Page  386 ï~~STABILITY/INSTABILITY OF PERIODIC SOLUTIONS AND CHAOS IN PHYSICAL MODELS OF MUSICAL INSTRUMENTS X. Rodet, Institut de Recherche et de Coordination Acoustique/Musique (IRCAM), Paris Tel: (1)44 78 48 68, Fax: (1) 42 77 29 47, e-mail: rod@ircam.fr Abstract We study multiple solutions and chaos in simple physical models and their stabilities. A path toward a unique solution can be based upon the low pass character of the linear element. Chaotic signals are shown to cover a large space ranging from a sound perceived as harmonic without noise up to an essentially noisy sound. This is a very important feature for sound synthesis since the synthesis of noise and irregularities of traditional instruments has always been difficult and unsatisfactory. Methods are proposed to find the solutions and their stabilities, and to obtain controllable systems with periodic and chaotic behaviors. 1. Introduction Real instruments and computer simulations, usually called physical models, often have more than one oscillating solution for a given setting of their parameters [Valimaki et al. 93], [Idogawa 92], [Rodet 93c]. On one hand, it can be argued that this is part of the richness of a musical instrument. On the other hand, unpredictable solutions can render the usage of a physical model rather difficult in a real-time musical performance. Only stable solutions should be considered because unstable ones do not persist. Therefore, the first goal of this paper is to study the multiple solutions of simple physical models and their stabilities. We explain that a path toward such a goal could eventually be based upon the low pass character of the linear element. As a second goal, it would be an interesting achievement to design a system which would model the usual playing behavior of an instrument but could avoid the other behaviors if requested. By the other behaviors, we mean those corresponding to stable solutions other than the usual playing solution. Finally, our last goal is to control the ratio of non sinusoidal components (noise) induced by chaos in synthetic signals, opening a new field of fascinating research and applications. Many systems which have been used as models of physical instruments for music synthesis can produce chaotic signals in some regions of their parameter space. This is related to instability since chaos often appears when all periodic solutions become unstable. The most innovative direction is probably in the area of the control of chaotic behavior of the models and thus of chaotic sounds. In the past it may have seemed that chaotic sounds could not be of any musical interest. To our great surprise we have found the contrary: these signals exhibit very interesting properties, such as a clearly perceived pitch or an intermittent type behavior. We have also found that a signal which is mathematically chaotic can be perceived in a very different way. Actually, chaos can happen within a more or less small neighborhood of a periodic trajectory. It can cover a large space from a a sound perceived as harmonic without noise, up to an essentially noisy sound. This is a very important feature for sound synthesis since the introduction in synthesis of noise and irregularities of natural instruments has always been difficult and unsatisfactory. Our models, by themselves, effectively generate a noise component that is so important for the sound quality of musical instruments. In more general terms it appears that one should not merely "build" models and deliver them to musicians. It is indispensable to go into the understanding of the models, to conceive abstractions of them and to po explanations helpful to the users. In particular, this comprehension is indispensable for elaborating the control of synthesis models, which are at the same time efficient and musically pertinent. 2. Uniqueness of solutions and low pass feedback loop We have mentioned in the introduction our interest for the design of a system which would model the usual playing behavior of an instrument but could avoid the other behaviors if requested. For the feedback systems studied here it is conceivable that a path toward such a goal could be based upon the low pass character of the linear element. In [Bergen 71], the low pass character above the oscillation frequency appears among the sufficient conditions for oscillation according to the describing function method. However it does not guarantee the uniqueness of the solution. The low pass character appears also in the formulation of the Hopf theorem in the frequency domain [Mees 79], [Rodet 93c] which gives sufficient conditions for oscillation and uniqueness of the solution. Similarly, the linear part of Chua's circuit [Chua 90], at least for some values of the parameters, can be viewed as a band pass filter, i.e. a low-pass filter above the frequency of a pair of conjugate poles, this frequency being approximately the oscillation frequency. This suggests some relation with the dominant mode of an acoustic instrument [Rodet 93b]. This idea has also been evoked in [Wawrzynek 83] but Sound Synthesis Techniques 386 1CMC Proceedings 1994

