A Mixed TimeâFrequency-Domain Approach for the ... - IEEE Xplore
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 9, SEPTEMBER 2005
525
A Mixed Time–Frequency-Domain Approach for the Analysis of a Hysteretic Oscillator Michele Bonnin, Marco Gilli, Fellow, IEEE, and Pier Paolo Civalleri, Life Fellow, IEEE
Abstract—The global dynamic behavior of a hysteretic oscillator is investigated. It is shown that the most significant bifurcation phenomena can be accurately detected through the joint application of two frequency-domain techniques, i.e., describing function and harmonic balance, and of a suitable time-domain method for computing limit-cycle Floquet’s multipliers. The proposed approach can be effectively applied for investigating nonlinear oscillators and circuits that do not admit of a simple Lur’e representation and that exhibit a complex dynamic behavior. Index Terms—Bifurcation, describing functions, harmonic balance (HB), hysteresis, nonlinear circuits.
I. INTRODUCTION
F
REQUENCY-DOMAIN techniques, like harmonic balance (HB) and describing function, are classical methods for studying and designing electronic oscillators and nonlinear microwave circuits [1]–[3]. In most applications, these techniques have been used for determining the steady-state behavior of nonlinear circuits that exhibit a single periodic attractor. On the other hand, the global dynamics of nonlinear networks and systems is usually investigated through time-domain techniques that require introducing rather complex and sophisticated concepts [4]. Recently, some HB-based techniques have been proposed for investigating bifurcation processes in nonlinear circuits that present several attractors [5]–[12]. In [13], an HB-based approach was presented for computing limit-cycle Floquet’s multipliers. The proposed method is applicable to Lur’e type oscillators, but it requires computing large matrix determinants that in some cases may affect the accuracy of some Floquet’s multipliers. In [14] and [15], the authors have still considered systems that admits of a Lur’e representation. They have shown that the describing function technique (i.e., HB with a single harmonic) is able to predict the occurrence of chaotic behavior and several bifurcation phenomena. Their approach presents the advantages of providing a simple and qualitative description of the system dynamics that can be effectively exploited for design purposes. In [12], the authors have investigated bifurcation processes occurring in the intermittence route to chaos; they have shown that Floquet’s multipliers can be effectively computed in the time domain by exploiting limit-cycle HB solutions. Manuscript received November 9, 2004. This work was supported in part by Ministero dell’Istruzione, dell’Università e della Ricerca under the FIRB Project RBNE012NSW. This paper was recommended by Associate Editor W. K.-S. Tang. The authors are with the Department of Electronics, Politecnico di Torino, I-10129 Turin, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2005.850518
Fig. 1. Hysteretic oscillator. The (i ; v ) diode characteristic is assumed to be i = I s(exp(v =V ) 1) with I = 10 A and V = 25:9 mV.
0
In this paper, we will examine a third-order hysteretic oscillator that cannot be described in the classical Lur’e form. We will show that its dynamics (including the most significant bifurcation processes) can be effectively investigated through a mixed time–frequency-domain approach. The proposed method exploits the describing function technique for estimating the whole set of periodic invariant limit sets and the HB technique for providing an accurate characterization of each limit cycle. Then, similarly to [12], the Floquet’s multipliers are computed (and the related bifurcation processes are identified) by exploiting the HB results and the time-domain numerical algorithm described in [16]. We remark that the proposed approach is more accurate than that developed in [13] because it does not require computing large matrix determinants. In addition, it allows one to detect some significant bifurcations that cannot be dealt with by the sole application of the describing function technique [14], [15]. II. MATHEMATICAL MODEL OF THE HYSTERETIC OSCILLATOR We consider the third-order hysteretic oscillator introduced in [17] and shown in Fig. 1. It is described by the following system of normalized state equations (see [17] for details): (1) (2) (3) where the dot denotes the derivative with respect to the normalized time , and . , and are expressed in term of the cirThe coefficients , and , i.e., cuit parameters , and .
