbling route to SNA: f 1 < 0:8400 – torus region, 0:8400 < f 1 < 0:8470 – multilayered bubble region, 0:8473 < f 1 < 0:8533 – SNA region and f 1 > 0:8533 – chaotic region. (a) the largest Lyapunov exponent in the ðkmax  f 1 Þ plane. Here kmax is negative and varies smoothly in the torus region, kmax is negative and varies irregularly in the SNA region, and kmax is positive and varies irregularly in the chaotic region and (b) its variance ðrÞ varies smoothly in the torus region, and varies irregularly in the SNA and chaotic regions.

fractal [15]. Plot of jYða; NÞj2 vs N for a ¼ X1 =4 and for the value of f 1 ¼ 0:8527, is shown in Fig. 6(a). The slope of the curve shown in Fig. 6(a), gives the scaling exponent as l ¼ 1:597. The scaling exponent value falls in between 1 < l < 2, because, the system dynamics exhibits neither irregular nor regular motion. The walk in the complex plane as found in Fig. 6(b) exhibits fractal nature, which is identified by the fractional dimensional value of the scaling exponent. The above results strongly suggest that the dynamics of the system is indeed strange and nonchaotic.

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Fig. 6. Singular-continuous spectrum analysis of the fy2 gk time series at f 1 ¼ 0:8527. (a) plot of jYða; NÞj2 vs N showing power-law scaling. The slope or the fractal dimension is =1.597, and (b) the orbit walker in the complex plane exhibiting fractal nature, and strongly suggesting the presence of SNA.

3.4. Distribution of local Lyapunov exponents

3.5. Separation of nearby points

Another important quantitative measure used to distinguish the type of route through which SNA appears, is the distribution of finite time Lyapunov exponents. It has been shown that a typical trajectory on SNA actually possesses positive Lyapunov exponents in finite time intervals, although the asymptotic exponents are negative [11,27]. As a consequence, it is possible to observe different characteristics of SNAs created via different mechanisms by a study of the differences in the probability (P) of distribution of finite time exponent PðN;  kÞ for positive and negative values. The distribution can be obtained by taking a long trajectory and dividing it into segments of length N, from which the finite time Lyapunov exponent can be calculated. We have calculated the distribution of finite time Lyapunov exponent PðN;  kÞ, with N = 2000, for both negative and positive values. In order to confirm the nature of transition to SNA, the results are plotted in Fig. 7. The distribution of the finite time Lyapunov exponents for the period-3 tori is shown in Fig. 7(a), in which the finite time Lyapunov exponents have their maximum at their negative values. Similarly, for bubbles, the distribution of finite time Lyapunov exponents have their maximum at their negative values, but the distribution exhibits a tiny tail on their positive values, indicating the slow transition of bubbles into SNA as shown in Fig. 7(b). For the case of SNA shown in Fig. 7(c), the distribution of finite time Lyapunov exponents have maximum at their positive values. Also, in Fig. 7(c), the distribution of finite time Lyapunov exponents for SNA exhibits an elongated tail in the negative values, because of the fact that, in the bubbling transition, parts of the strands of period-3 tori remain unaffected even after the birth of SNA, which contributes largely to the negative values. This confirms the existence of bubbling transition to strange nonchaotic attractor. The plot in Fig. 7(d) indicates the chaotic behavior, in which the distribution of finite time Lyapunov exponents are peaked about their positive values and the distribution shifts more towards the positive side of their values.

The SNAs exhibit complicated geometrical structure like chaotic attractors. One way to distinguish SNAs from chaotic attractors is to look for the sensitive dependence on initial conditions [23]. In order to verify the sensitive dependence on initial conditions, we analyze the separation D between two orbits starting with two nearby initial conditions. For this, two nearby points ðxi ; yi ; zi Þ and ðxj ; yj ; zj Þ on the attractor are chosen and their separation is calculated using the standard formula

D¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi  xj Þ2 þ ðyi  yj Þ2 þ ðzi  zj Þ2

