Page  48 ï~~FLEXIBLE YET CONTROLLABLE PHYSICAL MODELS: A NONLINEAR DYNAMICS APPROACH Xavier RODET IRCAM, Paris & Center for New Music and Audio Technologies (CNMAT) University of California, Berkeley, CA 94709 Tel: (510) 643 9990, Fax: (510) 642 7918, e-mail: Abstract For a physical model to be a useful instrument, it is necessary to control the dynamic behavior of the model in terms of stability, periodic oscillation, uniqueness of solutions, chaos, etc... It is hard to describe analytically the parameter space structure for control of the dynamic behavior of these nonlinear models. We focus here on the difficulties of this control, on different methods to solve the problem and on the results we have obtained. Stability, oscillation, and other conditions for the dynamic behavior of two classes of models are studied leading to a surprising family of sounds. 1. Introduction The final aim of a physical-model approach to sound synthesis ([Modeles 92],[Rodet 92b]) goes beyond the strict imitation of a specific instrument. Rather it aims at providing new simulated instruments with extended properties, such as broader range of sounds, improved playability or other properties sought by musicians. Simplified physical models can easily be simulated in real-time. They are represented by one or two feedback loops and a nonlinearity [Cook 92, Rodet 92a]. One of the favored equations, as we will show, is: x(t) = h * Y(x(t-t)) (1) where x e R, Y:R--,R (Fig. 1), r eR is some time delay, h:R -- R is the impulse response and * is the convolution operator. Complicated models are required to take into account all the subtleties of real instruments [Rodet 92b, Keef 93]. They consist of a system of integral and differential equations with delays and convolution terms. Furthermore, for contemporary music purposes, one wishes to have a flexible model that can be smoothly changed from one instrument to another by means of parameter changes. To obtain a useful instrument when simulating such a model, it is necessary to control the dynamic behavior of the model in terms of stability, periodic oscillation (frequency, harmonic content and stability of the oscillation), quasiperiodicity, chaos, etc..., each one being a function of the model parameters. Because of the nonlinear nature of the equations, even for simple models it is not easy to describe analytically the parameter space structure for control of the dynamic behavior of the model. We focus here on the difficulties of this control, on different methods to solve the problem and on the results we have already obtained on classes of useful models. A surprisingly rich and novel family of musical sounds has been obtained. Moreover, chaotic dynamics lead to sounds with important properties such as a combination of harmonic and noise components [Rodet 93a]. 2. Chua's and time-delayed Chua's circuits In [Rodet 93a] we have presented the sounds obtained by use of Chua's circuit [Chua 86, 90]. We recall that the basic oscillator circuit contains three linear energy-storage elements, a linear resistor, and a single nonlinear resistor Nr with a piece-wise linear v-i characteristic. Simple as it is, this circuit exhibits a surprisingly large variety of bifurcations and chaos [Chua 92]. We have done a real-time simulation of the circuit on a digital computer with audio capabilities and we have designed a graphical-user interface to control it interactively. According to the parameter values, harmonic and chaotic sounds can be obtained. The simultaneous presence of sinusoidal components and noise in the signal is very interesting since this occurs for the majority of classical instruments and because this is relatively difficult to model in a way which is useful for musical purposes. However, for musically interesting use, a synthesis algorithm has to provide control parameters allowing for expressive timbre modifications, i.e., essentially spectrum content modifications, as required by the performer. In order to get such flexibility, we now consider a slightly modified circuit, known as the time-delayed Chua's circuit [Sharkovsky 93]. Sharkovsky et al. add a dc bias voltage source in series with the Chua's diode and replace the capacitor C2 and the inductance L by a lossless transmission line, resulting in the time-delayed Chua's circuit. In a first simplification, two of the slopes of the