1057-7130/$20.00 © 2005 IEEE
526
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 9, SEPTEMBER 2005
According to [17], is defined as the function, obtained by finding for each the unique solution of the following transcendental equation:
where denotes the bias term, is the amplitude of the first order harmonic, and is the angular frequency. The following expressions can be readily computed:
(4) (9) We will consider and as bifurcation parameters, whereas we assume the following values for the other circuit nF; k , elements: A, and mV. As a preliminary step, we show that the above set of equations can be reduced to a scalar Lur’e-like system. The first two equations (1) and (2) allow one to derive a linear relationship and between (5) where and
denotes the first-order differential operator
(10) with
(11) By exploiting (11), it is also derived that expression can be considered as a function (hereafter named ) of the describing function parameters , and and of time
(6) By substituting (5) in (3), we obtain the following Lur’e-like model in terms of the scalar variable (12) (7)
The following first-harmonic approximation of function holds:
III. ANALYSIS OF THE GLOBAL DYNAMIC BEHAVIOR As pointed out above, we will show that the most significant dynamic properties of the hysteretic oscillator under study can be revealed through the joint application of two frequency-domain techniques (describing function and harmonic balance) and of a suitable time-domain method for computing limit-cycle Floquet’s multipliers. We assume that there exist circuit parameters for which the oscillator does not present a complex behavior, i.e., the only invariant limit sets are limit cycles and equilibrium points. We also assume that each periodic orbit can be detected by the describing function technique, i.e., that there exists a one-to-one correspondence between the number of periodic orbits and the solution provided by the application of the describing function technique. The rigorous conditions under which such an assumption holds are difficult to check [3]; however, for most oscillators, the describing function solutions can be considered reliable if the normalized distortion index, defined in [14] and [15], is sufficiently small. The occurrence of complex nonperiodic attractors can be subsequently revealed through the application of the second and third steps of the proposed spectral technique, i.e., by analyzing limit cycle bifurcations. According to the describing function method, each periodic is approximately represented by a signal of period bias term and a single harmonic, i.e., (8)
(13) where
(14) and the coefficients and (12), as
, and
are defined, according to (11)
(15) Even if the explicit expression of function is not known [because the latter is found as the solution of the transcendental equation (4)], the integrals above admit of an accurate analytical approximation. In order to show this, we firstly note that, after
BONNIN et al.: MIXED TIME–FREQUENCY-DOMAIN APPROACH FOR THE ANALYSIS OF A HYSTERETIC OSCILLATOR
some algebraic manipulations, the following fundamental inte, and grals can be derived (for any set of real coefficients ):
527
Since, according to (9) and (11), the following expression : stands for
(23) (16)
An analytical close form for integrals (22) can be readily derived by replacing , and in (16)–(18) with the expressions shown below:
(17)
(18) where and denote the modified Bessel function of the first kind of order zero and one, respectively. Then, by introducing the second side of the transcendental equation (4) as the argument of the integrals (14), by substituting with its harmonic approximation (13), and by exploiting , we (16)–(18) with obtain the following three approximate equations involving the , and coefficients
(24) By substituting into (7) the first-order harmonic approximations of and , derived in (13), (21), and (10), and by adding the three equations derived in (19), we obtain a nondifferential system of six equa, and (hereafter tions with six unknowns named the describing function system)
(25) where
(19) where (20) By following a similar procedure, the first harmonic ap[the other nonlinear function proximation of appearing in the scalar model (7)] can be found. We have (21) where
and are given by (11), is given by (20), and the coefficients are obtained by substituting (24) into (16)–(18). The describing function system can be solved in a very efficient way by exploiting standard numerical methods. Through the describing function technique, we have identified the most significant limit cycles in the various regions of parameters. Then the main features of each limit cycle have been investigated through the HB technique [18] by exploiting as input the single harmonic approximation provided by the describing function method. As we have already pointed out, some spectral techniques have been developed for investigating limit-cycle stability properties and their most significant bifurcation phenomena [13]–[15]. In this paper, we show that an accurate investigation of limit-cycle stability and bifurcations can be carried out through a mixed time–frequency-domain approach, i.e., by linearizing system (1)–(3) along the solution predicted by the HB technique and then by computing the Floquet’s multipliers of the corresponding variational equation. the harmonic balBy denoting with ance solution and with a generic perturbation of , the variational equation is readily derived (26)
(22)
(27)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 9, SEPTEMBER 2005
Fig. 2. Bifurcation process of the symmetric limit cycle in the hysteretic oscillator and related bifurcation curves. The explanation of the curves and of the symbols is given in the text.
