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Chaos and bifurcation in a third-order phase locked loop
Chaos, Solitons and Fractals 19 (2004) 667–672 www.elsevier.com/locate/chaos
Chaos and bifurcation in a third-order phase locked loop Bassam A. Harb a, Ahmad. M. Harb b
b,*
a Electrical Engineering Department, Yarmouk University, Irbid 21110, Jordan Electrical Engineering Department, Jordan University of Science & Technology, P.O. Box 3030, Irbid 21110, Jordan
Accepted 24 April 2003
Abstract The chaos induced in a new type of phase locked loop (PLL) having a second-order loop filter is investigated. The system under consideration is modeled as a third-order autonomous system with sinusoidal phase detector characteristics. The modern of nonlinear theory such as bifurcation and chaos is applied to a third-order of PLL. A method is developed to quantitatively study the type of bifurcations that occur in this type of PLLÕs. The study showed that PLL experiencing a Hopf bifurcation point as well as chaotic behaviour. The method of multiple scales is used to find the normal form near the Hopf bifurcation point. The point is found to be supercritical one. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction Phase locked loops (PLLs) are an important part of modern electronic communication and control systems [1,2]. They play a major role in systems where it is necessary to estimate the phase of a received signal. Also, they are used extensively in applications requiring the synthesis of highly stable sinusoidal signals. In such systems, it is often necessary to estimate the instantaneous phase of a received signal which has been contaminated by random noise/or other type of interference. Often, one is supplied with a sinusoidal reference voltage, and it is necessary to produce a second sinusoidal voltage having a frequency which is an integer multiple of the referenceÕs frequency. PLLs offer a practical method of performing these tasks. The basic configuration of a PLL is shown in Fig. 1. It consists of three building blocks; a phase detector (PD), a time-invariant loop filter and a voltage controlled oscillator (VCO) [3]. The PLL is considered to be phase locked when the loopÕs phase error is constant /0 and the loop is in stable equilibrium state [4,7]. This mean that small perturbation from /0 , in the phase error will eventually dampen out as a result of the closed loop dynamic. Equivalently, /0 is an asymptotically stable solution of the autonomous, nonlinear differential equation describing the closed loop phase error. This differential equation is given in Section 2. The PLL can be false locked [4,7,9]. When this occurs a periodic orbits exist for a certain loop parameter. The behavior of these periodic orbits is of a great interest. It is known that the PLL under consideration exhibits a chaotic behavior preceeded by a series of period doubling for the periodic orbit associated with false lock state of the PLL. Chaotic properties of the PLLÕs have been frequently studied in recent years [10–12]. Many methods were used to prove the existence of chaos in different dynamic systems. The Melnikov method is the most popular one by far [10–12]. Also, the harmonic balance technique [8] predicts the occurrence of chaotic behavior and several bifurcation phenomenon in nonlinear circuits. Watada et al. [10] use piecewise linear analysis to investigate the Shilnikov homoclinic bifurcation in a third-order phase locked loop with
*
Corresponding author. Tel.: +962-2-720-1000; fax: +962-2-709-5018. E-mail address: [email protected] (A.M. Harb).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00197-8
668
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672
Fig. 1. Block diagram of a phase-locked loop with second-order loop filter.
piecewise linear PD. They demonstrate the role of the homoclinic orbit in the global bifurcation of attractors. In this paper, a third order PLL with sinusoidal PD will be considered. The modern nonlinear theory [5,6], such as bifurcation and chaos theory, and the method of multiple scales [13], is used to analyze the dynamics of PLL. To investigate the type of bifurcations that occur in PLL, we found the normal form near the Hopf bifurcation point by using the method of multiple scales. A bifurcation diagram will be developed which shows the parameter (K0 ) for which a limit cycle (false lock state) exists. Moreover, the limit cycle will be computed numerically. It is shown that this limit cycle undergoes period doubling bifurcation which at the end drive the PLL to chaotic behavior. The parameter ranges for which chaos exist is determined. The outline of our paper is as follows: In Section 2, the classical model for the third-order loop will be derived. Also, the nonlinear differential equation that describes the closed loop phase error will be derived. While in Section 3, the equilibrium solutions and their stability will be discussed. In Section 4, the dynamical solution will be presented. Finally, in Section 5, some conclusions have been withdrew.
