Birth of strange nonchaotic attractors through type III intermittency

16 August 1999

Physics Letters A 259 Ž1999. 246–253 www.elsevier.nlrlocaterphysleta

Birth of strange nonchaotic attractors through type III intermittency A. Venkatesan, K. Murali 1, M. Lakshmanan

2

Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan UniÕersity, Tiruchirappalli - 620 024, India Received 29 April 1999; accepted 29 June 1999 Communicated by C.R. Doering

Abstract A new route and the associated mechanism are described for the creation of a strange nonchaotic attractor during the transition from two-frequency quasiperiodicity to a chaotic attractor through torus doubling bifurcation in quasiperiodically forced systems. The strange nonchaotic attractor arises when a a torus doubled attractor is interrupted by a subharmonic bifurcation, resulting in the inhibition of torus doubling sequence. This transition is shown to exhibit type III intermittent characteristic scaling behaviour. The scenario and mechanism are illustrated in detail through experimental and numerical studies of a quasiperiodically driven simple piecewise linear electronic circuit. q 1999 Elsevier Science B.V. All rights reserved. PACS: 05.45 q b; 07.50.Ek

A strange nonchaotic attractor ŽSNA. is one that exhibits complicated geometric structure but without any sensitive dependence on initial conditions Žthat is, no positive Lyapunov exponent.. Following the initial study of Grebogi et al. w1x, several theoretical as well as experimental studies pertaining to the existence and characterization of SNAs in different quasiperiodically driven nonlinear dynamical systems have elucidated many major features of these exotic but important class of attractors w2–14x. While the existence of SNAs is shown to be generic in quasiperiodically forced nonlinear sys-

1

Present address: Department of Physics, Anna University, Chennai - 600 025, India. 2 Corresponding author. E-mail: [email protected]

tems, a question that remains of considerable physical interest is what are the possible routes by which they arise and ultimately become chaotic and how do these attractors are born in a system. So far, a few routes and mechanisms have been identified for the creation of SNAs. The routes include, torus doubling to chaos via SNAs w3x, gradual fractalization of torus w4x, the appearance of SNAs via blowout bifurcation w5x, the occurrence of SNAs through type I intermittent phenomenon w6x, remerging of torus doubling bifurcations and the birth of SNAs w7x, and so on w8–12x. Some of the mechanisms identified are the following: Ž1. the birth of a SNA is due to the collision of a period doubled torus with its unstable parent w3x, Ž2. the increasing of wrinkling of a torus leads to the appearance of SNAs without any interaction with a nearby unstable periodic orbit w4x, Ž3. the

0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 5 6 - 9

A. Venkatesan et al.r Physics Letters A 259 (1999) 246–253

occurrence of SNAs is through the loss of transverse stability of a torus w5x, Ž4. SNAs arise in the neighbourhood of a saddle-node bifurcation whereby a torus is replaced by SNAs w6x. Recently, it has been found that a scenario which seems to be generic is a transition from two-frequency quasiperiodicity to SNAs through torus doubling bifurcation in the quasiperiodically forced systems w3,4x. A common feature in this scenario is that the birth of SNAs is either due to the collision of a period doubled torus with its unstable parent, so that a period 2 k-torus gives rise to a 2 ky1-band SNA w3x or a gradual fractalization of torus, in which a period 2 k-torus approaches a 2 k-band SNA w4x. However, in the present work, we describe a new scenario inwhich the torus doubling sequence is tamed due to subharmonic bifurcations leading to the creation of SNAs. During this transition, a growth of the subharmonic amplitude begins together with a decrease in the size of the fundamental amplitude, characteristics of type III behaviour which are typical in the standard intermittent phenomenon w15x. When the subharmonic amplitude reaches a high value, the attractor loses its regularity and a strange attractor appears. We also find that the dynamics at this transition possesses type III intermittent characteristic scaling behaviour. To illustrate our findings, we consider the simplest second order nonlinear dissipative nonautonomous circuit introduced recently w16–18x and shown in Fig. 1. The circuit is a classic configuration