characteristic of Nr are set equal, leaving two slopes only, called sl and s2 The study of this dynamical system is difficult, but with the capacitor C1=0, it reduces to a nonlinear difference equation: fi(t) = 7(O(t-2x)), where T is the time delay in the transmission line and y is a piecewise linear 1-D map which can be computed from the parameters of the circuit. In [Rodet 92a] we have shown that this system is a basic model of a clarinet-like reed instrument. We have also found that the (sl,s2) structure for 7 controls two important characteristics of the sound, transient onset velocity and richness as a function of the slopes s l and s2. It can easily be seen that Isi I controls the transient onset velocity, the greater IsI, the faster the onset. The frequency balance can be controlled by Isil and Is21: the closer they are to unity, the less high frequencies are in the signal. 5A.3 48 ICMC Proceedings 1993

Page  49 ï~~3. Some results on the role of the nonlinearity The most important assumption here is that the reed has no mass, leading to a memoryless nonlinearity. In this section, we present a few results on systems where the nonlinearity can be extracted as a memoryless element. 3.1. Piecewise-linear function We first look at some results of the time-delayed circuit with a piecewise-linear function y presented above, but without the filter h. By a proper affine change of variable, the invariant interval of the map can be set to the interval [0,1]. For certain parameter values, the map y is composed of two segments only in the invariant interval with slopes 10 and 11. In this particular case, Sharkovsky et al. have shown analytically that the time-delayed Chua's circuit exhibits the remarkable period-adding phenomenon shown on Figure 11 of [Sharkovsky 93]. In the (10, 11) space, the regions Ir2, t3, 1tI4, etc.... are those where the system has a stable limit cycle with period respectively 2, 3, 4, etc.... In between every two consecutive stable regions the system exhibits a chaotic behavior. To simulate this very interesting circuit, we have implemented the digital system shown on Fig. 1. The role of the convolution by h will be explained below, and for the moment we can ignore it. The time delay r allows us to easily control the fundamental frequency of the produced sounds. This delay is directly related to the structure and physics of many classical musical instruments. A rich variety of sounds can be produced by the system, and this is due to the combination of the dynamics of the nonlinear map together with the number of states represented by the delay line t. As an example, one can hear remarkable sounds by use of 10=0.99 and 11 between -1 and -10000. In [Rodet 93a] we show the short-time spectrum of signals from the digital system for some of these values. With sl = -1.3 and s2 = 2.0 we observe the simultaneous presence of sinusoidal components and noise in the signal as displayed on Fig. 2 To better show the noise component, Fig. 3 zooms on frequencies between 0 and 200 Hz. 3.2. Absence of nth partials It is relatively obvious that when the system has a stable limit cycle with period 2, of duration 2t, the limit cycle x(t) has the symmetry x(t+'r) = x(-t), i.e. has no even harmonics. What is more remarkable is that this result generalizes also to any region irk, k=3, 4, etc. In irk the system has a stable limit cycle x(t) with period k the harmonics k, 2k, 3k etc... of which are absent. The spectrum of such a limit cycle is given on Fig. 4. This is a rather interesting result from a musical point of view as well. To prove this result, let us take k=3 for instance. Then the system has a period 3 of duration 3t, and only three values of y(.) are used, say ai, i=l, 2 and 3. If x(t)=ai, then x(t+t)=ai+l and x(t+2 )=ai+2 where i+l and i+2 are taken modulo 3. Therefore each value al is represented during exactly a third of the total period 3i and the value of the 3kth Fourier coefficient is zero. 3.3. Drawback of a piecewise-linear function The piecewise-linear function used in Chua's circuit has a drawback. Consider the onset of the signal, i.e. the transient from zero. Observe that before a certain amplitude is reached, only a linear part of Y is used. The system behaves therefore like a linear system, that is there is no change in the short time spectrum of the signal other than an amplitude growth (this can