2. Dynamical model The basic configuration of a PLL is shown in Fig. 1. It consists of three building blocks; a PD, a time-invariant loop filter and a VCO. The VCOÕs instantaneous frequency is given by dhv ¼ x0 þ kv eðtÞ rad=s; dt
ð1Þ
where x0 is called the VCO quiescent or center frequency and kv is the VCO gain which has units of (rad/s V). The output of the PD is given by xðtÞ ¼ k1 ½sinðhi hv Þ þ sinðhi þ hv Þ:
ð2Þ
The term sinðhi þ hv Þ contains higher fundamental frequency components which are eliminated by the loop filter. Then, the output of the loop filter becomes Z t eðtÞ ¼ e0 ðtÞ þ xðt uÞf ðuÞdu; t P 0; ð3Þ 0
where f ðtÞ is the impulse response of the filter, and e0 ðtÞ is the filterÕs zero-input response which depends only on the initial conditions existing in the filter at t ¼ 0. If the filter is stable, e0 ðtÞ ! 0 as t ! 1 for any set of initial conditions. The loop filter which will be considered in this paper has the form F ðsÞ ¼
1 þ sz1 s 1 þ sz2 s ¼ F1 ðsÞF2 ðsÞ; 1 þ sp1 s 1 þ sp2 s
where F1 ðsÞ and F2 ðsÞ are two cascade lag-lead filters. Eqs. (1)–(3) can be combined to obtain Z t Z t dhv ¼ x 0 þ kv xðuÞf ðt uÞdu ¼ x0 þ k1 kv f ðt uÞ sinðhi ðuÞ hv ðuÞÞdu: dt 0 0
ð4Þ
ð5Þ
Define the closed loop phase error as / hi hv ;
ð6Þ
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672
669
and the closed loop gain as ð7Þ
k ¼ k1 kv : The results which follow are simplified by defining h1 ¼ hi x0 t;
ð8Þ
h2 ¼ hv x0 t:
ð9Þ
These quantities can be used with Eqs. (5)–(7) to write Z t d/ dh1 ¼ Ak f ðt uÞ sinð/ðuÞÞdu for t P 0: dt dt 0
ð10Þ
The differential equation that describes the closed loop phase error in the PLL is given by d/ dh 1 þ sz1 d=dt 1 þ sz2 d=dt ¼ k sinð/Þ; dt dt 1 þ sp1 d=dt 1 þ sp2 d=dt
ð11Þ
where sp1 , sp2 , sz1 and sz2 are the loop filters time constants. After simplications, the above equation becomes d3 / d2 / þ 2 dt3 dt ¼
sp1 þ sp2 ksz1 sz2 cos / þ sp1 sp2 sp1 sp2
þ
d/ dt
1 kðsz1 þ sz2 Þ cos / þ sp1 sp2 sp1 sp2
d2 h1 sp1 þ sp2 d2 h1 1 dh1 : þ þ sp1 sp2 dt dt2 sp1 sp2 dt2
ksz1 sz2 sin / sp1 sp2
d/ dt
2 þ
k sin / sp1 sp2 ð12Þ
If the input frequency is constant, then h1 ðtÞ ¼ ðxi x0 Þt þ h0 ;
ð13Þ
and by normalizing the time variable using t0 ¼ ðk=sp1 sp2 Þ1=3 t, Eq. (12) becomes hv þ a/€ þ b cosð/Þ/€ þ c/_ þ d cosð/Þ/_ e sinð/Þ/_ 2 þ sinð/Þ ¼ d;
ð14Þ
where ¼ d=dt0 , s ¼ ðk=sp1 sp2 Þ1=3 , a ¼ ðsp1 þ sp2 Þs2 =k, b ¼ sz1 sz2 s2 , c ¼ s=k, d ¼ ðsz1 þ sz2 Þs, e ¼ sz1 sz2 s2 , d ¼ xos =k and xos ¼ xi x0 . The above equation can be written as: /_ ¼ x;
ð15aÞ
x_ ¼ y;
ð15bÞ
y_ ¼ ay b cosð/Þy cx d cosð/Þx þ e sinð/Þx2 sinð/Þ þ d:
ð15cÞ
3. Equilibrium solutions and their stability v The equilibrium solution of the system of Eqs. (15a)–(15c) corresponding to /_ ¼ /€ ¼ / ¼ 0. Setting the right hand sides of Eqs. (15a)–(15c) equal to zero, we end up with a nonlinear algebraic equation. The solution of this equation as a function of one of the control parameters are defined by using a continuation scheme. The stability of an equilibrium solution depends on the eigenvalues of the Jacobian matrix of Eqs. (15a)–(15c) evaluated at the equilibrium point. In this paper