ð6Þ

and is monitored at each forward step. In the case of SNA, D will decay to zero as t ! 1, but for chaotic attractor, D will become irregular and sustains for a longer time. The plot for the separation D as time proceeds is shown in Fig. 8(a) and (b). In Fig. 8(a), it has been observed that D becomes zero for f 1 ¼ 0:8527 in a short interval of time and thus clearly supports the loss of sensitive dependence on initial condition, which corresponds to the strange nonchaotic attractor. Fig. 8(b) shows the variation of D for f 1 ¼ 0:9119, in which the irregular variation of D sustains for a long time ðt ! 1Þ, and indicates that the behavior corresponds to chaotic in nature. 3.6. Phase sensitivity exponent To characterize the dependence of dynamics on phases of driving force, Pikovsky and Feudel [15] introduced a quantity called the Phase Sensitivity Measure which gives an idea of how the phase of the external force influence the state variables of the system. For sake of convenience let us rewrite Eqs. (2) as

dxl ¼ F 1 ðx1 ; x2 ; /1 ; /2 Þ l ¼ 1; 2; dt

ð7aÞ

d/k ¼ Xk dt

ð7bÞ

k ¼ 1; 2;

A. Arulgnanam et al. / Chaos, Solitons & Fractals 75 (2015) 96–110

(a)

1

–

P(N, λ)

equations

dylk X @F l @F l ¼ yjk þ ; dt @x @/ j k j

0.01 0.001

ð8Þ

Solving these, we obtain the phase derivative ylk . The phase sensitivity measure is then defined as -5.5e-05

-3.5e-05

–

-1.5e-05

5e-06

λ

0.1

–

0.01 0.001

0.0001 -3.5e-05

-1.5e-05

5e-06

–

λ (c)

1

Ct ¼ minct ðyl ; /k Þ yl ;/

ð9Þ

where

(b)

1

P(N, λ)

where we have taken x ¼ x1 ; y ¼ x2 ; h ¼ /1 and / ¼ /2 , then the derivatives of the variables xl with respect to @xl are governed by the inhomogeneous the phases ylk ¼ @/ k

0.1

0.0001 -7.5e-05

103

ct ðyl ; /k Þ ¼ max 06i6t

@yli : @/k

ð10Þ

The quantity Ct takes on low values for a smooth torus for all instants of time t. However for SNA it shows a power law relation Ct ¼ t l , with l as the phase sensitivity exponent, and a fast exponential growth for chaos. For our system under consideration given by Eq. (7), we plot Ct as a function of t in Fig. 9, for the three different dynamical behaviors, namely (a) torus, with f 1 ¼ 0:7609, (b) SNA, with f 1 ¼ 0:8527 and (c) chaos, with f 1 ¼ 0:9119. The variation of Ct is in accordance with theoretical prediction for the torus, SNA and chaotic behaviors. This proves once again that SNA occurs in this system for a particular choice of system parameter of f 1 . 4. Experimental observation of multilayered bubbling route to SNA

–

P(N, λ)

0.1 0.01 0.001 0.0001 -2.5e-05

-5e-06

1.5e-05

–

λ (d)

1

–

P(N, λ)

0.1 0.01 0.001 0.0001 -3.5e-05

-1.5e-05

5e-06

–

2.5e-05

λ

Fig. 7. Distribution of finite time Lyapunov exponents for N = 2000. (a) period 3-tori for f 1 ¼ 0:7609, (b) three layered bubble for f 1 ¼ 0:8415, (c) SNA for f 1 ¼ 0:8527 and (d) chaotic attractor for f 1 ¼ 0:9119.