be observed very easily in the short time spectrum display of our real-time implementation as detailed below). On the contrary, the nonlinearity of the reed of a real instrument can be more realistically approximated by a quadratic function [Fletcher 91]. Therefore, during the transient, there is a constant transfer of energy between frequency components. As a consequence, we favor a quadratic or cubic nonlinearity of the form 7(x)= ax2+slx or ax3+slx. In the last case for instance, the value of a is determined according to the slopes sl at 0 and s2 at the point (x0,yo) such that yo = -xo = ax03+slx0. Note that as we vary s1, we determine the amplitude and spectral richness of the sound simultaneously. Then a greater amplitude leads to a richer sound (i.e. more high-frequency components and with larger amplitude) as generally happens with natural instruments. But by varying a and si we can still provide independant control of the two first sound qualities mentioned above. It should be noted that such a polynomial function may introduce other fixed points than the origin, thereby complicating the dynamics of the circuit'. An ideal function for our purpose should have the origin as the only fixed point. To guarantee this, the function should not cross the line y=xIH(0) where H(0) is the value of the transfer function of h at dc. 4. First results in presence of the linear element For strings, reed-woodwinds or brass, the delay term with the filter h plays an essential role [Fletcher 91]. In a simple clarinet model for instance the propagation of the sound wave and its reflection at the extremity is represented by a feedback through a delay line and a filter as in equation 1. In [Rodet 92a] we have given some simple results on stability, in the absence of the filter h, of the systems described by equation (1). Unfortunately, with the linear element h, the solutions to these equations and their stability are known only partially and in restricted cases (see for example [Chow 85], [Ivanov 92] and [Hale 91]). Chow et al [85] consider the particular case where h is constant on an interval [-s, +e] and zero elsewhere. 1 Such other fixed points exist for natural instruments. For a high enough blowing pressure, the reed of a clarinet will keep the mouthpiece closed, but this is usually an unwanted effect. ICMC Proceedings 1993 49 5A.3

Page  50 ï~~It is also restricted to functions y e C2 which are odd and such that y(1)=-l, '(x)<O and y"(x)>0 for x>O. In this case, Chow et al. have proven that the equation (1) has a stable period 2 solution composed of odd harmonics only. This is an interesting result since it corresponds to the usual playing condition of a clarinet. However we feel the need for a more general result since we show in section 7 that the odd nature of g and the odd-harmonicity of the solution are in some sort nongeneric. Ivanov and Sharkovsky [92] consider the solutions to singularly perturbed delay equation such as: ax'(t) + x(t) = g(x(t-'r)) (2) Note that the equation (2) is essentially equivalent to (1). 5. Single feedback loop systems with a memoryless nonlinearity Let us first consider our system with a filter h included in the feedback loop and described by the equation (1). Fig. 1 shows that the system can be decomposed into a memoryless nonlinearity Y(.) and a linear element including h and a delay. We define the class of single feedback loop systems of which we can easily determine the stability and some oscillation properties. Such a system is composed of a unique memoryless nonlinearity and a linear feedback loop. Note that the only restriction on the linear element is that its impulse response be stable [Vidyasagar 78]. In particular the transfer function of the linear element needs not to be a rational function and thus can include delays. Many systems can be redesigned to fall into this class. For more complex systems, a larger class is studied in section 8. The first point we consider is the condition for oscillation around a fixed point when such a filter h, with transfer function H, is introduced in the feedback loop. Without loss of generality, we can assume this fixed point to be the origin (see the example below). The open-loop transfer function is now: G(jo) = e-Jwt H(jco) Since this represents the transfer function of the physical instrument, we naturally suppose that its impulse response