we write our own program for calculating the fixed points and their bifurcations rather than use an available bifurcation software package, such as BIFOR2 and AUTO94. To this end, we follow Watada et al. [10] and consider the case of the sinusoidal PD. In Fig. 2, we show the bifurcation diagram, which is the variation of the control parameter K with the state variable /_ . The solid line denotes stable nodes, while the dashed line denotes unstable foci. There is only one Hopf bifurcation point H at K0 ¼ 7299:01.
670
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672 35 30 25
x
20 15 10
H
5 0
ko=7299.01
2000
4000
6000
8000
10000
12000
14000
k Fig. 2. Bifurcation diagram (variation of state variable versus the control parameter K).
Using the normal form near the Hopf bifurcation point H , we find that H is a supercritical point. So, as K was increased above 7299.01, a sequence of deformed (asymmetric) periodic solutions was observed, leading to chaos.
4. Dynamical solutions A representative bifurcation diagram for the case of sinusoidal PD is shown in Fig. 2. In this section we discuss dynamical solutions. Near the Hopf bifurcation point H , Eqs. (15a)–(15c) possess small limit-cycle solutions. The stable limit-cycles grow in size and break their symmetry and finally culminating in chaos. We used the method of multiple scales encoded in MATHEMATICA [13] to reduce Eqs. (15a)–(15c) into their normal form near the Hopf Bifurcation point H . The normal form is then used to determine the amplitudes of the small limit cycles and their stability. The normal form near Hopf Bifurcation is given by a_ ¼ b1 a þ b2 a3 ;
ð16Þ
ah_ ¼ b3 a þ b4 a3 ;
ð17Þ
where bi are functions of the system parameters. Limit cycles correspond to a_ . Then, it follows from Eq. (16) that the amplitudes of limit-cycles are given by sffiffiffiffiffi b1 : a¼ b2
ð18Þ
When b2 < 0, the limit-cycles is stable and hence the Hopf bifurcation is supercritical. When b2 > 0, the limit-cycle is unstable and hence the Hopf bifurcation is subcritical. In the case of sinusoidal PD, we found that b2 < 0, hence the Hopf bifurcation point H is supercritical point. As K increased above the supercritical Hopf bifurcation H at K ¼ 7299:01, a small and stable limit-cycle is born, as shown in Fig. 3 at K ¼ 7299:01. As K increased further, the limitcycle grows, deforms, and then undergoes to chaos at K ¼ 85299:01 as shown in Fig. 4.
5. Conclusions The modern nonlinear theory such as bifurcation and chaos theory has been used to study the stability of a third order PLL. The study shows that this kind of PLL experienced chaos through Hopf bifurcation. In other words, the PLL is unlocked after the Hopf bifurcation point H . Because for the equilibrium solution the PLL will be in locked situation. The method of multiple secales, perturbation method, has been used to find the normal form at the vicinity of the Hopf bifurcation point. The point is found to be supercritical one. That means, a small periodic solutions, limit
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672
Fig. 3. Time responses and state-plane representation.
