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International Journal of Electronics

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Chaos intermittency and control of bifurcation in a ZC

2

-DPLL

Tanmoy Banerjee a; Bishnu Charan Sarkar aaDepartment of Physics, University of Burdwan, Burdwan, India

To cite this ArticleBanerjee, Tanmoy and Sarkar, Bishnu Charan(2009) 'Chaos, intermittency and control of bifurcation ina ZC

2-DPLL', International Journal of Electronics, 96: 7, 717 732

To link to this Article: DOI: 10.1080/00207210902851431

URL: http://dx.doi.org/10.1080/00207210902851431

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Chaos, intermittency and control of bifurcation in a ZC2-DPLL

Tanmoy Banerjee and Bishnu Charan Sarkar*

Department of Physics, University of Burdwan, Burdwan 713104, India

(Received 14 July 2008; final version received 24 January 2009)

Nonlinear dynamics of a dual sampler-based zero crossing digital phase lock loop(ZC2-DPLL) has been investigated. Analysis supported by detailed numericalstudies shows that the system enters a chaotic state through a cascade of perioddoubling bifurcation. The dynamics of the system have been quantified with the

dynamical measures of Lyapunov exponent and correlation dimension. Further,it has been found that for certain system parameters intermittency occurs in thesystem. The occurrence of intermittency has been proved using the PomeauManneville principle. The phenomenon of bifurcation control in a ZC2-DPLLusing time delay feedback has been explored. It has been found that for somesuitably chosen control parameters bifurcation phenomena can be controlled suchthat the stable locked zone of a bifurcation controlled ZC2-DPLL can beextended, which enhances the application potentiality of a ZC2-DPLL.

Keywords: digital phase locked loop; stability analysis; bifurcation control;intermittency; chaos

1. Introduction

Studies on nonlinear dynamics of different nonlinear electronic systems have

attracted researchers for at least three decades. Occurrence of complex behaviours

like bifurcation, chaos, intermittency, etc. in electronic systems has been revealed

from these studies (Kilias, Kelber, Mogel and Schwarz 1995; Chen, Chau and Chan

1999; Giannakopoulos and Deliyannis 2005). Further, control of chaos and

bifurcation in electronic circuits and systems is an active field of research (Chen,

Hill and Yu 2003; Collado and Suarez 2005). By controlling chaos and bifurcation

one can suppress chaotic behaviour where it is unwanted (e.g. in power electronics

and mechanical systems) and on the other hand, in electronic systems, one can

harness the richness of chaotic behaviour in chaos-based electronic communication

systems. In fact, the possibility of exploiting the chaotic signal in chaos-based secure

communication systems has boosted up the research on the chaotic dynamics of

electronic circuits and systems (Kennedy, Rovatti and Setti 2000). Among all the

electrical systems, the phase locked loop (PLL) is probably the most widely studied

system owing to its application potentiality in synchronous communication systems

and rich nonlinear dynamical behaviour (Gardner 1979; Kudrewicz and Wasowicz

2007). At the advent of digital communication systems digital phase locked loops

(DPLLs) have rapidly replaced the conventional analogue PLLs because they

*Corresponding author. Email: [email protected]

International Journal of Electronics

Vol. 96, No. 7, July 2009, 717732

ISSN 0020-7217 print/ISSN 1362-3060 online

2009 Taylor & Francis

DOI: 10.1080/00207210902851431

http://www.informaworld.com


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overcome the problems of sensitivity to DC drift, periodic adjustment and the

building of higher order loops (Lindsey and Chie 1981). DPLLs are widely used in

frequency demodulators, frequency synthesisers, data and clock synchronisers,

modems, digital signal processors and hard disk drives to name a few (Zoltowski

2001; Mannino et al. 2006). A DPLL is a discrete time nonlinear feedback controlled

system: as such its nonlinear behaviour is difficult and it poses exact solutions only in

particular cases. To understand the complete behaviour of a DPLL we have to resort

to modern nonlinear dynamical tools of bifurcation and chaos theories. Also in this

regard bifurcation control of DPLL has not been explored yet. The study of

nonlinear dynamics of DPLLs has two-fold application. First, using the insight of

nonlinear behaviours of DPLLs one can design an optimum DPLL system. Second,

by characterising the chaos from DPLLs, one can explore the possibility of using

DPLLs in chaos-based secure electronic communication systems. So in these

contexts, motivations are always there to explore the nonlinear dynamics of DPLLs.