belongs to L1 and therefore is stable. As before we call sl the slope of T(.) at the origin. We can apply the Graphical Stability Test [Vidyasagar 78] to find the value of the slope s l above which the system is stable. By assuming that G(s) has no poles in the closed right half plane C.., the test is simplified in the following way: For the system to be stable at the fixed point, the limit value l/Si should lie to the left of all intersections of the Nyquist plot of G(jo0) with the real axis (Fig. 5). Let -q+jO denote the intersection point with the smallest value and let (lq be the value such that G(joq) =--q. Then the system becomes unstable when si1 < -l/q. Note that this only indicates that the system could oscillate. A possible proof that it actually oscillates is more involved and is postponed to section 8. 6. Example We illustrate the use of this test to find the limiting value of the slope Sl and the oscillation frequency on a simple case. Consider the simple physical brass wind instrument model proposed in [Cook 92] and shown on Fig. 6. The corresponding equation is: x(t) = h * (p + rpx2(t-'t)) (3), where h is the impulse response of the filter and t the time-delay value. Note that in this formulation, clarinet and trumpet have the same representation. Let us call H(jo) the transfer function of the filter and H0 = H(0). The fixed points are the solutions of: x = pH0 + rpHox2 To be precise, let us choose t = 0.005 sec., r = -0.95 and the poles of h correspond to a center frequency Fc = 100 Hz and a bandwidth Bw = 500 Hz, i.e. a Z transform g/(l+az-l+bz-2), where, for a sampling rate Sr = 5000 Hz., p = exp(-2tBw/Sr), a = -2pcos(2ltFcISr ) = -1.4493, b = p2 = 0.5335 and g = l/H(2iFc) = 0.0928. Now we choose the fixed point with positive value, -1 = 2H0rp. We let u=x-xl such that the equation (3) rewrites: u(t) = h * rp(u2(t-t) + 2xlu(t-t)) (2) which has a fixed point at the origin. To apply the Graphical Stability Test, we find that the slope at origin is Si = 2rpxl, and G(jo) = H(jo) e-Jrwx. The Nyquist plot of G(jo) is represented on Fig. 5. The leftmost intersection of the Nyquist plot of G(jo)) with the real axis is at G(joq) = -q+jO. From an enlarged view of G(jo)) around wq, Fig. 7, the value -q is found to be approximately -1.0275. Then the system becomes unstable when 2rpx > 1/-q, that is: 1-l-4rHo p2 1 l-(l+H/q)2 2rp <_- or p> = pm 2Horp -q 4rH0 In our case, Pm is approximately equal to 0.845. Therefore, we expect the system to oscillate when the blowing pressure p exceeds 0.845. A discrete simulation of equation (1) confirms that the system is stable at xi for p less than 0.845 and does oscillate for p greater than 0.845. A few periods of the waveform obtained for p=0.87 are shown on Fig. 8. A simple examination of the waveform shows that the fundamental frequency is approximately 84.88 Hz. This is in excellent agreement with the frequency corresponding to Oq, i.e. between 84.8 and 84.9 as we can read it from Fig 7. Note that in the absence of the filter h or with a zero phase filter, the delay would lead to an oscillation frequency of f0 = lt2"t = 100 Hz. It is the supplementary delay added by the filter which displaces the oscillation frequency to 84.88 Hz as explained below. The graphical test for the single loop system is a way to find whether the so called return difference 1+siG(s) is bounded away from zero as s varies in the 5A.3 50 ICMC Proceedings 1993

Page  51 ï~~closed right half plane C+. In the multidimensional case, where x is a vector, the criterion is that the determinant of the return difference matrix [Vidyasagar 78] is bounded away from zero. Therefore, the same graphical test can be applied. However, we prefer another procedure (see section 8) which also provides existence and estimates for the frequency and amplitude of the oscillation. 