Page  440 ï~~Chaos and Non-linear Models Ami Radunskaya Mathematics Department,Pomona College Claremont, CA 91711 aradunskaya@pomona. edu Abstract Non-linear differential or difference equations are often used as models of musical instruments for two reasons: these models are capable of producing a wide variety of sounds by changing one or two parameters, and they mimick the expressive potentials of physical instruments in their ability to move smoothly from the generation of periodic to chaotic signals. An understanding of the bifurcation sequence in parameter space is essential to control these models. This paper discusses the KAM theorem, the proof that the non-linear string model satisfies the hypotheses of the theorem, and the resulting description of phase-space. 1 Introduction Transitions from stable, low-period oscillations to chaotic motion has been studied for several classes of physical models of musical instruments. Among these are the models represented by non-linear delayed difference equations, [Rodet, 1992, Rodet 1994], which exhibit transitions to oscillations of higher and higher periods according to a period-adding or period-doubling series of bifurcations in parameter space. Another "route to chaos" appears in families of nearly-integrable Hamiltonians: perturbations of integrable Hamiltonian systems, where the size of the perturbation is now the control parameter. The KAM (Kolmogorov-Arnold-Moser) theorem describes the conditions necessary for a gradual breakup of the invariant curves of the integrable system under these perturbations, and the structure of the resulting phase space. One of the simplest physical models, the one-dimensional vibrating string, can be shown to satisfy this theorem when small non-linear force terms are introduced. 2 Background:Hamiltonians Suppose we have a dynamical system whose phase space, Q is coordinatized by (E, 0), where 0 R. Let H be a Hamiltonian, i.e. a function which is constant over the evolution of the system,and which satisfies: aG. and 0 - 0{j (1) Suppose further that H is completely integrable, depending only on the n varialbes, 1, 2, H = H0(i2,.pie which implies that J = 0 and 0 = constant = w. (2) Then the motion is quasi-periodic on the invariant tori, 9, = Ct with frequencies w3. If we add another term which is not necessarily integrable to Ho, requiring that the sum still satisfies the partial differential equations (1), which is not necessarily integrable we get: H = H0 +H(I,O) (3) In 1923, Fermi conjectured that most perturbations of integrable Hamiltonians ( - 0 in (3)) are ergodic, or undergo a process of thermal relaxation. In 1954, Fermi, Pasta and Ulam (FPU) numerically investigated the simplest example they could think of: a one-dimensional model of a vibrating string, in order to test this conjecture, [Fermi, Pasta, Ulam, 19551. 3 The model The string is discretized by concentrating its mass at N + 1 equally spaced nodes, connected by N massless springs. We label the masses 0 through Radunskaya 440 ICMC Proceedings 1996

Page  441 ï~~- ' "x.1 Figure 1: The discretized string model y A fm Figure 3: More energy is exchanged: E = 1 Figure 2: Energy exchange between modes: E=.25 N, and the springs 1 through N: if we let the total mass of the string be N + 1, x, be the displacement of the ith mass from its equilibrium position, and di the difference between the length of the ith spring and its equilibrium length, we get the following differential equation for the motion of each node: d2T=(t) F(d,+ ) - F(d), i= 1,2...N - 1 dt2(4 (4) where F(d) is the force exerted by each spring, and we assume that the ends are fixed xo(t) = xN(t) = 0 for all t. Notice that if the force is proportional to the displacement with proportionality constant 1: F(d) = d, and the differential equation (4) gives a discretization of the one-dimensional wave equation. In this case, the mechanical system is an integrable Hamiltonian as defined above in the square-energy coordinates: f1,..., N, where k is the square of the energy in the kth mode. In the linear case, no energy is exchanged between modes, since k is constant by equation (2). FPU perturbed the integrable Hamiltonian by adding a small non-linear term to the force exerted by the springs: F(d) = d + Ed' for r = 2 or 3. According to Fermi's 1923 conjecture, the energy should, after some time, be equally distributed among all modes, no matter