671
672
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672 (a1)
1.5
1
1
0.5 X3
X3
0.5 0
0
-0.5
-0.5
-1
-1
-1.5 0
(a2)
1.5
200
400
600
800 1000 1200 1400 1600 1800 2000 Time
-1.5 -2.5
-2
-1.5
-1
-0.5
0 X2
0.5
1
1.5
2
2.5
Fig. 4. The chaotic time response and state-plane for K ¼ 85299:01.
cycles, are borne at the Hopf bifurcation point. As the control parameter K increased, the limit cycles deformed and at the end culminating to chaos.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Viterbi AJ. Principle of Coherent Communications. New York: McGraw Hill; 1966. Blanchard A. Phase-Locked Loops. New York: John Wiley; 1976. Gardner FM. Phaselock Techniques. New York: John Wiley; 1979. Stensby JL. Saddle node bifurcation at a nonhyperbolic limit cycle in a phase locked loop. J Franklin Inst 1993;330(5):775–86. Perko L. Differential Equation and Dynamical Systems. New York: Springer Verlag; 1991. Hale J, Kocak H. Dynamics and Bifurcations. New York: Springer Verlag; 1991. Harb B, Stensby JL. The half-plane pull-in range of a second-order phase locked loop. J Franklin Inst 1996;333(B)(2):191–9. Piccardi C. Bifurcations of limit cycles in periodically forced nonlinear systems: The harmonic balance approach. IEEE Trans Circuits Syst I 1994;41:315–20. Stensby JL, Harb B. Computing the half-plane pull-in range in a second-order phase locked loop. IEE Electron Lett 1995;31: 845–6. Watada K, Endo T, Seishi H. Shilnikov orbits in an autonomous third-order chaotic phase-locked loop. IEEE Trans Circuits Systems––I: Fundamental Theory Appl 1998;45(9):1998. Endo T, Chua LO. Chaos from phase-locked loops. IEEE Trans Circuits Systems 1988;35:987–1003. Endo T, Chua LO. Synchronization of chaos in phase-locked loops. Part II––non-Hamiltonian case. IEEE Trans Circuits Systems 1989;36:255–63. Nayfeh AH. Introduction to Perturbation Techniques. New York: John Wiley; 1981.
Chaos and bifurcation in a third-order phase locked loop Bassam A. Harb a, Ahmad. M. Harb b
b,*
a Electrical Engineering Department, Yarmouk University, Irbid 21110, Jordan Electrical Engineering Department, Jordan University of Science & Technology, P.O. Box 3030, Irbid 21110, Jordan
Accepted 24 April 2003
Abstract The chaos induced in a new type of phase locked loop (PLL) having a second-order loop filter is investigated. The system under consideration is modeled as a third-order autonomous system with sinusoidal phase detector characteristics. The modern of nonlinear theory such as bifurcation and chaos is applied to a third-order of PLL. A method is developed to quantitatively study the type of bifurcations that occur in this type of PLLÕs. The study showed that PLL experiencing a Hopf bifurcation point as well as chaotic behaviour. The method of multiple scales is used to find the normal form near the Hopf bifurcation point. The point is found to be supercritical one. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction Phase locked loops (PLLs) are an important part of modern electronic communication and control systems [1,2]. They play a major role in systems where it is necessary to estimate the phase of a received signal. Also, they are used extensively in applications requiring the synthesis of highly stable sinusoidal signals. In such systems, it is often necessary to estimate the instantaneous phase of a received signal which has been contaminated by random noise/or other type of interference. Often, one is supplied with a sinusoidal reference voltage, and it is necessary to produce a second sinusoidal voltage having a frequency which is an integer multiple of the referenceÕs frequency. PLLs offer a practical method of performing these tasks. The basic configuration of a PLL is shown in Fig. 1. It consists of three building blocks; a phase detector (PD), a time-invariant loop filter and a voltage controlled oscillator (VCO) [3]. The PLL is considered to be phase locked when the loopÕs phase error is constant /0 and the loop is in stable equilibrium state [4,7]. This mean that small perturbation from /0 , in the phase error will eventually dampen out as a result of the closed loop dynamic. Equivalently, /0 is an asymptotically stable solution of the autonomous, nonlinear differential equation describing the closed loop phase error. This differential equation is given in Section 2. The PLL can be false locked [4,7,9]. When this occurs a periodic orbits exist for a certain loop parameter. The behavior of these periodic orbits is of a great interest. It is known that the PLL under consideration exhibits a chaotic behavior preceeded by a series of period doubling for the periodic orbit associated with false lock state of the PLL. Chaotic properties of the PLLÕs have been frequently studied in recent years [10–12]. Many methods were used to prove the existence of chaos in different dynamic systems. The Melnikov method is the most popular one by far [10–12]. Also, the harmonic balance technique [8] predicts the occurrence of chaotic behavior and several bifurcation phenomenon in nonlinear circuits. Watada et al. [10] use piecewise linear analysis to investigate the Shilnikov homoclinic bifurcation in a third-order phase locked loop with
*
Corresponding author. Tel.: +962-2-720-1000; fax: +962-2-709-5018. E-mail address: [email protected] (A.M. Harb).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00197-8
668
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672
Fig. 1. Block diagram of a phase-locked loop with second-order loop filter.