In the numerical study, SNA is observed via different routes namely fractalization, Heagy-Hammel, intermittency and multilayered bubbling route. In order to support the numerically simulated multilayered bubbling route to SNA, an experimental circuit is constructed as shown in Fig. (10), where L represents the connection towards the decade inductance box. Circuit elements with the following values are chosen. C ¼ 9 nF, L ¼ 41 mH, R ¼ 2:151 kX, the frequency of the external quasiperiodic signals are set at m1 ¼ 28:220 kHz, m2 ¼ 6:671 kHz, the amplitude of the external quasiperiodic driving force is fixed at F 2 ¼ 50 mV and F 1 , the amplitude of the external is treated as the control parameter. To begin the experiment, the first external signal is switched on and the control parameter (F 1 ) was slowly increased and the response of the circuit was observed. It was found that the circuit progressed through a series of transitions from quasiperiodic motion to aperiodic motion via strange nonchaotic attractors. On increasing the amplitude (F 1 ), the circuit was found to exhibit 1-torus, 2-tori, 3-tori, multilayered bubbling, and SNA followed by chaos in the range 1:990 V < F 1 < 2:20 V. Since we are interested only in the multilayered bubbling route to SNA experimentally, the Poincaré surface of section in the ðt  V T 2 Þ plane is plotted by varying the control parameter, F 1 . Here the suffix T 2 represents the time period of the driving signal. Live pictures of the corresponding Poincaré map of the projected attractor have

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(b) 1

0.1

0.1

Δ

Δ

(a) 1

0.01

0.01

0.001

0.001

0.0001

0.0001

1e-05

1e-05 0

10000

20000

0

Time

10000

20000

Time

Fig. 8. Time variation of the separation D of two nearby points: (a) For SNA when f 1 ¼ 0:8527, the separation decays to zero in a short span of time and (b) in the case of chaotic attractor when f 1 ¼ 0:9119 the separation shows irregular and sustained oscillation.

40

10

30

(c)

Γt

10

20

10

10

(b)

0

(a)

10

10

102

103

104

105

t Fig. 9. Plot of the phase sensitivity exponent for multilayered bubbling route to SNA. (a) torus at f 1 ¼ 0:7609, (b) SNA at f 1 ¼ 0:8527 and (c) chaos at f 1 ¼ 0:9119.

been produced using Poincaré map circuit as shown in Fig. 11. The stroboscopic Poincaré maps are produced by triggering the beam of the oscilloscope at two different driving frequencies namely m1 ð¼ X1 =2pÞ and m2 ð¼ X2 =2pÞ (z-modulation) [32]. Since the multilayered bubbling route appears only in the narrow region, it is possible to observe this route by fine tunning the control parameter. The

oscillographs are shown in Fig. 12. On increasing the control parameter from F 1 ¼ 1:9 V, bubbles started appearing in the strands of period 3 tori at F 1 ¼ 1:99 V, The one layered bubble appears at F 1 ¼ 1:992 V as shown in Fig. 12(a). Further increase in the control parameter, 2-layered and 3-layered bubble appear at F 1 ¼ 1:999 V and 2.005 V respectively as in Fig. 12(b) and (c). The three layered bubble get destructed (wringles) at F 1 ¼ 2:008, as shown in Fig. 12(d). This destructed three layered bubble gives birth to SNA on further increase of the control parameter and SNA forms at F 1 ¼ 2:010 V as found in Fig. 12(e). When we increase the control parameter little further, finally chaos appeared at F 1 ¼ 2:120 V, as shown in Fig. 12(f). Several phenomena, as seen in numerical studies, such as increase of bubble size and the disappearance of three, two layers are not easily observed experimentally in the laboratory. But over all, the multilayered bubble route to SNA is experimentally exhibited successfully. Oscillographs of the power spectrum of the corresponding attractors like 3-tori, one layered, two layered, three layered, SNA and chaos were also taken. There is not much variation in the power spectra of 3-tori, one, two and three layered bubbles, and all of them have discrete set of frequencies representing the periodic nature. Hence only 3-tori and three layered bubbles were represented in

Fig. 10. Photograph of experimental circuit realization of the quasiperiodically forced series LCR circuit along with the Poincaré map circuit (shown in PCB).

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A. Arulgnanam et al. / Chaos, Solitons & Fractals 75 (2015) 96–110

1M 10 K

+

+ 9V

1/6 7414 470 Ω

0.1 μf

1/6 7414

PULSES TO CRO Z- INPUT

- 9V 0.1 V Zener

220 Ω

0.1 μf

Fig. 11. Experimental circuit realization of the Poincaré map circuit.