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of a forced negative resistance oscillator, where N denotes a voltage-controlled nonlinear resistor described by i s g Ž Õ ., which in this case is a Chua’s diode w19x, C is the capacitor, R is a linear resistor, L is an inductor while R s is a sensing resistor. f 1Ž t . and f 2Ž t . are the external forcing functions delivered from external function generators ŽHP 33120A series.. Then, the governing equations of this circuit for the voltage Õ across the capacitor C and the current i L through the inductor L are given by the following set of two first order nonautonomous differential equations: C L

dÕ dt diL dt

siL yg Ž Õ. , s y Ž R q R s . i L y Õ q F1 sin V 1 t q F2 sin V 2 t ,

Ž 1. where F1 and F2 are the amplitudes and V 1 and V 2 are the angular frequencies of the forcing functions f 1Ž t . and f 2Ž t ., respectively, of the circuit Ž1.. In the absence of F2 , the dynamics of the circuit Ž1. has been studied in detail in Refs. w16–18x. In the presence of F 2, the dynamics of Ž1. has been studied numerically and experimentally w20,21x on some of the aspects of strange nonchaotic trajectories on torus. In order to realise the novel dynamical phenomena underlying Ž1. completely, we will first consider the results of numerical simulations and then confirm them experimentally. The expression g Ž Õ . s G b Õ q 0.5Ž Ga y G b . w < Õ q Bp < y < Õ y Bp
Fig. 1. Circuit realization of the simplest nonautonomous circuit. Here, N is the Chua’s diode, Rs1340 V , Ls18 mH, C s10 nf, R s s 20 V . f 1Ž t . s F1 sin V 1 t and f 2Ž t . s F2 sin V 2 t are the function generators ŽHP 33120A.. The values of V 1 and V 2 are chosen as 23706.667 Hz and 7325.763 Hz respectively.

y˙ s y Ž 1 q n g . b y y b gx q f 1 sin u q f 2 sin f ,

u˙s v 1 , f˙ s v 2 ,

Ž 2.

A. Venkatesan et al.r Physics Letters A 259 (1999) 246–253

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where the rescaled quantities are defined by the relations Õ s xBp ,

v2 s f1 s

i L s GyBp ,

V2C G

F1 b Bp

,

,

ts

f2 s

tC G

Gs

Bp

R

bs

,

F2 b

1

v1 s

, C

LG 2

V 1C G

,

n s GR s ,

,

,

and the overdot corresponds to

d

ž /

. Obviously dt g Ž x . s bx q 0.5Ž a y b .w < x q 1 < y < x y 1
cally by observing the phase trajectory and power spectrum. For our experimental study of the circuit given in Fig. 1, a two dimensional projection of the attractor is obtained by measuring the voltage Õ across the capacitor C and the current i L through the inductor L in the form of voltage drop across the current sensing resistor R s Ž Õs s R s i L . and connected to the X and Y channels of an oscilloscope. A live picture of the corresponding power spectrum Žobtained from a digital storage oscilloscope – HP 54600 series. of the projected attractor has also been used to distinguish the different attractors. At first, the phase trajectory obtained in the experiment is compared with the numerical trajectory. Then, the Fast Fourier Transform option in the oscilloscope has been used to distinguish different attractors. Further to identify the different attractors the dynamical transitions are traced out by two scanning procedures, both numerically and experimentally: Ž1. varying f 1 Žor F1 . at a fixed f 2 Žor F2 ., and Ž2. varying f 2 Žor F2 . at a fixed f 1 Žor F1 . in a 1000 X 1000 grid. The resulting phase diagram in the Ž f 1 y f 2 . parameters space in the region f 1 g Ž0,0.2. and f 2 g Ž0,0.2. is shown in Fig. 2, which has also been verified in the corresponding Ž F1 y F2 . parameters

Fig. 2. Phase diagram in the f 1 y f 2 parameter space for the model Ž2.. 1T, 2T and 3T correspond to torus, doubled torus and torus of period 3 respectively. SI3, and SF denote the formation of SNAs through type III intermittency and gradual fractalization respectively. C1 and C2 represent chaotic attractors. Tr stands for transient regions.