piecewise linear PD. They demonstrate the role of the homoclinic orbit in the global bifurcation of attractors. In this paper, a third order PLL with sinusoidal PD will be considered. The modern nonlinear theory [5,6], such as bifurcation and chaos theory, and the method of multiple scales [13], is used to analyze the dynamics of PLL. To investigate the type of bifurcations that occur in PLL, we found the normal form near the Hopf bifurcation point by using the method of multiple scales. A bifurcation diagram will be developed which shows the parameter (K0 ) for which a limit cycle (false lock state) exists. Moreover, the limit cycle will be computed numerically. It is shown that this limit cycle undergoes period doubling bifurcation which at the end drive the PLL to chaotic behavior. The parameter ranges for which chaos exist is determined. The outline of our paper is as follows: In Section 2, the classical model for the third-order loop will be derived. Also, the nonlinear differential equation that describes the closed loop phase error will be derived. While in Section 3, the equilibrium solutions and their stability will be discussed. In Section 4, the dynamical solution will be presented. Finally, in Section 5, some conclusions have been withdrew.
2. Dynamical model The basic configuration of a PLL is shown in Fig. 1. It consists of three building blocks; a PD, a time-invariant loop filter and a VCO. The VCOÕs instantaneous frequency is given by dhv ¼ x0 þ kv eðtÞ rad=s; dt
ð1Þ
where x0 is called the VCO quiescent or center frequency and kv is the VCO gain which has units of (rad/s V). The output of the PD is given by xðtÞ ¼ k1 ½sinðhi hv Þ þ sinðhi þ hv Þ:
ð2Þ
The term sinðhi þ hv Þ contains higher fundamental frequency components which are eliminated by the loop filter. Then, the output of the loop filter becomes Z t eðtÞ ¼ e0 ðtÞ þ xðt uÞf ðuÞdu; t P 0; ð3Þ 0
where f ðtÞ is the impulse response of the filter, and e0 ðtÞ is the filterÕs zero-input response which depends only on the initial conditions existing in the filter at t ¼ 0. If the filter is stable, e0 ðtÞ ! 0 as t ! 1 for any set of initial conditions. The loop filter which will be considered in this paper has the form F ðsÞ ¼
1 þ sz1 s 1 þ sz2 s ¼ F1 ðsÞF2 ðsÞ; 1 þ sp1 s 1 þ sp2 s
where F1 ðsÞ and F2 ðsÞ are two cascade lag-lead filters. Eqs. (1)–(3) can be combined to obtain Z t Z t dhv ¼ x 0 þ kv xðuÞf ðt uÞdu ¼ x0 þ k1 kv f ðt uÞ sinðhi ðuÞ hv ðuÞÞdu: dt 0 0
ð4Þ
ð5Þ
Define the closed loop phase error as / hi hv ;
ð6Þ
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672
669
and the closed loop gain as ð7Þ
k ¼ k1 kv : The results which follow are simplified by defining h1 ¼ hi x0 t;