Fig. 12. Oscillographs of Poincaré surface of section plots of Eqs. (2) in the ðt  V T 2 Þ plane, where T 2 is the time period of the second driving signal. (a) single layered bubble when F 1 ¼ 1:992 V, (b) two layered bubble when F 1 ¼ 1:999 V, (c) three layered bubble when F 1 ¼ 2:005 V, (d) wringling at F 1 ¼ 2:008 V, (e) SNA when F 1 ¼ 2:010 V, and (f) chaos when F 1 ¼ 2:120 V.

Fig. 13(a) and (b). The oscillograph in Fig. 13(c) represents SNA, since it is having harmonics along with discrete set of frequencies and also there is a decrease in amplitudes of the spectrum. Fig. 13(d) showing the broad band nature indicates the chaotic behavior which exactly matches with the numerically observed one found in Fig. 4. In order to characterize the SNAs of multilayered bubbling route experimentally, the two dimensional projections of the corresponding attractor are obtained by measuring the voltage v across the capacitor C and the current iL through the inductor L which are connected to the X and Y channels of an oscilloscope, respectively. Then, the experimental data of the corresponding attractor was recorded using a 14-bit data acquisition system (Agilent U2531A 4ch 14 bits) at a sampling rate of 2MS/s. This experimental data was then analyzed quantitatively

using the quantitative measure namely singular continuous spectrum analysis as given in Section 3.3. A similar plot of jYða; NÞj2 vs N for a ¼ Xð1=4Þ which has the scaling exponent l ¼ 1:561 at F 1 ¼ 2:010 V, for this multilayered bubbling route is shown in Fig. 14(a). The values of the scaling exponents clearly indicates the existence of SNAs in the experimentally observed multilayered bubbling route for that particular choice of control parameter. A plot of Re[Y] vs Im[Y] is shown in Fig. 14(b) for the above route at F 1 ¼ 2:010 V. The orbit walker in the complex plane exhibits fractal nature, strongly suggesting the presence of SNA. From this figure, it is observed that they exactly match with the numerical figures and, strongly suggesting that the experimentally observed dynamics is indeed strange and nonchaotic for that particular choice of control parameter.

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(a)

(b)

(c)

(d)

200

6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1

(a)

(b)

150 100 50

Im Y

log10 |Y(α, N)|2

Fig. 13. Oscillographs of power spectrum. (a) 3-tori for F 1 ¼ 1:990 V, (b) three layered bubble for F 1 ¼ 2:005 V, (c) strange nonchaotic attractor for F 1 ¼ 2:010 V and (d) chaotic attractor for F 1 ¼ 2:120 V.

0 -50 -100 -150 -200

3

3.5

4

4.5

5

5.5

6

-250 -600

-500

-400

-300

-200

-100

0

100

Re Y

log10 N

Fig. 14. Experimental SNA data of multilayered bubbling route. Singular-continuous spectral analysis of the fy2 gk time series at F 1 ¼ 2:010 V. (a) A plot of 2 jYða; NÞj vs N showing power-law scaling. The slope or the fractal dimension is l ¼ 1:561, and (b) The orbit walker in the complex plane exhibits fractal nature, strongly suggesting the presence of SNA.

D ¼ fðx; yÞj x 6 1g:

5. Dynamics through analytical study The quasiperiodic dynamics of the present system is understood first by making analytical calculations.

ð12cÞ

The equilibrium points (x0 ; y0 ) for the three regions are explicitly given by

Pþ ¼ ð0; b  aÞ 2 D1 ;

ð13aÞ

O ¼ ð0; 0Þ 2 D0 ;

ð13bÞ

P ¼ ð0; a  bÞ 2 D1 :

ð13cÞ

5.1. Linear stability It may be noted that the nonlinear function gðxÞ given in Eq. (2e) is symmetric with respect to the origin so that it is invariant under the transformation

ðx; yÞ ! ðX; YÞ ¼ ðx; yÞ:

ð11Þ

In this case, the equilibrium points can be obtained by making the right hand side of Eqs. (2a) and (2b) to be zero at the equilibrium points. It follows from the form of gðxÞ that, Eq. (2e) has a unique equilibrium in each of the following three regions.