Fig. 3. Bifurcation processes of the origin in the hysteretic oscillator and related bifurcation curves. The explanation of the curves and of the symbols is given in the text.
where can be computed as denotes the with respect to its argument, whose derivative of function expression can be obtained from (4) as (28) is known, the Floquet’s multipliers can be Once matrix effectively computed by using the time-domain numerical algorithm described in [16]. The application of the mixed time–frequency-domain technique described above allows one to identify and characterize the main dynamic features of the hysteretic oscillator under study. The system dynamics can be described by resorting to the bifurcation diagrams shown in Figs. 2 and 3. They have been obtained via a continuation method by exploiting conductances and as bifurcation parameters. For small values of the , the system exhibits only two invariant limit conductance sets: the origin, which is an unstable equilibrium point and a stable symmetric limit cycle. By increasing , the latter undergoes a series of bifurcations that are schematically shown in the upper part of Fig. 2. First, it becomes unstable, through a pitchin Fig. 2) from which a pair fork bifurcation (denoted by of asymmetric stable limit cycles originates. Then each asymmetric cycle exhibits a period-doubling bifurcation (denoted by PD in Fig. 2) giving rise to a Feigenbaum cascade and eventually to a chaotic attractor. The corresponding bifurcation curves, obtained by computing the Floquet’s multipliers, are shown in the lower part of Fig. 2. In the upper part of Fig. 3, the origin bifurcation process is outlined. First, it exhibits a pitchfork bi) that gives birth to a pair of unstable furcation (denoted by equilibrium points. By increasing , each of these points undergoes a couple of supercritical Hopf bifurcation (denoted by and ) that makes it stable between and . The two Hopf bifurcations give rise to a pair of asymmetric limit cycles and , that disappear through fold bifurcations (denoted by respectively). The corresponding bifurcation curves are reported and curves in the lower left part of Fig. 3, where the cannot be distinguished, because they occur for a very small in-
Fig. 4. Top: Most significant Floquet’s multiplier versus G , for the symmetric limit cycle that undergoes the pitchfork bifurcation PF (G = 105 s). Bottom: Most significant Floquet’s multiplier versus G for the two asymmetric limit cycles, that exhibit the period doubling bifurcation PD (G = 38 s).
terval of the bifurcation parameter . Such curves are shown in detail in the lower right-hand side of Fig. 3. In Fig. 4, we have reported the most significant Floquet’s multiplier for the symmetric and asymmetric limit cycles, shown in the upper part of Fig. 2. It is seen that the symmetric limit cycle becomes unstable (and the corresponding Floafter the pitchfork bifurcation quet’s multiplier equals 1), whereas the asymmetric limit cycles become unstable after the period doubling bifurcation PD (and ). the related Floquet’s multiplier equals IV. CONCLUSION We have shown that a hysteretic oscillator that does not have a classical Lur’e representation can be studied through the application of a mixed time–frequency-domain approach, based on
BONNIN et al.: MIXED TIME–FREQUENCY-DOMAIN APPROACH FOR THE ANALYSIS OF A HYSTERETIC OSCILLATOR
the describing function and the HB technique, and on a suitable time-domain algorithm for computing the Floquet’s multipliers. We make the following remarks regarding the main characteristics of our approach in comparison with previous works on spectral techniques [13], [15] and on hysteretic oscillators [17]. Remark 1: The proposed technique applies to a hysteretic oscillator that cannot be described as a classical Lur’e system. Then it represents a sort of extension of the general results presented in [13] and [15]. Remark 2: In comparison to the spectral techniques described in [13] and [15], the proposed method is more accurate, because the state is represented through a suitably large number of harmonics (and not via a single harmonic, as in [15]) and the time-domain algorithm allows one to avoid the computation of complex and possibly inaccurate large matrix determinants (as is required in [13]). Remark 3: The proposed spectral technique can easily detect the existence of unstable limit cycles that play an important role for understanding bifurcation phenomena. Hence, it represents an effective alternative to time-domain methods for investigating complex dynamics in hysteretic oscillators [17].