ð8Þ
h2 ¼ hv x0 t:
ð9Þ
These quantities can be used with Eqs. (5)–(7) to write Z t d/ dh1 ¼ Ak f ðt uÞ sinð/ðuÞÞdu for t P 0: dt dt 0
ð10Þ
The differential equation that describes the closed loop phase error in the PLL is given by d/ dh 1 þ sz1 d=dt 1 þ sz2 d=dt ¼ k sinð/Þ; dt dt 1 þ sp1 d=dt 1 þ sp2 d=dt
ð11Þ
where sp1 , sp2 , sz1 and sz2 are the loop filters time constants. After simplications, the above equation becomes d3 / d2 / þ 2 dt3 dt ¼
sp1 þ sp2 ksz1 sz2 cos / þ sp1 sp2 sp1 sp2
þ
d/ dt
1 kðsz1 þ sz2 Þ cos / þ sp1 sp2 sp1 sp2
d2 h1 sp1 þ sp2 d2 h1 1 dh1 : þ þ sp1 sp2 dt dt2 sp1 sp2 dt2
ksz1 sz2 sin / sp1 sp2
d/ dt
2 þ
k sin / sp1 sp2 ð12Þ
If the input frequency is constant, then h1 ðtÞ ¼ ðxi x0 Þt þ h0 ;
ð13Þ
and by normalizing the time variable using t0 ¼ ðk=sp1 sp2 Þ1=3 t, Eq. (12) becomes hv þ a/€ þ b cosð/Þ/€ þ c/_ þ d cosð/Þ/_ e sinð/Þ/_ 2 þ sinð/Þ ¼ d;
ð14Þ
where ¼ d=dt0 , s ¼ ðk=sp1 sp2 Þ1=3 , a ¼ ðsp1 þ sp2 Þs2 =k, b ¼ sz1 sz2 s2 , c ¼ s=k, d ¼ ðsz1 þ sz2 Þs, e ¼ sz1 sz2 s2 , d ¼ xos =k and xos ¼ xi x0 . The above equation can be written as: /_ ¼ x;
ð15aÞ
x_ ¼ y;
ð15bÞ
y_ ¼ ay b cosð/Þy cx d cosð/Þx þ e sinð/Þx2 sinð/Þ þ d:
ð15cÞ
3. Equilibrium solutions and their stability v The equilibrium solution of the system of Eqs. (15a)–(15c) corresponding to /_ ¼ /€ ¼ / ¼ 0. Setting the right hand sides of Eqs. (15a)–(15c) equal to zero, we end up with a nonlinear algebraic equation. The solution of this equation as a function of one of the control parameters are defined by using a continuation scheme. The stability of an equilibrium solution depends on the eigenvalues of the Jacobian matrix of Eqs. (15a)–(15c) evaluated at the equilibrium point. In this paper we write our own program for calculating the fixed points and their bifurcations rather than use an available bifurcation software package, such as BIFOR2 and AUTO94. To this end, we follow Watada et al. [10] and consider the case of the sinusoidal PD. In Fig. 2, we show the bifurcation diagram, which is the variation of the control parameter K with the state variable /_ . The solid line denotes stable nodes, while the dashed line denotes unstable foci. There is only one Hopf bifurcation point H at K0 ¼ 7299:01.
670
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672 35 30 25
x
20 15 10
H
5 0
ko=7299.01
2000
4000
6000
8000
10000
12000
14000
k Fig. 2. Bifurcation diagram (variation of state variable versus the control parameter K).
Using the normal form near the Hopf bifurcation point H , we find that H is a supercritical point. So, as K was increased above 7299.01, a sequence of deformed (asymmetric) periodic solutions was observed, leading to chaos.