Dþ ¼ fðx; yÞj x P 1g;

ð12aÞ

D0 ¼ fðx; yÞj jxj 6 1g;

ð12bÞ

The stability determining eigenvalues ðk1 ; k2 Þ for the region D0 are calculated from the stability matrix

A0 ¼ Aðb; A1 Þ ¼  A1  k1;2 ¼



A1

1

b

bð1 þ mÞ

 ð14Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A21  4B1 2

;

ð15Þ

where A1 ¼ ðb þ bm þ aÞ and B1 ¼ ðb þ ba þ bmaÞ. As the term ðA21  4B1 Þ in Eq. (15) is always positive, i.e.,

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ðA21  4B1 Þ > 0, because it depends on the circuit parameters b; a; m, and b, and these parameters are chosen in such a way that ðA21  4B1 Þ is always positive. Therefore, k1 is a positive real value and k2 is a negative real value. This indicates that (x0 ; y0 ) = (0, 0) 2 D0 is an unstable hyperbolic fixed point, which will lead to chaotic dynamics. Similarly, from the stability matrix, for the regions Dþ and D

 A ¼ Aðb; B2 Þ ¼



B2

1

b

bð1 þ mÞ

2

yðtÞ ¼ C 1 em1 t þ C 2 em2 t þ E1 þ E2 sinx1 t þ E3 cosx1 t þ E4 sinx2 t þ E5 cosx2 t;

A1 

m1;2 ¼ ;

ð17Þ

where, A2 ¼ ðb þ bm þ bÞ and B2 ¼ ðb þ bb þ bmbÞ. Upon using the circuit parameter values as in Fig. 1, used in the experiment and in Eq. (17), we obtain a two pair of negative eigenvalues (k3;4 , and k5;6 ) for the regions Dþ and D respectively, and hence the fixed points Pþ and P  are stable nodes. Naturally, these fixed points can be observed based on the initial values x(0) and y(0) in Eqs. (2a) and (2b), in the absence of the external forces f 1 ; f 2 . Thus, one finds that as long as the initial conditions are confined to the region D0 , due to the unstable focus nature of the fixed point O, a stable period one torus results, when f 1 ; f 2 > 0. With a fixed value of f 2 , on increasing the control parameter f 1 further, it leads to an interaction between the period one torus and the external periodic signal, resulting in a torus doubling or truncation of torus doubling, which give rise to multilayered bubbling route to SNA as observed in the experiment. However, if the initial conditions are chosen in the regions D , the system will end up in one of the fixed points P þ or P  as the case may be, since they correspond to stable nodes. In fact, Eqs. (2) can be integrated explicitly in terms of elementary functions in each of the three regions D0 ; D and the resulting solutions can be matched across the boundaries to obtain the full solution as given below: 5.2. Explicit analytical solutions

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A21  4B1 2

ð21bÞ

E2 ¼

f 1 x21 ðA1  aÞ þ af1 B1  2 ; A21 x21 þ B1  x21

ð21cÞ

E3 ¼

  f 1 x1 aA1  B1 x21 ;   2 A21 x21 þ B1  x21

ð21dÞ

E4 ¼

  f 2 x22 ðA1  aÞ þ aB1 ;   2 A21 x22 þ B1  x22

ð21eÞ

E5 ¼

  f 2 x2 aA1  B1 x22  2 : A21 x22 þ B1  x22

ð21fÞ

In Eq. (20), C 1 and C 2 are arbitrary constants of integration which are to be fixed using the initial conditions. In order to keep the circuit in the negative conduction region to get the chaotic dynamics, as learned from the experiment, the circuit parameters are chosen in such a way that A21 < 4B1 . This makes the quantity ðA21  4B1 Þ to be always negative, which leads to positive real value for m1 and m2 will be a negative real value. This indicates that ðx0 ; y0 Þ = (0,0) 2 D0 is an unstable hyperbolic fixed point, which will lead to chaotic dynamics, whose solution can be written in the standard form as

pffiffiffiffiffiffiffiffiffi ffi 2

(i) Region D0 ðjxj 6 1Þ: In this region, gðxÞ ¼ ax and hence Eqs. (2a) and (2b) become

x_ ¼ y  ax;