A. Venkatesan et al.r Physics Letters A 259 (1999) 246–253

Fig. 3. Maximal Lyapunov exponent L versus f 2 for f 1 s 0.08.

space. The various features indicated in the phase diagram are summarised and the dynamical transitions are elucidated in the following. For low f 2 and any f 1 values in the chosen range, the system exhibits two-frequency quasiperiodic oscillations denoted by 1T in Fig. 2. When the value of f 2 exceeds a certain critical value for a fixed f 1 , the two-frequency quasiperiodic attractor undergoes a torus doubling bifurcation and the region of torus doubled orbit is denoted by 2T in Fig. 2. It may also

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be observed that this bifurcation is geometrically very similar to that of period doubling bifurcation in three dimensional flows. One then expects as f 2 is increased further that the period doubling sequence has to continue as in the case of period doubling phenomenon. However in the present case, the doubling cascade is interrupted by an intermittent SNA Žalong the region SI3 in Fig. 2., which then finally transits to chaotic attractor ŽC1. as f 2 is increased further Žsee Fig. 3.. To understand the mechanism of the interruption of the doubling cascade let us consider a more specific parameter value of f 1 as 0.08, while f 2 is varied. For f 2 s 0.05, the attractor is a two-frequency quasiperiodic attractor as shown in Figs. 4–6. As f 2 is increased to f 2 s 0.0585, the attractor undergoes a torus doubling bifurcation and the corresponding orbit is shown Figs. 4–6. Increasing the f 2 value further, a second period doubling of the doubled torus does not take place as in the usual period doubling route to chaos. Instead, a new dynamical behaviour, namely intermittent phenomenon starts appearing at f 2c s 0.07454785 where most points remain near the doubled torus with sporadic large

Fig. 4. Projections of attractors of system Ž2. in the Žx, f . plane with f modulo 2p for f 1 s 0.08 and several values of f 2 : Ža. quasiperiodic attractor for f 2 s 0.05; Žb. torus doubled attractor for f 2 s 0.0585; Žc. intermittent strange nonchaotic attractor for f 2 s 0.07454785; Žd. chaotic attractor for f 2 s 0.07454804.

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A. Venkatesan et al.r Physics Letters A 259 (1999) 246–253

Fig. 5. Attractors corresponding to Figs. Žnumerical integration of Eq. Ž2..: Ži. phase trajectory Ž x y y .; Žii. power spectrum.

deviations, as shown in Figs. 4–6. One also finds that the amplitude of the subharmonic component of the period doubled orbit increases while the amplitude of the fundamental component decreases when a transition from doubled torus to intermittent phenomenon takes place. At the intermittent transition, the amplitude variation loses its regularity and a

burst appears in the regular phase Žquasiperiodic orbit trajectory.. This behaviour repeats as time increases as observed in the usual type III intermittent scenario. The duration of laminar phases in this state is random. Obviously, the corresponding Fourier spectrum in Figs. 5 and 6 confirms this behaviour where the amplitude of the subharmonic component

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maximal Lyapunov exponent ŽFig. 7Ža.. and its variance ŽFig. 7Žb.. during the transition from doubled torus attractor to SNA. To verify further that the attractor depicted in Figs. 4c, 5c & 6c is strange but nonchaotic, we proceed to quantify the changes in the power spectrum. When we compute the so called spectral distribution N Ž s ., which is defined as the number of peaks in the Fourier amplitude spectrum larger than some value say s , the SNAs satisfy a scaling power law relationship N Ž s . s syb , 1 - b - 2. The approximate straight line in the log–log plot shown in Fig. 8a obeys the power-law relationship with a value of b ; 1.14, affording an important characteristic signature by means of which SNA is distinguished from a chaotic attractor. Further, we have also checked that the path between the real and imaginary Fourier amplitude values exhibits selfsimilarity structure Žindicated in Fig. 8b.. These results strongly confirm that the attractor in Figs. 4c, 5c & 6c is indeed a strange nonchaotic one. More-

Fig. 6. Attractors obtained experimentally Žfrom circuit given in Fig. 1. corresponding to Figs. Fig. 5.