Page  387 ï~~was not fully developed. It can be shown [Rodet 93e] that there is a precise correspondence between Chua's circuit and a model of an acoustic instrument where the feedback loop is limited to the first mode(s) (the first mode could easily be replaced here by the dominant 2nd Sec together with the number of states represented by the delay line. In particular, Sharkovsky et al. [1993] have shown a remarkable period-adding property. In some regions the system has a stable limit cycle with periods 2, 3, 4, etc.... In between every two consecutive stable regions the system exhibits a chaotic behavior. We also observe the simultaneous presence of sinusoidal components and noise in the signal. By a proper affine change of variables, the invariant interval of the nonlinear function used in the circuit can be set to the interval [0,1]. For certain parameter values, the function is composed of two segments only in the invariant interval [Rodet 93a] with slopes 11 and 12. When 11 = 1.025 and 12 is varied from 1 to 2 and then to 40, we observe the progressive appearance of non sinusoidal components superimposed on the harmonic sinusoidal components (Fig. 2). Fig. 1. Feedback loop limited to the first mode(s). mode in the same way). Naturally, the system so obtained has a unique oscillatory solution in a certain range of parameter values. Fig. 1 shows a possible implementation and the control of each partial through a second order section. 3. NLO systems with chaotic behavior Many systems which have been used as models of physical instruments for music synthesis can produce chaotic signals in some regions of their parameter space. Chua's circuit [Chua 90] is a very simple circuit which exhibits a surprisingly large variety of bifurcations and chaos. As mentioned in [Madan 92] and [Rodet 92a], signals with period-I, period-2, etc..., can be obtained and lead to harmonic sounds. In the case of chaotic signals that can be obtained with some other values, the corresponding sound can be qualified as noisy. Some short-time spectra of periodic and chaotic signals from Chua's circuit are shown in [Rodet 93b]. For certain parameter values noisy signals are obtained together with some harmonic sinusoidal components. Some properties and musical uses of the Time Delayed Chua's circuit [Sharkovsky 93] have been presented in [Rodet 92c] and [Rodet 93d]. Compared to the original Chua's circuit, the time delayed version includes a delay line which is directly related to the structure and physics of many classical musical instruments. This delay allows for much better control of the circuit in terms of musical usage. The delay can also be seen as a stabilizing feedback loop applied to the original circuit. The Time Delayed Chua's Circuit exhibits a very interesting variety of sounds which we have analysed in [Rodet 93c]. This is due to the combination of the rich dynamics of the nonlinear map Fig. 2. Spectra showing presence of sinusoidal components and noise, 12 = 1.7 (top) and 30 (bottom). This property is very promising since it opens the possibility of a precise control of the proportion of noise added to the harmonic components, which is essential for a musical usage. This is the typical property that we want to extend to nonlinear functions and systems beyond the very specific two segment function of Sharkovsky 's case. We focus here on the class of systems composed of an instantaneous nonlinearity and a linear feedback ICMC Proceedings 1994 387 Sound SynthesisTechniques