In the previous studies on the nonlinear dynamics of DPLLs, positive zero crossing

DPLLs (ZC1-DPLL) (Bernstein, Liberman and Lichtenberg 1989; Leonov andSeledzhi 2005; Banerjee and Sarkar 2005ac; Banerjee and Sarkar 2008a), uniform

sampling DPLLs (Zoltowski 2001), bang-bang DPLLs (Dalt 2005) and tanlock

DPLLs (Hussain and Boashash 2002) were of main interest but work on the

nonlinear dynamics of dual sampler-based zero crossing DPLLs (ZC2-DPLL) is yet

to be reported. Unlike a ZC1-DPLL, in a ZC2-DPLL sampling is done at the positive

and negative zero crossings of the input signal. For this particular sampling

technique it has wide frequency acquisition range in comparison with a ZC 1-DPLL

and that is why ZC2-DPLLs have drawn the attention of researchers for a long time

(Majumdar 1979; Frias and Rocha 1980; Banerjee and Sarkar 2006).

This article investigates the nonlinear dynamics of a ZC2-DPLL using nonlineardynamical theoretical and computational tools. The bifurcation and chaos in the

system have been explored. Also the chaotic behaviour is quantified through

dynamical measures namely, the Lyapunov exponent and correlation dimension.

Further, the occurrence of intermittency in a ZC2-DPLL has been investigated and a

proper scaling behaviour has been explored using PomeauManneville principle.

Subsequently, we explore the possibility of bifurcation control in a ZC2-DPLL using

the time delay feedback control method (Pyragas 1992). It has been found that for

some suitably chosen control parameters, a bifurcation controlled ZC2-DPLL

(BCDPLL) has an extended stability zone and larger frequency acquisition range in

comparison with a conventional ZC2-DPLL (CDPLL), which will make it more

suitable in practical applications.

This article is organised in the following manner. The following section describes

the system structure and system equation formulation. In section 3, a bifurcation

analysis has been carried out. The nonlinear dynamical measures such as the

Lyapunov exponent and correlation dimension of a ZC2-DPLL have also been

reported in this section. Section 4 examines the effect of bifurcation control

technique on a ZC2-DPLL using analytical and numerical tools. Finally, section 5

summarises the outcome of the whole study.

2. System description and system equation formulationFigure 1 shows the block diagram of a ZC2-DPLL. It contains two positive edge

triggered samplers. The input signal is fed directly into sampler-1 and a p-shifted

718 T. Banerjee and B.C. Sarkar


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version of the input signal is fed into sampler-2. To formulate the system equation

consider e(t) to be the noise-free analogue input signal to the system with a phase

angleyi(t) relative to the loop DCO phase. Mathematically,

et A0sin o0t yit 1

whereyi(t) (oio0)t y0andA0,oi,y0are the amplitude, angular frequency andphase of the input signal, respectively. o0 is the nominal angular frequency of the

DCO having time period T. Writing the sampled version ofe(t) at the kth sampling

instant (SI)t(k) asx(k), one can write the output signals of sampler-1 and sampler-2,

respectively as follows (Majumdar 1979):

x1 k0 A0sin o0t k

0 yi k0 k0 2k 2

x1

k00

A

0sin o

0t k00

y

i k00

p

k00

2k

1

3

wherek 0,1,2,3 . . . . Here SIs are occurring at the end of each half period of theDCO (unlike positive zero crossing DPLL (ZC1-DPLL), where SIs occur at the end

of each full period of DCO).

The sequence {x(k)}, k 0,1,2 . . . is filtered digitally by a loop digital filter(LDF). In a first-order loop, the transfer function of LDF is written as a constant

gain G1 (s/V). The LDF output sequence is given by yk G1x(k). The sequences{y(k)} are used to control the next half period of the DCO. The kth half periodT0(k)

of DCO can be written as,

T0k Tk=2 tk1 tk; 4

Figure 1. Functional block diagram of a ZC2-DPLL.