7. Role of the linear element in single feedback loop systems We can extract more information from the polar plot of G(jo). Suppose for the moment that Hw)0) is real positive (without loss of generality we can at least choose the delay t such that for a given wo, G(joo0) = -q). Then the intersection of G(jo) with the negative real axis occurs for cokor---+2kn, i.e. for frequencies fk= (1+2k)/2t. Observe that f0 = 112v is the frequency corresponding to twice the delay t necessary for a sound wave to propagate from the reed to the end of the bore and back to the reed. The values fk, k=0, 1, 2,.....are the frequencies of the modes of the instrument. Therefore, G(jOq) and (iq/2n can be simply interpreted as the amplitude and the frequency of the strongest mode of the instrument2. Observe that the frequencies fk are the odd harmonic partials of the fundamental f0, but the oscillation frequency may be different from f0, since it generally is the frequency of the strongest mode rq/2n. Suppose for simplicity that the oscillation frequency is (Oq/2n and is equal to f0. Assume now that the argument of G(j2kcoq) is different from zero. Then the modes can be moved away from harmonic positions as we have shown above. To show that the oscillation frequency may be on the highest mode, let us change Fc to 300 Hz and Bw to 300 Hz in the previous example. The Nyquist plot of G(jco) appears on.Fig. 9 and shows that the highest mode is not the first around 1/2t = 100 Hz but the one around 300 Hz. With a precise examination of the plot, the leftmost intersection is found at a frequency of 274 Hz. The simulation shows that the system actually has an oscillatory solution with a frequency of approximately 266 Hz. In sections 3 and 4 we have noted the possible absence of odd partials. In our experiments we have observed that neither the non-odd character of y, nor the inclusion of a filter h in the feedback loop are sufficient alone to produce stable solutions with even harmonic partials. When simultaneously y, is not odd symmetric and there is a filter h, then even partials can appear (Fig. 10). Note that a very slight breaking of the symmetry of y, is sufficient. This is why, as we mentioned in section 4, we tend to consider the result of [Chow 85] 2 In the case of the trumpet for instance, the mouthpiece acts as a resonator wich boosts some modes with number greater than 1, thereby allowing an easy oscillation at the frequency of one of these modes [Fletcher 91 ]. of little applicability in our case, all the less since natural instruments will not have perfectly odd symmetric nonlinearities. Finally, it seems also that when y is not very far from odd symmetry, the even harmonic partials are of small amplitude (clarinet) if the argument of G(j2kcoq) is zero (Fig. 10), and can be of large amplitude (saxophone) when the argument of G(j2ko.q) is different from zero (Fig. 11). This latter case, the argument of G(j2kcoq) different from zero, can lead to surprising results which could be taken for quasi-periodicity or noise. Fig. 12 for instance displays the spectrum of such a signal. The enlarged portion displayed on Fig. 13 shows that a very low fundamental frequency (17 Hz) has been obtained even though the delay corresponds to a relatively high frequency (200 Hz.). Similarly on Fig. 14 and Fig. 15 with corresponding frequencies of 6 and 300 Hz. 8. Hopf Bifurcation and periodic solutions The Graphical Stability Test given above is valid as long as we can partition our system into a memoryless nonlinearity and a linear feedback loop. This encompasses more models than the simple one we have studied here. But since we are interested in periodic oscillation, we mention here a more general method which allows us to prove of the existence of a periodic solution when it occurs, and provides estimates for the frequency and amplitude of the oscillation. It also applies to an even more general class of systems encountered with the most sophisticated physical models of instruments such as in [Rodet 92b] and [Keef 93]. The Graphical Hopf Theorem and its algebraic version [Mess 79] apply to a nonlinear multiple feedback loop system as shown in Fig. 16 where Y is C4. Note in particular that G may include delays. Then under certain conditions on Y and G, the system has a unique stable periodic solution. Even though it is straightforward, we will not state this theorem in detail since it is rather lengthy. We merely emphasize that it provides the existence, uniqueness and stability test of the solution required for our application. Furthermore, the graphical interpretation is analogous to the graphical test applied in sections 5 and 6. However, the periodic solution is guaranteed only in a limited neighborhood of the bifurcation value. Therefore, other stable solutions may appear under more general playing conditions. This occurs in natural instruments [Idogawa 92] but can be a serious inconvenience for an electronic instrument. 