Page  388 ï~~loop, which is a valid basic model of reed instruments when the mass of the reed is considered negligible. But in the case of the trumpet [Rodet 92b] for example, this assumption cannot be maintained: as we noticed in [Rodet 93c], the clarinet and trumpet would then be represented by the same model. On the contrary, in [Rodet & Steinecke 94a] we propose a minimal one mass model of the trumpet in the form of a two-loop system. This system can also exhibit chaotic behaviors. For speech production, various models have been proposed. The most common is a two mass model of vocal folds [Flanagan 72]. Using the impedance of the opening in the glottis and the driving point impedance of the vocal tract, air flow pressure and velocity in the glottis are computed and used to drive two coupled non-linear oscillators representing a vibrating vocal fold. This model usually exhibits bifurcations and chaos often related to the desynchronisation of lower order modes, e.g. of lateral and vertical motion, or to a left-right asymmetry [Steinecke & Herzel93], [Rodet & Steinecke 94b]. 4. Stability of 2-periodic solutions of a class of systems We consider the following equations which represent the same discrete system: xn= h * Y(xnT) (1), or xn = Y(h * Xn.T) (2), where h is the impulse response of a filter in the linear feedback loop, * is the convolution operator and j(.) is the nonlinear function representing for instance the influence of the reed in a basic clarinet model [Rodet 92a]. The system can be decomposed into an instantaneous nonlinearity Y(.) and a linear element including h and a delay. This defines the class of single feedback loop systems with an instantaneous nonlinearity of which we can easily determine the stability and some oscillation properties [Rodet 93c]. Note that the only restriction on the linear element is that its impulse response be stable. In particular the continuous transfer function of the linear element does not need to be a rational function and thus can include delays. Many systems can be redesigned to fall into this class. Equation (1) can also be written: m2 xn = hmT'(xn-m-T) (3) m=-m1 (3) defines a map Q:Rm2+T+1 Rm2+T+l. Since we are dealing with physical systems, it is natural to suppose that (hmj is causal, m 1 < T and finite and m. <@o. The sample n depends on the past samples n-m2-T to n+m1-T. Therefore the set of samples from n-in2 to n+T-1 depends only on the values xnm2.T tO xn+ml. themselves depending on the values xn.m2.T to xn- only which defines the map Q. Such a system may have many periodic solutions. Since they are the most important for the normal playing conditions of a musical instrument, we study here only 2-periodic solutions, i.e. fixed points of T=QÂ~Q with period length about 2T. But finding the 2-periodic solutions of Q and their stabilities in general is not easy. Therefore we are going to introduce another system. Let (sn} be a 2-periodic solution of (3). Assume now that (hn} is even symmetric. We have noticed in [Rodet 93c] that in this case the period 2 is of length 2T exactly, i.e. such that sn = =s+2T for all integer n. Let us take the Fourier series of one period of sn and of T(sn) and call S=SkJ and r'(S)=--r k(S)) respectively their Fourier coefficients which are vectors of length 2T. Then for (snj equation (3) can be written: Sk = (- l)k Hk rk(S) (4), where the term (-l)k accounts for the time delay T in (3), and H=(Hk) is the Fourier series taken after time aliasing of (hn) on an interval of length 2T and overlap-adding. From equation (4) we define a map M:B2T -- B2T where B C C is the set of Fourier series of real periodic discrete signals of period length 2T: Mk(S) = (-1)k Hk rk(S), k=l,2,...2T (5) where r'k(S) is the kth Fourier coefficient of Y(s) when s is the inverse Fourier series of S. In the time domain, we call P:R2T -- R2T the map associated with M and defined by: P(s)=Fourier-(M(S)) (6) where Fourier"1 designates the inverse Fourier series, or equivalently: P(s)=q Â~ (s-T) (7), where 1l=Fourier'l(H k) and 0 is the circular convolution. The reason for introducing the maps M and P is that they have the same fixed points as 'I'=QÂ~Q. They are simpler than 'P in the sense that they have smaller dimension and that they map exactly the signal of one time interval of length 2T into the signal of the next time interval. The dynamical system that they define is different from the original one and we call it the period-after-period (p.a.p.) system as opposed to the original one (1). Let D'Ix be the derivative of 'P at the point x e Rm2+T+1. The stability of a fixed point s of ' is given by the eigenvalues of D's. The problem is therefore to find all the fixed points of 'P or equivalently of P and M. 5. Case of a polynomial nonlinearity The case where 7' is a polynomial has inteesig properties. The first one is that it is easy to implement and can be efficiently computed on a digital processor. The control of the important characteristics of T is easy. The slope ofTÂ~ at a fixed point, assumed to be at the origin, controls the oscillating/non oscillating behavior of the system (Rodet 93b]. This slope is easily changed by a modification of the linear Sound SynthesisTechniques 388 ICMC Proceedings 1994