what the initial energy distribution. What they observed, in fact, was that the energy stayed in only a few modes, moving between them with an observable "super-period". Figure 2 shows the energy exchanged between the initially excited first mode and a few of the other lower modes with a small amount of quadratic nonlinearity (F(d) = d +.25d2). The super-period is just under 7000 cycles. As the amount of nonlinearity is increased, energy is shared between more modes, and the super-period becomes less well-defined. When the non-linear term grows large enough, ergodic behaviour can be observed: the energy becomes evenly distributed throughout the modes. We get the three ingredients necessary for chaos: infinitely many periodic orbits, generic dense orbits, and sensitivity to initial conditions, [Devaney,1989J. These results, which were a mystery to FPU at the time of their original experiment, can now be partially explained. 4 The KAM theorem and transitions to chaos In 1954, the same year as the original FPU experiment, Kolmogorov stated the following theorem, which was independently proved in 1963 by Arnold and Moser: [Arnold, 1961, Moser, 19661 Theorem 1 (KAM) If the unperturbed Hamiltonian, HQ is non-degenerate: a2 Ho Detj a 2 I# 0 (5) and if the perturbation, H1 is small eough and periodic in the variables Oj, then for Lebesgue almost every initial frequency vector, w*, there exists an invariant torus, T(w*) of the perturbed system which is close to the corresponding invariant torus of the unperturbed system, To(w*). Furthermore, the measure of the complement of T(w*) - 0 as (Hl [ -0. We explain: in the unperturbed case of the linear string, the energy remains completely in the initially excited modes. The set of points with these initial energies are surfaces in phase-space on which the orbits of the system move periodically, i.e. they are tori. What this theorem says is that,when a small amount of non-linearity is added, most of these invariant surfaces persist, but they are slightly deformed. They are no longer surfaces of constant energy, but they are made up of orbits which return again and again to the same spot in phase space, never wandering very far away: ICMC Proceedings 1996 441 Radunskaya

Page  442 ï~~,''' i! i!1 J i, ___________ __ ___ IIIIII,(II I, Figure 5: Waveforms creased as the non-linearity is in Figure 4: a KAM fractal torus they move off the initial energy surface, but repeatedly come back to it, hence the "super-period" observed in the non-linear string model. Also, some of the invariant surfaces DO disappear, creating seas of chaotic behavior between the invariant surfaces. Inside these chaotic seas are smaller invariant islands, each containing smaller chaotic seas, and so on. Figure 4 depicts some of this fractal structure in the case when the phase space has dimension 4, so that the constant energy states describe three-dimensional subsets of this phase space. As the size of the perturbation is increased the invariant tori gradually disappear until all of phase space is one connected chaotic sea. This scenario is in contrast to the period-doubling bifurcation sequences seen in one-dimensional dynamics, since chaotic and quasi-periodic behavior co-exist in phase space for all perturbations, but the size of their respective domains changes as the parameter changes. 5 FPU and KAM In order to check that the FPU model does indeed undergo a KAM transformation as the non-linearity is increased, we must check the non-degeneracy and periodicity conditions of the theorem. The correct coordinates are obtained by first changing to the spectral domain: determinant of the Hessian: Deti H o b2om0. The complete Hamiltonian becomes: H = Ho + EH1 k = _N-1 Â~ +H1(,O) (8) where we can think of H1 as the "energy exchange" term: Eexch = Ekak Ei,kAkijaiaj The Ak's are obtained by solving the system of simultaneous equations: sin(Q)q, +... sin(NN qN-1 sin( - )ql + '... sin(2(NN 1)7r )qN -1 sin(('N-1)Â~)ql +.. sin((N-N)qN a1 = a2 -aN-1 Since this system is non-singular for every N, there is a unique solution for every N. Also, note that the result will be a function of sin Oj and s, and thus will be periodic in all the Oj's. Thus, the FPU model with a quadratic nonlinearity satisfies the hypotheses of the KAM theorem, and the observed recurrence in the energyexchange function for small non-linearities, the socalled "super-period", can now be explained. We remark that a cubic non-linearity can be treated similarly. The waveforms resulting from this model with increasing amounts of non-linearities are shown in Figure 5. 