ð18aÞ

y_ ¼ by  mby  bx þ f 1 sin x1 t þ f 2 sin x2 t;

ð18bÞ

Here overdot represents derivative with respect to time. Differentiating Eq. (18b) once with respect to time and using Eq. (18a) in the resultant equation, we get

€ þ A1 y_ þ B1 y ¼ f 1 asinx1 t þ f 2 asinx2 t y ð19Þ

ð21aÞ

;

E1 ¼ 0;

A1 þ

Since Eq. (2e) represents the piecewise linear nature, the solution in each of the regions can be obtained explicitly as follows:

þ f 1 x1 cosx1 t þ f 2 x2 cosx2 t;

ð20Þ

where

ð16Þ

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2  A22  4B2 k3;4 & k5;6 ¼

where A1 ¼ ðb þ bm þ aÞ and B1 ¼ ðb þ ab þ ambÞ. As Eq. (19) is a linear second order inhomogeneous differential equation with constant coefficients, its solution can be found using standard method. The form of the solution obtained is written as

yðtÞ ¼ C 1 e

A 4B1 1 t

2

þ C2e

A1 

pffiffiffiffiffiffiffiffiffi ffi 2

A 4B1 1 t

2

þ E1

þ E2 sinx1 t þ E3 cosx1 t þ E4 sinx2 t þ E5 cosx2 t:

ð22Þ

Having found yðtÞ; xðtÞ can be obtained using Eq. (18b) as

1 ½ðr þ m1 ÞC 1 em1 t  ðr þ m2 ÞC 2 em2 t b ½rE3 þ E2 x1 cosx1 t

xðtÞ ¼

½rE2  E3 x1  f 1 sinx1 t ½rE5 þ E4 x2 cosx2 t ½rE4  E5 x2  f 2 sinx2 t;

ð23Þ

where, r ¼ ðb þ mbÞ. The arbitrary constants C 1 and C 2 are now evaluated by using the initial conditions ðt; x; yÞ ! ðt0 ; x0 ; y0 Þ in Eqs. (22) and (23). Upon solving the values of C 1 and C 2 as

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ðÞem1 t0 C 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðb þ mb þ m2 Þy0 þ bx0 A21  4B1

In Eq. (27), C 3 and C 4 are arbitrary constants of integration which are to be fixed using the initial conditions. Since in m3 and m4 , ‘b’ represents the value of the positive slope,

þðE2 x2  E3 m2 Þcosx1 t 0

2

ðE3 x1 þ E2 m2 þ f 1 Þsinx1 t 0 þðE4 x2  E5 m2 Þcosx2 t 0 ðE5 x2 þ E4 m2 þ f 2 Þsinx2 t 0 ;

ð24aÞ

em3 t0 C 3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðb þ mb þ m4 Þy0  bx0 A22  4B2

ðÞem2 t0 C 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðb þ mb þ m1 Þy0 þ bx0 A21  4B1

ðE7 x1  E8 m4 Þcosx1 t0

þðE2 x1  E3 m1 Þcosx1 t 0

þðE8 x1 þ E7 m4 þ f 1 Þsinx1 t 0

ðE3 x1 þ E2 m1 þ f 1 Þsinx1 t 0

ðE9 x2  E10 m4 Þcosx2 t 0

þðE4 x2  E5 m1 Þcosx2 t 0 ðE5 x2 þ E4 m1 þ f 2 Þsinx2 t 0 :

½ð1 þ bÞ  4b remains positive. As a consequence, m3 and m4 become negative real values. Using the initial conditions, ðt; x; yÞ ! ðt 0 ; x0 ; y0 Þ, the arbitrary constants C 3 and C 4 are determined as