Žwith frequency W2r2. is greater than the amplitude of fundamental component Žwith frequency W2.. Although the attractor shown in Figs. 4–6 possesses geometrically complicated structure with the correlation dimension having a value d G s 1.34, the maximal Lyapunov exponent is only y0.003 ŽFig. 7.. Hence, the attractor is strange but nonchaotic. We also note that there is an abrupt change in the

Fig. 7. The transition from torus doubled attractor to intermittent SNA: Ža. the behaviour of the Lyapunov exponent Ž L.; Žb. variance Ž s ..

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A. Venkatesan et al.r Physics Letters A 259 (1999) 246–253

Fig. 8. Ža. Spectral distribution function for the spectrum shown in Fig. 4c; Žb. the path between the Re XŽ v . and Im X Ž v ..

over, the plots of mean laminar length ² l : as a function of the derived bifurcation parameter ´ s Ž f 2c y f 2 ., where f 2c is a critical parameter for the occurrence of the intermittent transition, for this attractor reveals a power-law relationship of the form ² l : s ´ya with an estimated value of a ; 0.81 ŽFig. 9.. This analysis also confirms that such an attractor is associated with standard intermittent dynamics of type III described in Ref. w15x. Further, a question may be asked as to how robust is the phenomenon of type III intermittent type SNA as it occurs in a narrow range of the parameter space. To check this, we added a small noise term to the right hand side of Eq. Ž2. and examined the influence of it on the SNA. We find that the SNA still survives even in the noise environment. In fact, the range of SNA increases, as can be seen from the advancement of Ž f 2c y f 2 . value in Fig. 10, where the occurrence of SNA starts.

Fig. 9. Ža. Mean laminar length ² l : versus e s f 2c y f 2

On further increase of the value of f 2 beyond 0.07454803, we find the emergence of a chaotic attractor as shown in Figs. 4–6, which though visibly similar to the SNA, Figs. 4–6, has a positive Lyapunov exponent Žsee Fig. 7.. Increasing f 2 value even further, we have observed that apart from the above mechanism there is another kind of mechanism Žpointed out in the beginning. for the creation of SNAs that is operative in this system in the region of the parameters considered. Such an SNA is created through a process of fractalization which occurs along the entire lower edge of the region C2, which is marked by the curve SF ŽFigs. 2 and 3. In the upper region of C1, the transition from chaotic behaviour to torus occurs through transient phenomenon ŽTr..

Fig. 10. The behaviour of the Lyapunov exponent at the transition to intermittent SNA from torus doubled attractor Ža. in the absence of noise Žsolid curve. Žb.in the presence of noise Žclosed circles..

A. Venkatesan et al.r Physics Letters A 259 (1999) 246–253

In conclusion, we have described a new mechanism for the inhibition of torus doubling sequence and creation of strange nonchaotic attractors through intermittency, whereby the torus doubled quasiperiodic orbit is eventually replaced by chaotic attractor via SNAs. Particularly, the birth of SNAs is linked with subharmonic bifurcation which destabilizes the period doubled torus and the intermittent behaviour appears. Also the intermittent nature is shown to have Type III Pomeau–Manneville intermittent characteristic scaling behaviour. We have also presented a circuit model for which the subharmonic bifurcation route to SNAs can be observed experimentally and numerically. Finally we wish to mention that we have observed the phenomenon reported in this letter also in quasiperiodically driven Duffing oscillator and some maps, which clearly shows that the routes and the associated mechanism are generic ones. The fuller details will be presented separately.

Acknowledgements This work forms part of a Department of Science and Technology, Government of India research project. A.V. wishes to acknowledge the Council of Scientific and Industrial Research, Government of India, for financial support.

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