International Journal of Electronics 719


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in terms of thekth and (k 1)-st SIs, respectively. The DCO periodT(k 1) at thatinstant is governed by the relation

Tk1 T

2

yk 5

Assuming t(0) 0, we get

tk kT

2

Xk1i0

yi; 6

thus the sampler output at the kth instant is,

xk A0sin fk 7

where,

jk yik o0Xk1i0

yi 8

is the phase error between the input signal and the DCO output at t(k). Now, one

can get the phase governing equation of the ZC2-DPLL as,

f k1 f k p z1 12

zK1sinf k ; 9

where z has been substituted in place of (oi/o0). Also where K1 A0o0G1 is theclosed loop gain of ZC2-DPLL.

3. Nonlinear dynamics of ZC2-DPLL

3.1. Analytical bifurcation analysis

System (9) can be treated as a one-dimensional sine circle map fk1 f(f) and westudy its dynamics near the fixed point. A fixed point x* is defined as a point for

which the condition,f(x*) x* is satisfied. In the present case, fixed point is a stablelocked state characterised by the steady-state phase errorfsfrom Equation (9). The

steady-state phase error can be calculated as,

fs sin1 2L0

zK1

; 10

whereL0 p(z 7 1). For stable loop operation the conditionL0 zK1/2 should beobeyed. The system will converge to a fixed point if the following condition is

maintained (Osborne 1980),

f0jsj j< 1; 11



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which in turn gives,

1zK1

2 cosfs


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Equations (13), (16) and (17). Similarly, Figure 2b shows the bifurcation diagram for

a frequency step input (with z 1.1). It is seen that bifurcation occurs at

(K1z)2

7 4L02

16 (In the numerical simulations for frequency step input, wehave taken the values of the initial phase inside thebasin of attractionof the system,

i.e. after a transient state finally all the initial values give bound solutions for the

system.). It can also be seen from Figure 2 that, with the increase ofK1 the system

goes to chaotic state through a period doubling bifurcation. Simulation results

confirm that ZC2-DPLL has no other route to chaos other than the period doubling

route. The chaotic behaviour of the loop has been studied in the next section.

3.2. Quantifying chaos in a ZC2 DPLL

Because the present system is a one-dimensional system it has only one Lyapunov

exponent. The Lyapunov exponent of a ZC2-DPLL can be defined as,

l limN!1

1

N

XNi1

ln 1zK1

2 cosfi

: 18

We calculate the Lyapunov exponent for the ZC2-DPLL with the help of Equation

(18) by taking the number of iterationsN 50,000, which is sufficiently large for thepresent system. Figure 3a shows the plot of the Lyapunov exponent (l) with the

control parameterK1forz 1. At K1 2, it shows a super stable cycle (i.e. l goes

to negative infinity). At K1 4, l becomes zero, it indicates that first bifurcationoccurs at this value of system parameter. Also at K1 6.28, period four oscillationoccurs. A positive Lyapunov exponent is a sign of chaos and K17lplot shows that

Figure 2. Bifurcation diagram of a ZC2-DPLL for (a) a phase step input (z 1) and (b) afrequency step input (z 6 1) with K1 as the control parameter.



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the system becomes chaotic at K1 7.1. Sudden jumps of l into the negativeLyapunov region indicate the presence of periodic windows interspersed into the

chaotic zone. The Lyapunov spectrum of the system is well agreed with

the bifurcation diagram of Figure 2a. Similarly, Figure 3b shows the spectrum of

the largest Lyapunov exponent of ZC2-DPLL for z 1.1.The correlation dimension is also a quantitative measure of chaos. To compute it

first we have to find the correlation integral defined as,

Cd

2

NN1XNi0

XNji1 H ddij

: 19

Nis the number of data points used in the time series (here we use N 50,000),Histhe Heaviside function, dijis the spatial separation between two data points labeled i

andjin the phase space anddis a set of pre-assigned values. A detailed procedure for

finding the correlation dimension with the GrassbergerProcaccia algorithm can be

found in Grassberger and Procaccia (1983). Since the present system is embedded in

one dimension with dij jfi7fjj and we have the opportunity of knowing therelevant variables, we do not have to resort to time delay technique (Sprott 2003).

Correlation dimension (D2) o f Z C2-DPLL at chaotic region (K1 8) can be

computed from the slope of the log C(d) vs. log dcurve (Figure 4). The correlationdimension of the system is found to be D2 0.83933 + 0.00215, which shows thatthe resulting chaos from a ZC2-DPLL is well defined and low dimensional.