REFERENCES [1] K. S. Kundert and A. Sangiovanni-Vincentelli, “Simulation of nonlinear circuits in the frequency domain,” IEEE Trans. Comput.-Aided Des, Integr. Circuits Syst., vol. 5, no. 4, pp. 521–535, Oct. 1986. [2] A. Ushida, T. Adachi, and L. O. Chua, “Steady-state analysis of nonlinear circuits, based on hybrid methods,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 39, no. 8, pp. 649–661, Aug. 1992. [3] A. I. Mees, Dynamics of Feedback Systems. New York: Wiley, 1981. [4] Y. A. Kuznetov, Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995.
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[5] C. Piccardi, “Bifurcations of limit cycles in periodically forced nonlinear systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 41, no. 4, pp. 315–320, Apr. 1994. [6] , “Bifurcation analysis via harmonic balance in periodic systems with feedback structure,” Int. J. Contr., vol. 62, pp. 1507–1515, 1995. , “Harmonic balance analysis of codimension-2 bifurcations in pe[7] riodic systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 43, no. 12, pp. 1015–1018, Dec. 1996. [8] V. Rizzoli and A. Neri, “State of the art and present trends in nonlinear microwave techniques,” IEEE Trans. Microw. Theory Tech., no. 2, pp. 343–365, Feb. 1988. [9] V. Rizzoli, A. Neri, and D. Masotti, “Local stability analysis of microwave oscillators based on Nyquist’s theorem,” IEEE Microw. Guided Wave Lett., vol. 7, no. 10, pp. 341–343, Oct. 1998. [10] A. Suarez, J. Morales, and R. Quéré, “Syncronization analysis of autonomous microwave circuits using new global-stability analysis tools,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 494–504, May 1998. [11] D. W. Berns, J. L. Moiola, and G. Chen, “Predicting period-doubling bifurcations and multiple oscillations in nonlinear time-delayed feedback systems,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 7, pp. 759–763, Jul. 1998. [12] S. Sancho, A. Suárez, and P. S. Román, “Floquet analysis of an intermittence route to chaos through a pitchfork bifurcation,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 48, no. 3, pp. 374–376, Mar. 2001. [13] F. Bonani and M. Gilli, “Analysis of stability and bifurcations of limit cycles in Chua’s circuit through the harmonic balance approach,” IEEE Trans. Circuits Syst. I, Fundam Theory Appl., vol. 46, no. 8, pp. 881–890, Aug. 1999. [14] R. Genesio and A. Tesi, “A harmonic balance approach for chaos prediction: Chua’s circuit,” Int. J. Bifurc. Chaos, vol. 2, no. 1, pp. 61–79, 1992. [15] M. Basso, R. Genesio, and A. Tesi, “A frequency method for predicting limit cycle bifurcations,” Nonlinear Dynam., vol. 13, pp. 339–360, 1997. [16] M. Farkaj, Periodic Motions. Berlin, Germany: Springer-Verlag, 1994, pp. 58–59. [17] F. Bizzarri and M. Storace, “Coexistence of attractors in an oscillator based on hysteresis,” in Proc. ISCAS, Scottsdale, AZ, May 2002. paper 1713. [18] K. S. Kundert, A. Sangiovanni-Vincentelli, and J. K. White, Steady-State Methods for Simulating Analog and Microwave Circuits. Boston, MA: Kluwer, 1990.
525
A Mixed Time–Frequency-Domain Approach for the Analysis of a Hysteretic Oscillator Michele Bonnin, Marco Gilli, Fellow, IEEE, and Pier Paolo Civalleri, Life Fellow, IEEE
Abstract—The global dynamic behavior of a hysteretic oscillator is investigated. It is shown that the most significant bifurcation phenomena can be accurately detected through the joint application of two frequency-domain techniques, i.e., describing function and harmonic balance, and of a suitable time-domain method for computing limit-cycle Floquet’s multipliers. The proposed approach can be effectively applied for investigating nonlinear oscillators and circuits that do not admit of a simple Lur’e representation and that exhibit a complex dynamic behavior. Index Terms—Bifurcation, describing functions, harmonic balance (HB), hysteresis, nonlinear circuits.