Page  389 ï~~term of the polynomial. Another property is that the equation (1) or (2) leads to a system of m2+T-I polynomial equations in m2+T-1 variables, and that the map P becomes a system of 2T polynomials in 2T variables. But this is also true for the map M associated with P in the Fourier domain: Note first that the Fourier series of x2, xE R2T, is the circular convolution XÂ~X which is a polynomial in the variables (Xk, k=1,2,...2T}. Then rk(X) is also a polynomial in the same variables and (5) for k=1,2,...2T defines a system of 2T polynomial equations. The portion of Y which we are interested in for easy control of oscillations is that of negative slope around the origin and greater slope further from the origin [Rodet 92a]. A drawback of the choice of a polynomial y is that it is not possible to avoid the existence of other points of functioning than the origin, which is a serious inconvenience for musical usage [Rodet 93d]. We will see in section 10 that a rational function allows us to avoid that difficulty and can be studied with the same tools which we are using for the polynomials. 6. Solutions of the p.a.p. system: a homotopy method In general, it is possible to find the solutions of a system of polynomial equations by a homotopy method [Chow 79]. For two maps Mo and M1t:RZT - R2T, a homotopy is a smooth map 4):R2T X [0,1] -* R2T, such that 4)(., 0) is M0(.) and b(., 1) is MI (.). More precisely, we are looking for the zeros of the system of polynomials Mi(s)=s-PÂ~P(s). Suppose that Mo(s) has known zeros, then under certain conditions, it is possible to follow the trajectories of the zeros of 4)(s, p) starting from the zeros of Mo(s) and going to those of MI(s) when p is varied from 0 to 1. However this is possible provided that no such zero escape to infinity for some value of p in the interval [0,1] and that 0 is a regular value of 0 on R2T X [0,1].The first requirement is easily fulfilled by making sure that the coefficients of the highest terms of 0 in sn do not vanish. The second requirement guarantees that 4-1(0) is a submanifold of dimension 1, i.e. consists only of arcs from a zero of Mo(s) to a zero of MI(s), but it may not be fulfilled for some trajectories in our case. In our problem, an interesting choice for Mo is the one for which, in the equation (3), h0 = I and hm=0 for m 0, Mo(s)=s-l'Â~T(s), i.e. there is no filtering in the feedback loop. For simplicity we assume from now on that 'Y(.) is odd symmetric. Let fl be the set of solutions of: Among the solutions of Mo(s)=s-T'V(s)=0, s E R2T, we retain only the vectors s such that Sn E Â~2 for n=l,2,..., 2T. The other solutions imply a value x E R, x 0, such that '(x)=x which we want to avoid since it corresponds to non-zero fixed points of the original system, which give no sound. Note that for the solutions which we retain sn= - Sn-T for n=T+l,T+2,...2T. When a signal sn has this odd symmetry of the period with respect to its middle, we say that the signal is odd harmonic (o.h.) because its even harmonics are null. The initial solutions for the homotopy are easy to build since they are the combinations of values from 0t. Another reason for this choice of Mo is as follows. The homotopy can simply be chosen as: 4)(s, p) = (1-p)Mo(s) + pM1 (s). (9) Therefore, when varying p from 0 to 1, we follow the evolution of the solutions according to the increase of low-pass filtering in the feedback loop, which is one of the main goals we mentioned earlier. 