9. Digital simulation For more flexibility, we have simulated the timedelayed Chua's circuit on a Silicon Graphics Indigo workstation. Real-time simulations were implemented using HTM [Freed 92]. Various graphs are displayed in real-time: the output signal, its Short Time Fourier Transform (STFT) or the function T. In particular, the possibility of looking at the STEJT in real-time is very useful for better understanding of the circuit and of the role of the various parameters [Rodet 93a]. The structure of the periodic and chaotic regions in the (10, 1i) space as displayed in Figure 11 of [Sharkovsky 93] is ICMC Proceedings 1993 51 5A.3

Page  52 ï~~interesting from a sonic point of view. The analytical computation is possible because the characteristic of the nonlinear element is piecewise linear. The computation would not be possible for more complex characteristics. But, by listening to the sound of the circuit, one can easily determine these regions and their frontiers. Let us take as example the values for which histograms have been represented in [Sharkovsky 93], i.e. s1=-18 and s2 varies from 0.04 to 0.49 or more. One can listen to the sound while changing parameter s2. In ltn regions, the periodic signal is clearly heard as a harmonic sound and the changes in periodicity are easily found by ear. In the intermediate chaotic regions, the sound is unstable or even noisy and it is not difficult to find approximate values for the frontiers between these regions. It is remarkable that this audification of the local properties of the space allows an easy determination of very complex structures which in some cases can not be computed analytically and are not simple to determine by other ways. 10. Conclusion We have studied here some problems stemming from physical models of musical instruments for the purpose of sound synthesis. In particular we have shown that the time-delayed Chua's circuit is a model of the basic behavior of an interesting class of musical instruments, namely those, like the clarinet, consisting of a memoryless nonlinearity coupled to a passive linear system. This circuit allows for an easy control of a large variety of dynamical behaviors. In the different regions of the parameter space, periodic and chaotic signals provide novel musical sounds. We have found conditions for periodic oscillations and relationships between parameter values and important properties of the produced signal such as onset time and spectral balance. We have also proposed an-analysis of the role of the linear part of the circuit in terms of the amplitude of the harmonic partials of periodic solutions. In the case of circuits that cannot be reduced to the simple form cited above, the Graphical Hopf Theorem provides a test for existence, stability and uniqueness of periodic solutions. A real-time implementation of the circuit on a digital workstation has allowed interactive parameter changes while simultaneously listening to the corresponding sounds and easy experimentation with the properties and behaviors of the circuit and sounds. It has revealed a rich and interesting family of sounds for musical applications. The real-time interaction also provides unusual insights on properties of the circuit that would not be as easily discovered using other means. We expect to extend our results to other instruments such as brass, voice, flute, and strings. It appears that such models are essential for the development and musical use of physical models of classical or new instruments. For instance, we have noted that several stable solutions may appear in general playing conditions. This occurs in natural instruments but can be a serious inconvenience for an electronic instrument. 