6 Application In order to make the model playable, we add a damping term with damping constant p and a variable restoring force, f3. The equations of motion for each node in a general one-dimensional string model with an rth order non-linearity then becomes: d2Xi(t) d2 = -[F,r(di Â~ 1)- F,r(di)I]- pÂ~i(t) dt2 N (kjr ak =.= 1sin N) x, dak ak = (6) and then changing to square-energy coordinates: 2 22 where wk =4 /2sin(r) k Â~kakWk2N (7) Ok = arctan akwk It is easy to see that in these coordinates, with H0 = EN-1 in equations (3) and (5), that the Radunskaya 442 ICMC Proceedings 1996

Page  443 ï~~where Fc,r(d) = d + cdT. The duration, pitch, and timbre of the sounds produced by the model can then be controlled by the parameters p, f3, and c, respectively. The more masses in the model, the richer the sounds, since the number of possible vibrational modes is determined by the number of masses. A wide variety of sounds can be obtained this way and, by controlling the parameter E, the probability of producing chaotic orbits can also be controlled. We remark that the natural vibrational modes of this string are not harmonically, since the natural frequencies are wk = 4v/2sin( ) where k ranges from 1 to N. 7 Summary The KAM theorem predicts the persistance of invariant energy surfaces in phase space with small non-linear perturbations. However, it is not very specific: it does not predict the amount of nonlinearity required to completely destroy these invariant surfaces, signifying the onset of complete ergodicity or chaos. Furthermore, experiment has shown that the orbits in phase space are even more regular than predicted by the theorem. It appears that soliton-like waveforms are generated for small non-linearities. Indeed, if the model is taken to the continuous-space limit, letting the number of masses go to infinity and the space between them go to zero, Zabusky and Kruskal show one gets something like the KdV equation. This equation, with periodic boundary conditions (the ends of the string would wrap around and connect to each other), has been shown to have soliton solutions for short time intervals [Zabusky, 1962]. On the other hand, what the theorem loses in specificity it gains in generality. ANY perturbation of an integrable system of the right form will share the FPU model's rich phase-space structure. For systems with a small number of dimensions it is possible to completely analyse the parameter space of this system by using Poincare sections [Henon, Heiles, 19641 and to anticipate the breakdown of the quasiperiodic orbits. [Poggi and Ruffo, 1995]. The description of the evolution of the system in the energy coordinates of each mode is also useful for its perceptual significance. [Devaney, 1989] [Fermi, Pasta, Ulam, motions under small perturbations of the Hamiltonian. Uspehi Mat. Nauk 181(113), pp.13-40, 1963. Robert Devaney. An introduction to chaotic dynamical systems. pp.50. AddisonWesley Publishing Company. 1989. 1955] E. Fermi, J. Pasta, S. Ulam. Studies of non linear problems. Los Alamos report LA-1940, May, 1955. In E. Fermi. Collected Works Vol. 2, pp.978-88. Univ. of Chicago Press. 1965. [Henon, Heiles, 1964] M. Henon and C. Heiles. The applicability of the third integral of motion: some numerical experiments. Astrophys. J. 69, pp.73-9, 1964. [Moser, 19661 Jurgen Moser. On the theory of quasiperiodic motions. SIAM Review 8(2), pp.145 -172, 1966. [Poggi and Ruffo, 1995] P. Poggi and S. Ruffo. Exact soluions in the FPU oscillator chain. preprint Departimento di Energetica, Universita di Firenze. 1995. [Rodet, 1992] [Rodet 1994] Xavier Rodet. Nonlinear oscillator models of musicalinstrument excitation. Proc. International Computer Music Conference, San Jose, pp.412-413, Oct. 1993. Xavier Rodet. Stability/Instability of Periodic Solutions and Chaos in Physical Models of Musical Instruments. Proc Int. Computer Music Conference. Copenhaguen, 1994. Norman Zabusky. Exact solutions for the vibrations of a nonlinear continuous model string. Journal of Mathematical Physics. 3(5), pp.1028 -1039. 1962 [Zabusky, 1962) References [Arnold, 1961] V.I. Arnold. Proof of A.N. Kolmogorov's theorem on the preservation of quasiperiodic ICMC Proceedings 1996 443 Radunskaya