7. Example of the homotopy method Let us detail the application of the method on a very simple example. Let T=3 and Y(x)=x3+aix for al = 1.1. We also choose the most simple low-pass filter, h1=h1=0.25, h0=1-2h1, otherwise hm=0. More precisely, this means that when the homotopy variable p goes from 0 to 1, (h0, hl) goes from (1, 0.), i.e. no filtering, to (0.5, 0.25), progressively applying more and more low-pass filtering. The solutions of x3+aix=-x are {0, Â~4-a1-1 7 =Q. The initial solutions which we retain for the homotopy are obtained as the o.h. signals with sample values in Q, so that s - rY'(s)=0. Two solutions are equivalent if they are identical by sign inversion or time delay. Then, apart from the identically null solution which is irrelevant, there are only 5 equivalence classes of solutions. These are shown in Fig. 3 and are designated by the letters A, B, C, D, and E. Starting from one of these initial solutions and from p=0, we increase p by a small Ap, estimate the new solution by the Newton method from the previous solution and iterate this procedure until p=l. Using prefixed superscripts to index successive steps of the algorithm, let Os be the initial solution, P s the solution estimated for the value p of the homotopy variable. The solution P+APs is obtained as follows. Let z0=Ps be the initial value for the Newton method, and DO the differential of 4). We iterate: Zk1= Zk - [D4)(zk, p+Ap)]'X4)(zk, p+Ap), (10) until when I j~+1 - zk Iis sufficiently small. Then P+APs gets the last value zk+l. It is easy to derive expressions for the coefficients of the polynomials of '(u)= -u, u E R. (8) ICMC Proceedings 1994 389 Sound SynthesisTechniques

Page  390 ï~~4 and to show that these coefficients never vanish for p in [0,1] so that no solutions escape to infinity. 0.2... 0.2[ 7 ] 0z 01 0 1oV o:.1 -0.2 -0.2 0 5 0 5 Solution A Solution 6 0.2 0.2 -0 0 Q 0 -0.4 LIII -0.4 Lilili -0.2 -0.2 0 5 0 5 Solution C Solution D 0.27 -0.4 0 5 Solution E Fig. 3. Equivalence classes of initial solutions Fig. 4 shows the successive values of Ps for the equivalence class C, al = -1.35 and p from 0 to 1. Note that Is is a solution of the original system (1), which is otherwise not easy to find in general. However, in case of class A the numerical estimation procedure does not succeed to go further than p = 0.965 for ali=-1.l. The reason is that the arcs issued from two different initial solutions join together at that value of p and disappear for larger values as explained in section 9. But as long as we can follow a real solution s, we can find its stability by computing the eigenvalues of D'YS. Since 'l'=QÂ~Q and the solutions under study are o. h., it is sufficient to compute the eigenvalues of Q and to compare them to unity. Solution A for instance rapidly becomes unstable. 8. Odd harmonic (o. h.) solutions We have mentioned in section 6 that the initial solutions of the homotopy have an o. h. symmetry. We now show that this is true for the trajectories:A to E. These curves are sets of points where *>(s, p)=0. Therefore they are integral curves of the ordinary diffrentia equation: D~d(sp) + -p>sp =0 (11) where Ds and Dp denote partial derivatives and X denotes arc length. If D54s, p) is non singular, then. ds D sp (2 * =- [Ds(s,p)f1 D (sp) (12) It is straightforward to verify that when s is o. h., Dp(s,p) is o. h. also. [Dso(s,p)]-1 preserves this property, which implies that [Dso(s,p)f1Dpo(s,p) is o. h. as well. From (12) it is then clear that the curves A to E are composed of o. h. solutions. In consequence, any solution issued from the initial ones is o. h.. We have observed in [Rodet 93f] that a slight breaking of the odd symmetry of 7 can lead to the appearance of even partials, with large amplitudes if {hn) is not even symmetric. 0.6 0.4 0.2 0 -0.2 -0.4 2 4 6 Fig. 4. Successive values of Ps for tie equivalence class C of solutions, al = -1.35 and p from 0 to 1. 9. Direct computation on the example The method consists of finding the equation of manifolds on which the solutions lie and then finding the intersections of these manifolds as 1-dimensional manifolds in a parametric form so that they can easily be explored for solutions. Here we use the o. h. property of the solutions to find the solutions analytically in the previous example. The system of polynomial equations (7) is: x = h1Y(x-3) + h0(x-2) + h-1(x-1) (13) x2 = hlf(x-2) + h0(x-1) + hILIY(xO) (14) x3 = hf(x-1) + h0i(xo) + h-lY(xl) (15) Using the periodicity of the solution x, the symmetry h1=h1, and the odd symmetry of 7, we can rewrite the equations in X., x2 and x3: xl= -h0Y(Xl) - hl'Y(x2) + hlY(X3) (16) x= -hlY(Xl) - h0'Y(x2) - hlY(X3) (17) X3 =f hiY(xi) -h1'Y(X2) - hoY(x3) (18) This can be written in matrix notation x = N T(x), (19), i.e. Nlx = Y(x), (20) where: Sound SynthesisTechniques 390 ICMC Proceedings 1994