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. Acknowledgments We are grateful to Prof. Chua for his interest and support, to CNMAT where this work has been done and to IRCAM for their support. We would also like to thank Silicon Graphics for the use of their Iris Indigo workstation. References [Cook 92] P. Cook, "A meta-wind-instrument physical model", Proc. International Computer Music Conference, San Jose, pp. 273-276, Oct. 1992. [Chow 85] S. N. Chow & D. Green Jr., "Stability, Multiplicity and Global Continuation of Symmetric Periodic Solutions of a Nonlinear Volterra Integral Equation", Japan Journal of Applied Mathematics, Vol. 2, No. 2, pp. 433-469, Dec. 85. [Chua 86] L. 0. Chua, M. Komuro and T. Matsumoto, "The Double Scroll Family", IEEE trans. Circuits & Syst., Vol. CAS-33 (Nov. 1986) No. 11, pp 1073-1118. [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. [Chua 92] L.O. Chua, "A zoo of strange attractors from the Chua's circuit", Proc. 35th Midwest Symposium on Circuits and Systems, Washington, D.C., August 9-12, 1992, pp. 916-926. [Fletcher 91] N.H. Fletcher & T. D. Rossing, "The Physics of Musical Instruments", Springer Verlag, 1991. [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. [Hale 91] J.K. Hale, "Dynamics and Delays", in Delay Differential Equations and Dynamical Systems, Proc., 1990, S. Busenberg & M. Martelli (Eds.), Lecture Notes in Mathematics 1475, Springer Verlag, 1991. [Idogawa 92] T. Idogawa, M. Shimizu & M. Iwaki, "Acoustical behaviors of an oboe and a soprano saxophone artificially blown", Proc. of the symposium "Some Problems on the Theory of Dynamical Systems in Applied Science", pp. 71-93, World Scientific, 1992. [Ivanov 921 A.F. Ivanov & A.N. Sharkovsky, "Oscillations in Singularly Perturbed Delay Equations", in Dynamics Reported, C.K.R.T. Jones, U. Kirchgraber & H.O. Walther editors, Springer Verlag, pp. 164-224, 1992. (Keef 93] D. Keefe, "Physical Modeling of Wind Instruments", Computer Music Journal, MIT Press, Vol 16 No. 4, pp. 57-73, Winter 1992. [Mees 79] A. Mees & L. Chua, "The Hopf Bifurcation Theorem and Its Applications to Nonlinear 5A.3 52 ICMC Proceedings 1993

Page  53 ï~~Oscillations in Circuits and Systems", IEEE Transactions on Circuits and Systems, Vol. Cas-26, No. 4, April 1979, pp. 235-254. [Modeles 921 "Modeles Physiques, Creation Musicale et Ordinateurs", Proceedings of the Colloquium on Physical Modeling, ACROE, Genoble,France, Oct. 1990, Editions de la Maison de Sciences de l'Homme, Paris, France, 1992. [Rodet 92a] X. Rodet, "Nonlinear Oscillator Models of Musical Instrument Excitation", Proc. International Computer Music Conference, San Jose, pp. 412-413, Oct. 1992. [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", to appear in IEEE Trans. on Circ. and Syst., Special Issue on Chaos in nonlinear electronic circuits, Sept. 1993. [Sharkovsky 93] A.N. Sharkovsky, Yu. Mastrenko, Ph. Deregel, and L.O.Chua, "Dry Turbulence from a time-delayed Chua's Circuit", to appear in Journal of Circuits, Systems and Computers, Special Issue on Chua's Circuit: a Paradigm for Chaos, Vol. 3, No. 2, June 1993. [Smith 86] J.O. Smith, "Efficient simulation of the reed-bore and bow-string mechanism", Proc 1986 Int. Computer Music Conf., P. Berg, eds., Computer Music Assoc., San Francisco, pp. 275-280, 1986. [Vidyasagar 78] M. Vidyasagar, "Nonlinear System Analysis", Prentice Hall, 1978. -4In Fig. 2: Spectrum of a signal showing the simultaneous presence of sinusoidal components and noise. d3 Fig. 3: Detail of Fig. 2 between 0 and 200 Hz. Fig. 1: A simple time-delayed nonlinear system, also a basic clarinet model Fig. 4: Spectrum of a period 3 signal showing the absence of 3th partials. ICMC Proceedings 1993 53 5A.3

Page  54 ï~~bir(G}w)1 Fig. 9: Nyquist plot of G(jo) for a p on another mode than th Fig. 6: A simple physical brass wind instrument model proposed in [Cook 92]. Im(Gfi#)1 Re(Gw)} )ossible oscillation e first l, 00 -0.o -0.0 -0.0 -o.. _-D11 )2 ]2 13 34 35 I. I I Fig. 10: Spectrum of a signal obtained with a non-odd nonlinearity and a zero-phase filter. lal -U U ' I 'J'.3285 -1.028 -1.0275 -1.027 -1.0265 -1.025 Re(G(jw)) Fig. 7: Detail of Fig. 5 around the crossing of real axis. U U.UI U.Uz U.U U4 U.U U.Ub S Fig. 8: A few oscillations from the simple physical brass wind instrument model of Fig. 6. I....,... -......... Fig. 11: Spectrum of a signal obtained with a non-odd nonlinearity and a non-zero-phase filter. 5A.3 54 ICMC Proceedings 1993

Page  55 ï~~-1* Fig. 12: Spectrum of a signal with complicated structure. 1 1 ' ' 1 1 + 1 ' 1 1, ' +, 1 / 1 + 1 1 + 1 / /; 1 1 1 1 + 1. + " 1 ' 1 1 1 1 1 1 " 1. 1 /... "" "1' "" "" 1 "" "" 1" "" " Ile I.. " 1" "" "" 1" "" Fig. 15: Detail of Fig. 12 for frequencies between 0 and 500 Hz. I.. Fig. 13: Detail of Fig. 12 for frequencies up to 400 Hz. I. I -U. --.~ S I Fig. 16: A nonlinear multiple feedback loop system, yeRm, ueR1. Fig. 14: Spectrum of a signal with complicated structure. ICMC Proceedings 1993 55 5A.3