I. INTRODUCTION
F
REQUENCY-DOMAIN techniques, like harmonic balance (HB) and describing function, are classical methods for studying and designing electronic oscillators and nonlinear microwave circuits [1]–[3]. In most applications, these techniques have been used for determining the steady-state behavior of nonlinear circuits that exhibit a single periodic attractor. On the other hand, the global dynamics of nonlinear networks and systems is usually investigated through time-domain techniques that require introducing rather complex and sophisticated concepts [4]. Recently, some HB-based techniques have been proposed for investigating bifurcation processes in nonlinear circuits that present several attractors [5]–[12]. In [13], an HB-based approach was presented for computing limit-cycle Floquet’s multipliers. The proposed method is applicable to Lur’e type oscillators, but it requires computing large matrix determinants that in some cases may affect the accuracy of some Floquet’s multipliers. In [14] and [15], the authors have still considered systems that admits of a Lur’e representation. They have shown that the describing function technique (i.e., HB with a single harmonic) is able to predict the occurrence of chaotic behavior and several bifurcation phenomena. Their approach presents the advantages of providing a simple and qualitative description of the system dynamics that can be effectively exploited for design purposes. In [12], the authors have investigated bifurcation processes occurring in the intermittence route to chaos; they have shown that Floquet’s multipliers can be effectively computed in the time domain by exploiting limit-cycle HB solutions. Manuscript received November 9, 2004. This work was supported in part by Ministero dell’Istruzione, dell’Università e della Ricerca under the FIRB Project RBNE012NSW. This paper was recommended by Associate Editor W. K.-S. Tang. The authors are with the Department of Electronics, Politecnico di Torino, I-10129 Turin, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2005.850518
Fig. 1. Hysteretic oscillator. The (i ; v ) diode characteristic is assumed to be i = I s(exp(v =V ) 1) with I = 10 A and V = 25:9 mV.
0
In this paper, we will examine a third-order hysteretic oscillator that cannot be described in the classical Lur’e form. We will show that its dynamics (including the most significant bifurcation processes) can be effectively investigated through a mixed time–frequency-domain approach. The proposed method exploits the describing function technique for estimating the whole set of periodic invariant limit sets and the HB technique for providing an accurate characterization of each limit cycle. Then, similarly to [12], the Floquet’s multipliers are computed (and the related bifurcation processes are identified) by exploiting the HB results and the time-domain numerical algorithm described in [16]. We remark that the proposed approach is more accurate than that developed in [13] because it does not require computing large matrix determinants. In addition, it allows one to detect some significant bifurcations that cannot be dealt with by the sole application of the describing function technique [14], [15]. II. MATHEMATICAL MODEL OF THE HYSTERETIC OSCILLATOR We consider the third-order hysteretic oscillator introduced in [17] and shown in Fig. 1. It is described by the following system of normalized state equations (see [17] for details): (1) (2) (3) where the dot denotes the derivative with respect to the normalized time , and . , and are expressed in term of the cirThe coefficients , and , i.e., cuit parameters , and .