Page  391 ï~~go -g1 "I = -gl go gl -g1 g81 -gl go collapse, leading to imaginary solutions. The other solutions are easily found in the intersections of the planes with the ellipses or between the planes. Finally we get the classes A to E previously found (Fig. 7). (21) h02-h12 -andg= h02+h0h wih g0 det(N) an g det(N) Using 'Y(x) = x3+aix, equation (20) leads to x13-(go-al)xl + glx2- gix3 =0 (22) x23 + glxi - (go-al)x2 + glx3 = 0 (23) x33- g1x1+glx2-(go-al)x3=0 (24) Let a = go-al-gl. Using two equations to eliminate one variable, we get: (xl+x2)(xl2+x22- XlX2-a) = 0 (25) (xl-x3)(x12+x32- Xlx3-a) = 0 (26) The two left hand terms of (25) and (26) define a plane and a cylinder with an ellipsoid basis provided a > 0. Each ellipse can be described by a parametric equation with r = 24a'73: x 1= r cos( + /6), x2 =r cos()- n/6) (27) x = r cos(t+lr/6), x2 = r cos(tt-ir16) (28) The solutions are found on the intersection of the planes and cylinders. The intersection of the cylinders Fig. 6. Left hand term of (22) ho= 1.,.98,.965,.955. 0.02 0.01 0 -0.01 -0.02 0 A D / 1 2 3 Fig. 5. Left hand term of (22) to (24) versus o. is a curve obtained with u = - wor = tw--r3. Then we can follow this curve by using the parameter wo and evaluating left hand terms of (22) to (24) for zero crossings. The values of these terms versus to from 0 to it are shown on Fig. 5 for al = -1.1, h1 = 0.98 and hp = 1-2h1 (two terms are identical). We observe common zero crossing for three values of wo which correspond to the solutions designated previously by A, D and E. The value of the left hand term of (22) versus w is shown on Fig. 6 for h1= l-hl/2 and h0= 1., 0.98, 0.965, and 0.955. This explains how solutions designated by D and E join together when hi reaches 0.965 and then disappear as discussed in section 7. This occurs when a becomes negative because the ellipses Fig. 7. The five classes of solutions of the example for h0= 1 toO. 10. A rational function nonlinearity As we mentioned in section 5, the choice of a polynomial y forces the existence of other points of functioning than the origin, which is serious inconvenience for a musical usage [Rodet 93d]. We present here a rational function nonlinearity which avoids that difficulty and can be studied with the same tools which we are using for the polynomials. The most simple polynomial for our purpose is the one which we have looked after in the previous sections, 1(x) = x3 + aix. Apart from the origin, it intersects the line y=x at two fixed points: x=Â~1-a1 (29) ICMC Proceedings 1994 391 Sound SynthesisTechniques

Page  392 ï~~which in general are unstable. Furthermore, as x--o, the slope of 7-oo, which implies that the system can become unstable and escape to infinity, this is rather annoying for a musical instrument. We correct the behavior of Y by dividing the polynomial by another polynomial of degree less or equal to 3. With an even degree the odd symmetry is kept: Y(x) = (x3 + alx)/(l + bbx2) (30) Then the system (1) has no spurious fixed points provided b2>1 and can be studied with the same tools as the polynomial case. This function is shown on Fig. 8 for b2 = 2 and al=-1.7. To keep things simple, let us consider the case where there is no filtering in the 1.5 1 0.5 0 -0.5 -1 -1.5 "' " - -D '.).". - ' " \ Â~" -..... im.-'"-''/: '" --. \" " 0" r i1 -1 0 to follow the solutions from those of a trivial system, i.e. the one without a filter in the feedback loop, to the solutions of the system with a specific filter. Having determined all the 2-periodic solutions, it is easy to compute their stabilities. Finally we have proposed that the nonlinearity be implemented as a rational function. This avoids any spurious functioning point and guarantees that the oscillation of the system does not grow to infinity. Furthermore, this rational function system exhibits a chaotic behavior with the simultaneous presence of harmonic sinusoidal components and of non sinusoidal components which are heard as noise added to the harmonic tone. This is a first step towards the control of physical models. Acknowledgements We would like to thank Philippe Depalle, Adrian Freed, Denis Matignon, Ami Radunskaya and Ina Steinecke for many useful comments and discussions. We are grateful to CNMAT where this work was begun. References [Bergen 71] A. Bergen & R. Franks, "Justification of the Describing Function Method", SIAM J. Control, Vol. 9, No. 4, pp. 568-589, Nov. 71. [Chow 83] S.N. Chow, J. Mallet- aret & J. Yorke, "A homotopy method for locating all zeros of a system of polynomials", Lect. Notes in Math. 730, SpringerVerlag, 1979 Chua 90] L. 0. Chua and G.