Figure 3. Lyapunov exponent (l) spectrum of ZC2-DPLL with different gain parameters(K1) for a (a) phase step input (z 1) and (b) frequency step input (z 1.1). PositiveLyapunov exponent indicates the presence of a chaotic attractor.



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3.3. Intermittency in a ZC2-DPLL

In its route to chaos, a system may show back and forth switching between

apparently regular behaviour and chaotic behaviour. This switching occurs even

though the control parameters remain constant and no significant external noise is

present (Schuster and Just 2005). This interesting phenomenon is known as

intermittency. Intermittency of a ZC1-DPLL has been reported elsewhere (Banerjee

and Sarkar 2008b). In the present work a numerical searching reveals that a ZC2-DPLL shows intermittency for 7.4500 5 K1 5 7.45101. In this range of values of

K1, a ZC2-DPLL switches back and forth between period-6 and chaotic behaviour.

The switching appears to occur randomly even though a ZC2-DPLL is described by

a deterministic iterated map. Figure 5 depicts this phenomenon for four different

values of K1. In the first trace, periodic behaviour occurs randomly in between

chaotic behaviour at K1 7.4500. In the second trace of Figure 5, the value ofK1has been slightly changed to K1 7.4505; the trace shows that periodic behaviourdominates over chaotic behaviour. The third trace (K1 7.4509) shows that thechaotic behaviour occurs for a very small time duration. Thus, the system dynamics

changes rapidly in this region with K1. The lower trace shows that at K1

7.45101

system behaviour becomes periodic. Occurrence of intermittency can be explained

with the help of a cobweb diagram. Figure 6 shows a blow-up view of cobweb

diagram plotting sixth iterate (f6(f)) along the y-axis and f along the x-axis. At

K1 7.4505, a gap has been seen in between the map and the 458 diagonal line. Thetrajectory spends certain duration of time in the gap, i.e. in the vicinity of previously

stable period-6 fixed point. As the gap size decreases (i.e. as theK1values approaches

7.45101), the trajectory spends more time in the gap. Qualitatively, it can be said that

more time is spent in periodic behaviour as the parameter approaches the critical

value Kc, where Kc is the value of K1 at which point the gap vanishes and the

behaviour becomes exactly periodic. In the present case, Kc 7.45101, i.e. at this

point the system shows exactly period-6 behaviour. Further, we examine theintermittency occurring in a ZC2-DPLL from the view point of the Pomeau

Manneville (Manneville and Pomeau 1979) principle. According to this principle the

Figure 4. Plot between log C(d) vs. log dfor ZC2-DPLL (K1 8, z 1).



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Figure 6. Blowup view of cobweb diagram for different values of K1. Along x-axis f andalongy-axis f6(f) have been plotted.

Figure 5. Intermittency in a ZC2-DPLL for different values ofK1.



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average length of the laminar region (i.e. the average number of iterations spent in

the periodic region) varies with the parameter e (Kc7K1) with a certain scalefactor. To investigate this we have determined the average length hli of the laminarregion for different values of e. Figure 7 shows the plot of hli versus e70.5, whichshows that they have linear dependence. This result obeys PomeauManneville

principle for a type-I intermittency, which confirms that the intermittency that

occurs in a ZC2-DPLL is a type-I intermittency.

4. Bifurcation control in a ZC2-DPLL

4.1. Time delay feedback in a ZC2-DPLL

We investigate the feasibility of bifurcation control in a ZC2-DPLL using Pyragass

time delay feedback technique (Pyragas 1992). Earlier Sarkar and Chattopadhyay

(1988) proposed this technique for improving the transient behaviour of DPLLs. In

this article, our aim is to explore the full nonlinear dynamical behaviour of time delay

feedback-based ZC2-DPLLs from the perspective of bifurcation control. The proposedmodel of bifurcation controlled ZC2-DPLL (BCDPLL) has been shown in Figure 8. It

has an additional error signal generated as xa(k) P[x(k)7x(k71)]. P is anadditional design parameter that quantifies the coupling strength of error signal

with the original sampled signal x(k). The DCO period control signal of BCDPLL at

the kth SI is generated using the digitally filtered version xc(k); where xc(k) is the sum of

x(k) andxa(k). The control signal atkth SI isy(k), given asG1xc(k). Now, defining the

signal phase relative to the DCO phase atkth SI asf(k) and using the same procedure

used in section 2, one can get the phase governing equation of the BCDPLL as,

f k1 f k p z1 12

zK1 1P sinf k 12

PzK1sinf k1 ; 20

Figure 7. Plot ofhli versus e70.5. Plot shows linear dependency.