ð24bÞ

(ii) Regions Dþ ðjxj P 1Þ and D ðjxj 6 1Þ: In these regions, where we have positive slopes, the piecewise linear function gðxÞ takes the value gðxÞ ¼ bx  ða  bÞ. Then, Eqs. (2a) and (2b) become

x_ ¼ y  bx  ða  bÞ;

ð25aÞ

y_ ¼ by  mby  bx þ f 1 sin x1 t þ f 2 sin x2 t:

ð25bÞ

þðE10 x2 þ E8 m4 þ f 2 Þsinx2 t 0 þ E6 m4 ;

ð29aÞ

em4 t0 C 4 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðb þ mb þ m3 Þy0 þ bx0 A22  4B2 þðE7 x1  E8 m3 Þcosx1 t 0  ðE8 x1 þ E7 m3 þ f 1 Þsinx1 t0 þðE9 x2  E10 m3 Þcosx2 t 0  ðE10 x2 þ E9 m3 þ f 2 Þsinx2 t 0 ð29bÞ E6 m3 : Having found yðtÞ; xðtÞ can be obtained using Eq. (25b) as

Differentiating Eq. (25b) once with respect to time, and using Eq. (25a) in the resultant equation, we get

ðE9 x2 þ rE10 Þcosx2 t þ ðE10 x2  rE9 þ f 2 Þsinx2 t  rE6 :

€ þ A2 y_ þ B2 y ¼ bða  bÞ þ f 1 bsinx1 t þ f 2 bsinx2 t y þ f 1 x1 cosx1 t þ f 2 x2 cosx2 t:

yðtÞ ¼ C 3 em3 t þ C 4 em4 t þ E6 þ E7 sinx1 t þ E8 cosx1 t ð27Þ

where

m3;4 ¼

A2 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A22  4B2 2

ð28aÞ

;

E6 ¼

bða  bÞ ; B2

ð28bÞ

E7 ¼

f 1 x21 ðA2  bÞ þ bf1 B2  2 ; A22 x21 þ B2  x21

ð28cÞ

  f 1 x1 B2  bA2  x21 E8 ¼ 2  2 ; A2 x21 þ B2  x21

ð28dÞ

E9 ¼

E10

f 2 x22 ðA2  bÞ þ bf2 B2 A22 x22 þ ðB2  x22 Þ

2

;

  f 2 x2 B2  bA2  x22 ¼ 2  2 : A2 x22 þ B2  x22

ð30Þ

ð26Þ

where A2 ¼ ðb þ bm þ bÞ and B2 ¼ ðb þ bb þ bmbÞ. Eq. (26) is also an inhomogeneous second order linear differential equation for which the solution can be obtained as in the previous region, whose general solution of the form

þ E9 sinx2 t þ E10 cosx2 t;

1 ðm3 þ rÞC 3 em3 t  ðm4 þ rÞC 4 em4 t b ðE7 x1 þ rE8 Þcosx1 t þ ðE8 x1  rE7 þ f 1 Þsinx1 t

xðtÞ ¼

ð28eÞ

ð28fÞ

5.3. Observations of analytical study Now, let us briefly explain how the analytical solutions can be generated in the ðx  yÞ plane. If we start with the initial condition xðt ¼ 0Þ ¼ x0 , yðt ¼ 0Þ ¼ y0 in the region D0 , the arbitrary constants C 1 and C 2 are evaluated at t ¼ 0 using Eqs. (24a) and (24b). Then xðtÞ evolves as given by Eq. (23) up to either t ¼ T 1 , when xðT 1 Þ ¼ 1 or t ¼ T 01 when xðT 01 Þ ¼ 1. The value of T 1 and T 01 are obtained numerically. Knowing whether T 1 > T 01 or T 1 < T 01 , we can determine the next region of interest (Dþ or D ). The arbitrary constants of the solutions of that region namely C 3 and C 4 can be evaluated at time t ¼ T 1 or in the time t ¼ T01 ; at which the solution just enters into the region Dþ with xðT 1 Þ; yðT 1 Þ as initial conditions in Eqs. (27) and (30) or time at which the solution just enters into the region D , with xðT 01 Þ; yðT 01 Þ and the solution evolves. This procedure can be continued for each successive crossing. The solutions of Eqs. (22) and (23) for the inner region and Eqs. (27) and (30) for the two outer regions are matched for each successive crossing and the phase portraits are represented in the ðx  yÞ plane by slowly varying the control parameter ‘f 1 ’ or ‘f 2 ’. One can easily verify the formation of bubbles through this explicit analytical solutions by choosing the following set of parameter values. L ¼ 50:2 mH, C ¼ 10:0 nF, R ¼ 2:4 kX, frequency m1 ¼ 10001 kHz, frequency m2 ¼ 40000 kHz, amplitude f 2 ¼ 0:8, and the amplitude f 1 is the control parameter.