1057-7130/$20.00 © 2005 IEEE
526
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 9, SEPTEMBER 2005
According to [17], is defined as the function, obtained by finding for each the unique solution of the following transcendental equation:
where denotes the bias term, is the amplitude of the first order harmonic, and is the angular frequency. The following expressions can be readily computed:
(4) (9) We will consider and as bifurcation parameters, whereas we assume the following values for the other circuit nF; k , elements: A, and mV. As a preliminary step, we show that the above set of equations can be reduced to a scalar Lur’e-like system. The first two equations (1) and (2) allow one to derive a linear relationship and between (5) where and
denotes the first-order differential operator
(10) with
(11) By exploiting (11), it is also derived that expression can be considered as a function (hereafter named ) of the describing function parameters , and and of time
(6) By substituting (5) in (3), we obtain the following Lur’e-like model in terms of the scalar variable (12) (7)
The following first-harmonic approximation of function holds:
III. ANALYSIS OF THE GLOBAL DYNAMIC BEHAVIOR As pointed out above, we will show that the most significant dynamic properties of the hysteretic oscillator under study can be revealed through the joint application of two frequency-domain techniques (describing function and harmonic balance) and of a suitable time-domain method for computing limit-cycle Floquet’s multipliers. We assume that there exist circuit parameters for which the oscillator does not present a complex behavior, i.e., the only invariant limit sets are limit cycles and equilibrium points. We also assume that each periodic orbit can be detected by the describing function technique, i.e., that there exists a one-to-one correspondence between the number of periodic orbits and the solution provided by the application of the describing function technique. The rigorous conditions under which such an assumption holds are difficult to check [3]; however, for most oscillators, the describing function solutions can be considered reliable if the normalized distortion index, defined in [14] and [15], is sufficiently small. The occurrence of complex nonperiodic attractors can be subsequently revealed through the application of the second and third steps of the proposed spectral technique, i.e., by analyzing limit cycle bifurcations. According to the describing function method, each periodic is approximately represented by a signal of period bias term and a single harmonic, i.e., (8)
(13) where
(14) and the coefficients and (12), as
, and
are defined, according to (11)
(15) Even if the explicit expression of function is not known [because the latter is found as the solution of the transcendental equation (4)], the integrals above admit of an accurate analytical approximation. In order to show this, we firstly note that, after
BONNIN et al.: MIXED TIME–FREQUENCY-DOMAIN APPROACH FOR THE ANALYSIS OF A HYSTERETIC OSCILLATOR
some algebraic manipulations, the following fundamental inte, and grals can be derived (for any set of real coefficients ):
527
Since, according to (9) and (11), the following expression : stands for
(23) (16)
An analytical close form for integrals (22) can be readily derived by replacing , and in (16)–(18) with the expressions shown below:
(17)
(18) where and denote the modified Bessel function of the first kind of order zero and one, respectively. Then, by introducing the second side of the transcendental equation (4) as the argument of the integrals (14), by substituting with its harmonic approximation (13), and by exploiting , we (16)–(18) with obtain the following three approximate equations involving the , and coefficients
(24) By substituting into (7) the first-order harmonic approximations of and , derived in (13), (21), and (10), and by adding the three equations derived in (19), we obtain a nondifferential system of six equa, and (hereafter tions with six unknowns named the describing function system)
(25) where
(19) where (20) By following a similar procedure, the first harmonic ap[the other nonlinear function proximation of appearing in the scalar model (7)] can be found. We have (21) where
and are given by (11), is given by (20), and the coefficients are obtained by substituting (24) into (16)–(18). The describing function system can be solved in a very efficient way by exploiting standard numerical methods. Through the describing function technique, we have identified the most significant limit cycles in the various regions of parameters. Then the main features of each limit cycle have been investigated through the HB technique [18] by exploiting as input the single harmonic approximation provided by the describing function method. As we have already pointed out, some spectral techniques have been developed for investigating limit-cycle stability properties and their most significant bifurcation phenomena [13]–[15]. In this paper, we show that an accurate investigation of limit-cycle stability and bifurcations can be carried out through a mixed time–frequency-domain approach, i.e., by linearizing system (1)–(3) along the solution predicted by the HB technique and then by computing the Floquet’s multipliers of the corresponding variational equation. the harmonic balBy denoting with ance solution and with a generic perturbation of , the variational equation is readily derived (26)
(22)
(27)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 9, SEPTEMBER 2005
Fig. 2. Bifurcation process of the symmetric limit cycle in the hysteretic oscillator and related bifurcation curves. The explanation of the curves and of the symbols is given in the text.