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(2) For a frequency step input (z 6 1), the steady-state phase error fs can beobtained from Equation (20) as, fs sin

71[2L0/(K1)z] 2np, where,n 0,1, 2 . . . andL0 p(z71). It means that, with higher values ofK1, fscan beless. The condition for convergence with a frequency step input is obtained

using the aforesaid method as,

0< K1=2 2

z2 L20

< 4 12P 2 23

with K1z 4 0. Here also P can be chosen in a way to enlarge the range ofz

leading to a stable loop operation in a BCDPLL. If one can extend the lock

range (i.e. period-1 state) then this would imply that the chaotic behaviour

would be shown by the system at a larger value of system parameter.

Also, the frequency acquisition range of BCDPLL can be obtained

as, pp0:5K1

< z < pp0:5K1

, which is broader than that of a CDPLL as

maximumK1 is more in this case for the new stability condition.

4.3. Lyapunov exponent of BCDPLL

To find out the Lyapunov exponent of BCDPLL we have employed Sprotts (2003)

method for finding the largest Lyapunov exponent of a 2D map. The Lyapunov

exponentl is given by,

l limN!1

1

2NXN1n0

ln abY0n

2 cdY0n2

1Yn2

" # 24

aand dare the diagonal terms of the Jacobian matrix J and b and c are off-diagonal

terms of the Jacobian matrix J. In the present case, from Equation (21) we get a 0,d 170.5K1z(1 P)cosy(k), b 1 and c 0.5K0zP cos x(k). Y

0 is the tangent of

the direction of maximum growth (tangent vector) that evolves in the following

manner,

Y0

n1

c dY0

n

abY0n: 25

For large numbers of iteration tangents, vector is independent of the initial value

of Y00.

4.4. Numerical results

The system response has been studied through extensive numerical simulation of

system equations. Figure 9a depicts the bifurcation diagrams of a BCDPLL in the

face of phase step input (z 1) for different values of control parametersP. It can be

seen that suitably chosen values of P controls the occurrence of bifurcationphenomenon as predicted in Equation (22). Figure 9b shows that period 2

bifurcation occurs at K1 8 for z 1 by choosing P 70.25 (unlike K1 4 for



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Figure 9. (a) Bifurcation diagram of BCDPLL with K1 for different values of bifurcationcontrol parameters P (for z 1). Bifurcation diagram of BCDPLL (P 70.25) for a (b)phase step input (z 1) and (c) frequency step input (z 1.1) withK1as the control parameter.



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a CDPLL; see Figure 2a). Bifurcation diagram of a BCDPLL (P 70.25) withfrequency step input (z 1.1) has been shown in Figure 9c. The obtained results arein complete agreement with the analytical predictions of Equations (22) and (23).

Also the largest Lyapunov exponent spectrum of BCDPLL with P 70.25 for aphase step (z 1) and frequency step input (z 1.1) have been shown in Figure 10aand b, respectively. These two results agree with bifurcation diagrams. Thus, it can

be concluded that time delay feedback technique can effectively control the

occurrence of bifurcation in a ZC2-DPLL.

5. Conclusion

The present work reports the nonlinear dynamical studies of a ZC2-DPLL. The

outcomes of the study can be summarised as follows:

(1) It has been found that a ZC2-DPLL goes to chaotic state through a cascade

of period doubling bifurcations. The chaotic state is quantified by dynamical

measures namely Lyapunov exponent and correlation dimension. It has been

shown that chaos from a ZC2-DPLL is low dimensional and well

characterised. This well-defined nature of chaos from ZC2-DPLL indicates

the application potentiality of a ZC2-DPLL in the field of chaos-basedcommunication systems.(2) It has been shown that a ZC2-DPLL depicts

intermittent behaviour for some values of system parameters. Applying the

Figure 10. Lyapunov exponent (l) spectrum of BCDPLL (P 70.25) with different K1values for a (a) phase step input (z 1) and (b) frequency step input (z 1.1).



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