A. Arulgnanam et al. / Chaos, Solitons & Fractals 75 (2015) 96–110

6. Conclusions In this paper, we reported the birth of strange nonchaotic attractors through a new novel route which we termed as the multilayered bubbling route to SNA in a quasiperiodically forced series LCR circuit with a simple nonlinear element. In this route, bubbles appear in the strands of the torus as a function of the amplitude of one of the signals f 1 , which is used as the control parameter. The size of the bubbles is increased with increase in the value of the control parameter and subsequently three layers of bubbles are formed one by one. After certain threshold value of the control parameter f 1 , the strands of the bubbles are increasingly wrinkled resulting in the birth of SNA (while the remaining parts of the strands of the torus outside the bubbles remain largely unaffected). The mechanism for this route is that, the quasiperiodic orbit becomes increasingly unstable in the transverse direction as a function of the control parameter which is induced by the active elements associated with the other system parameters. As the control parameter f 1 is increased slowly, the layers disappear one by one, leading to a single bubble. Then, this single bubble increasingly wrinkles and gives birth to SNA. The bubbles observed through Poincaré surface of cross section plots and the corresponding power spectra are used to distinguish torus, SNA and chaos qualitatively in the system. Lyapunov exponents and their variance are computed as a function of the control parameter f 1 , to distinguish the dynamical region of the torus, SNA and chaos. To confirm the presence of SNA, singular-continuous spectrum and the distribution of finite time Lyapunov exponents are computed. An analysis of the separation of nearby points shows that the separation becomes zero for a short span of time for SNA in the case of the multilayered bubble route. From the above observations, it is noted that the newly formed route to SNA is significantly different from the well known mechanisms in the literature. The experimental observations also validate this new mechanism as seen in the Figs. 11–13. We also examined quantitatively the largest Lyapunov exponent and its variance, and made a singular-continuous spectrum analysis. Thus, with the parallel LCR circuit with our simple nonlinear element, we are able to generate a new route to SNA, namely multilayered bubble route along with Heagy-Hammel and fractalization route to SNA (not reported herewith) apart from the standard period doubling route to chaos upon introduction of a second quasiperiodic force. Given the ubiquity of SNA dynamics in the quasiperiodically driven system, one of the main issues with respect to the observation of SNAs is that this dynamical behavior occurs in a very narrow range of values of the control parameter. While identifying these attractors from numerical analysis, one may wonder whether they occur due to numerical artifacts and whether they may get smeared out if the inherent noise or parameter mismatch is included. For this purpose, it is important to verify the underlying phenomena experimentally to be sure about the existence of the type of transitions to SNAs as discussed in the present paper. We are so successful in doing the experimental

109

verification of the multilayered bubble route, HeagyHammel and fractalization route to SNA and the results will be published elsewhere, very shortly. To further confirm our results, explicit analytical solution is developed and is presented, which indicates, this multilayered bubbling route is mathematically tractable. Identifying different routes to SNA in a quasiperiodically forced parallel LCR circuit is kept as the future work and work is already started in this direction.

Acknowledgments AA thanks the University Grants Commission (UGC), Government of India sponsored, Teacher Fellowship under the Faculty Development Program. AP thanks the Department of Science and Technology, Government of India for the support of this work. KT and MD acknowledges the University Grants Commission (UGC) – India, for financial support under Special Assistance Programme (SAP).

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