Fig. 3. Bifurcation processes of the origin in the hysteretic oscillator and related bifurcation curves. The explanation of the curves and of the symbols is given in the text.
where can be computed as denotes the with respect to its argument, whose derivative of function expression can be obtained from (4) as (28) is known, the Floquet’s multipliers can be Once matrix effectively computed by using the time-domain numerical algorithm described in [16]. The application of the mixed time–frequency-domain technique described above allows one to identify and characterize the main dynamic features of the hysteretic oscillator under study. The system dynamics can be described by resorting to the bifurcation diagrams shown in Figs. 2 and 3. They have been obtained via a continuation method by exploiting conductances and as bifurcation parameters. For small values of the , the system exhibits only two invariant limit conductance sets: the origin, which is an unstable equilibrium point and a stable symmetric limit cycle. By increasing , the latter undergoes a series of bifurcations that are schematically shown in the upper part of Fig. 2. First, it becomes unstable, through a pitchin Fig. 2) from which a pair fork bifurcation (denoted by of asymmetric stable limit cycles originates. Then each asymmetric cycle exhibits a period-doubling bifurcation (denoted by PD in Fig. 2) giving rise to a Feigenbaum cascade and eventually to a chaotic attractor. The corresponding bifurcation curves, obtained by computing the Floquet’s multipliers, are shown in the lower part of Fig. 2. In the upper part of Fig. 3, the origin bifurcation process is outlined. First, it exhibits a pitchfork bi) that gives birth to a pair of unstable furcation (denoted by equilibrium points. By increasing , each of these points undergoes a couple of supercritical Hopf bifurcation (denoted by and ) that makes it stable between and . The two Hopf bifurcations give rise to a pair of asymmetric limit cycles and , that disappear through fold bifurcations (denoted by respectively). The corresponding bifurcation curves are reported and curves in the lower left part of Fig. 3, where the cannot be distinguished, because they occur for a very small in-
Fig. 4. Top: Most significant Floquet’s multiplier versus G , for the symmetric limit cycle that undergoes the pitchfork bifurcation PF (G = 105 s). Bottom: Most significant Floquet’s multiplier versus G for the two asymmetric limit cycles, that exhibit the period doubling bifurcation PD (G = 38 s).
terval of the bifurcation parameter . Such curves are shown in detail in the lower right-hand side of Fig. 3. In Fig. 4, we have reported the most significant Floquet’s multiplier for the symmetric and asymmetric limit cycles, shown in the upper part of Fig. 2. It is seen that the symmetric limit cycle becomes unstable (and the corresponding Floafter the pitchfork bifurcation quet’s multiplier equals 1), whereas the asymmetric limit cycles become unstable after the period doubling bifurcation PD (and ). the related Floquet’s multiplier equals IV. CONCLUSION We have shown that a hysteretic oscillator that does not have a classical Lur’e representation can be studied through the application of a mixed time–frequency-domain approach, based on
BONNIN et al.: MIXED TIME–FREQUENCY-DOMAIN APPROACH FOR THE ANALYSIS OF A HYSTERETIC OSCILLATOR
the describing function and the HB technique, and on a suitable time-domain algorithm for computing the Floquet’s multipliers. We make the following remarks regarding the main characteristics of our approach in comparison with previous works on spectral techniques [13], [15] and on hysteretic oscillators [17]. Remark 1: The proposed technique applies to a hysteretic oscillator that cannot be described as a classical Lur’e system. Then it represents a sort of extension of the general results presented in [13] and [15]. Remark 2: In comparison to the spectral techniques described in [13] and [15], the proposed method is more accurate, because the state is represented through a suitably large number of harmonics (and not via a single harmonic, as in [15]) and the time-domain algorithm allows one to avoid the computation of complex and possibly inaccurate large matrix determinants (as is required in [13]). Remark 3: The proposed spectral technique can easily detect the existence of unstable limit cycles that play an important role for understanding bifurcation phenomena. Hence, it represents an effective alternative to time-domain methods for investigating complex dynamics in hysteretic oscillators [17].
REFERENCES [1] K. S. Kundert and A. Sangiovanni-Vincentelli, “Simulation of nonlinear circuits in the frequency domain,” IEEE Trans. Comput.-Aided Des, Integr. Circuits Syst., vol. 5, no. 4, pp. 521–535, Oct. 1986. [2] A. Ushida, T. Adachi, and L. O. Chua, “Steady-state analysis of nonlinear circuits, based on hybrid methods,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 39, no. 8, pp. 649–661, Aug. 1992. [3] A. I. Mees, Dynamics of Feedback Systems. New York: Wiley, 1981. [4] Y. A. Kuznetov, Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995.
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