-N. Lin, "Canonical Realization of Chua's Circuit Family", IEEE trans. Circuits & Syst., Vol. CAS-37 (July. 1990) No. 7, pp 885-902. [Freed 92] A. Freed, "Tools for rapid prototyping of Musical Sound Synthesis Algorithms", Proc. International Computer Music Conference, San Jose, pp. 178-181, Oct. 1992. [Idogawa 92] T. Idogawa, M. Shimizu & M.I waki, "Acoustical behaviors of an oboe and a soprano saxophone artificially blown", Proc. of the symposium "Some Problems on the Theory of Dyn. Systems in Applied Science", pp. 71-93, World Scientific, 1992. [Ishizaka 72] K. Ishizaka, J.L. Flanagan, Synthesis of Voiced Sounds from a Two-Mass Model of Vocal Cords, Bell Syst. Tech. J., 51, 1233-1268. [Mees 79] A. Mees & L. Chua, "The. Hopf Bifurcation Theorem and Its Applications to Nonlinear Oscillations in Circuits and Systems", IEEE Transactions on Circuits and Systems, Vol. Cas-26, No. 4, April 1979, pp. 235-254. [Rodet 92a] X. Rodet, "Nonlinear Oscillator Models of Musical Instrument Excitation", Proc. International Computer Music Conference, San Jose, pp. 412-413, Oct. 1992. Fig. 8. Rational function g for al = 1.8 and b2 = 2 feedback loop (h0 =1, hl = 0). Then the first 2-periodic solutions of (1) oscillate between points where 1(x)=-x: x=f"(-1- al)/ (1+b2)(31) Such a solution is stable if the derivative of (YM'(x)): ( YI(x))'f='Y'(x))Y(x)=Y'(-x).'Y'(x)=-(T'(x))2 (33r) is less than I in modulus at these points. This happens when -4 < al < -1. Fig. 8 shows 1(x), (1Â~1(x)) and (YM(x))' for at =-1.7. But for approximately -6.15 < al S -4, we get two 2-periodic solutions. Finally chaotic solutions can be obtained for al < -7.2 approximately as shown on Fig. 9. 11. Conclusion We have studied the class of single feedback loop systems with a single instantaneous nonlinearity. It appears that both the number of solutions and the number of stable solutions can be largely reduced by the presence of a low-pass filter in the feedback loop. For a system in the defined class we have defined an equivalent but simpler system named the period-afterperiod system which has the same 2-periodic solutions. In the case of a polynomial nonlinearity, it is possible Sound SynthesisTechniques 392 ICMC Proceedings 1994

Page  393 ï~~3 Z 1 0 -1., N... '.. '.. 's 4 Fig. 9. Bifurcation diagram of the rational function nonlinearity versus parameter al. [Rodet 92b] X. Rodet & P. Depalle, "A physical model of lips and trumpet", Proc. International Computer Music Conference, San Jose, pp. 132-135, Oct. 1992. [Rodet 93a] X. Rodet, "Sound and Music from Chua's Circuit"Journal of Circuits, Systems and Computers, Special Issue on Chua's Circuit: a Paradigm for Chaos, Vol. 3, No. 1, pp. 49-61, March 1993. [Rodet 93b] X. Rodet, "Models of Musical Instruments from Chua's Circuit with Time Delay", in IEEE Trans. on Circ. and Syst., Special Issue on Chaos in nonlinear electronic circuits, Sept. 1993. [Rodet 93c] X. Rodet, "Flexible yet controlable physical models", Proc. International Computer Music Conference, Tokyo, Sept. 1993. [Rodet 93d] X. Rodet, "Applications of Chua's circuit to sound, music and musical instruments", Proc. IEEE Workshop on Nonlinear Theory and Applications, Honolulu, Dec. 1993. [Rodet 93e] X. Rodet, "Chua's circuit and the first mode of the basic clarinet model", unpublished internal report, IRCAM, Nov. 1993. [Rodet & Steinecke 94a] X. Rodet & I. Steinecke, "One and two mass models oscillations for voice and instruments", unpublished internal report, IRCAM, March 1994. [Rodet & Steinecke 94b] X. Rodet & I. Steinecke, "Comparison and simulation of one and two mass physical models for voice and instruments", in preparation. [Sharkovsky & al 93] A.N. Sharkovsky, Yu. Mastrenko, Ph. Deregel, and L.O.Chua, "Dry Turbulence from a time-delayed Chua's Circuit", in Journal of Circuits, Systems and Computers, Special Issue on Chua's Circuit: a Paradigm for Chaos, Vol. 3, No. 2, June 1993. [Steinecke & Herzel 93] 1. Steinecke & H.H. Herzel, "Bifurcations in an asymmetric vocal fold model", Proc. Symposium on Computer Simulation in Biology, Ecology and Medecine", Copenhagen, Aug. 23-25, 1993 [Steinecke 93] 1. Steinecke, Untersuchungen an einem vereinfachten Stimmlippenmodell, Diplomarbeit, Humboldt-UniversitAt, Berlin, 1993 [Valimaki et al. 93] V. Valimaki, M. Karjalainen & T.I. Laakso, "Modeling of Woodwind Bores with Finger Holes", Proc. International Computer Music Conference, Tokyo, Sept. 1993. [Wawrzynek 83] PhD dissertation, Berkeley, 1983. ICMC Proceedings 1994 393 Sound Synthesis Techniques