4. Dynamical solutions A representative bifurcation diagram for the case of sinusoidal PD is shown in Fig. 2. In this section we discuss dynamical solutions. Near the Hopf bifurcation point H , Eqs. (15a)–(15c) possess small limit-cycle solutions. The stable limit-cycles grow in size and break their symmetry and finally culminating in chaos. We used the method of multiple scales encoded in MATHEMATICA [13] to reduce Eqs. (15a)–(15c) into their normal form near the Hopf Bifurcation point H . The normal form is then used to determine the amplitudes of the small limit cycles and their stability. The normal form near Hopf Bifurcation is given by a_ ¼ b1 a þ b2 a3 ;
ð16Þ
ah_ ¼ b3 a þ b4 a3 ;
ð17Þ
where bi are functions of the system parameters. Limit cycles correspond to a_ . Then, it follows from Eq. (16) that the amplitudes of limit-cycles are given by sffiffiffiffiffi b1 : a¼ b2
ð18Þ
When b2 < 0, the limit-cycles is stable and hence the Hopf bifurcation is supercritical. When b2 > 0, the limit-cycle is unstable and hence the Hopf bifurcation is subcritical. In the case of sinusoidal PD, we found that b2 < 0, hence the Hopf bifurcation point H is supercritical point. As K increased above the supercritical Hopf bifurcation H at K ¼ 7299:01, a small and stable limit-cycle is born, as shown in Fig. 3 at K ¼ 7299:01. As K increased further, the limitcycle grows, deforms, and then undergoes to chaos at K ¼ 85299:01 as shown in Fig. 4.
5. Conclusions The modern nonlinear theory such as bifurcation and chaos theory has been used to study the stability of a third order PLL. The study shows that this kind of PLL experienced chaos through Hopf bifurcation. In other words, the PLL is unlocked after the Hopf bifurcation point H . Because for the equilibrium solution the PLL will be in locked situation. The method of multiple secales, perturbation method, has been used to find the normal form at the vicinity of the Hopf bifurcation point. The point is found to be supercritical one. That means, a small periodic solutions, limit
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672
Fig. 3. Time responses and state-plane representation.
671
672
B.A. Harb, A.M. Harb / Chaos, Solitons and Fractals 19 (2004) 667–672 (a1)
1.5
1
1
0.5 X3
X3
0.5 0
0
-0.5
-0.5
-1
-1
-1.5 0
(a2)
1.5
200
400
600
800 1000 1200 1400 1600 1800 2000 Time
-1.5 -2.5
-2
-1.5
-1
-0.5
0 X2
0.5
1
1.5
2
2.5
Fig. 4. The chaotic time response and state-plane for K ¼ 85299:01.
cycles, are borne at the Hopf bifurcation point. As the control parameter K increased, the limit cycles deformed and at the end culminating to chaos.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Viterbi AJ. Principle of Coherent Communications. New York: McGraw Hill; 1966. Blanchard A. Phase-Locked Loops. New York: John Wiley; 1976. Gardner FM. Phaselock Techniques. New York: John Wiley; 1979. Stensby JL. Saddle node bifurcation at a nonhyperbolic limit cycle in a phase locked loop. J Franklin Inst 1993;330(5):775–86. Perko L. Differential Equation and Dynamical Systems. New York: Springer Verlag; 1991. Hale J, Kocak H. Dynamics and Bifurcations. New York: Springer Verlag; 1991. Harb B, Stensby JL. The half-plane pull-in range of a second-order phase locked loop. J Franklin Inst 1996;333(B)(2):191–9. Piccardi C. Bifurcations of limit cycles in periodically forced nonlinear systems: The harmonic balance approach. IEEE Trans Circuits Syst I 1994;41:315–20. Stensby JL, Harb B. Computing the half-plane pull-in range in a second-order phase locked loop. IEE Electron Lett 1995;31: 845–6. Watada K, Endo T, Seishi H. Shilnikov orbits in an autonomous third-order chaotic phase-locked loop. IEEE Trans Circuits Systems––I: Fundamental Theory Appl 1998;45(9):1998. Endo T, Chua LO. Chaos from phase-locked loops. IEEE Trans Circuits Systems 1988;35:987–1003. Endo T, Chua LO. Synchronization of chaos in phase-locked loops. Part II––non-Hamiltonian case. IEEE Trans Circuits Systems 1989;36:255–63. Nayfeh AH. Introduction to Perturbation Techniques. New York: John Wiley; 1981.