Analytical prediction of delayed Hopf bifurcations in a simplified stochastic model of reed musical instruments

Abstract

This paper investigates the dynamic behavior of a simplified single reed instrument model subject to a stochastic forcing of white noise type when one of its bifurcation parameters (the dimensionless blowing pressure) increases linearly over time and crosses the Hopf bifurcation point of its trivial equilibrium position. The stochastic slow dynamics of the model is first obtained by means of the stochastic averaging method. The resulting averaged system reduces to a non-autonomous one-dimensional Itô stochastic differential equation governing the time evolution of the mouthpiece pressure amplitude. Under relevant approximations the latter is solved analytically treating separately cases where noise can be ignored and cases where it cannot. From that, two analytical expressions of the bifurcation parameter value for which the mouthpiece pressure amplitude gets its initial value back are deduced. These special values of the bifurcation parameter characterize the effective appearance of sound in the instrument and are called deterministic dynamic bifurcation point if the noise can be neglected and stochastic dynamic bifurcation point otherwise. Finally, for illustration and validation purposes, the analytical results are compared with direct numerical integration of the model in both deterministic and stochastic situations. In each considered case, a good agreement is observed between theoretical results and numerical simulations, which validates the proposed analysis.

Appendices

A The 1-dimensional Itô’s formula

Let the following Itô differential equation

$$\begin{aligned} \mathrm {d}x_t=m(x_t,t) \mathrm {d}t+\sigma (x_t,t) \mathrm {d}W_t, \end{aligned}$$

(68)

where m and $\sigma$ are real functions and $W_t$ is the so-called Wiener process.

Let $f(x_t,t)\in \mathcal {C}^{2}(\mathbb {R},\mathbb {R}^+)$ (i.e., f is twice continuously differentiable on $(\mathbb {R},\mathbb {R}^+)$). Then $f(x_t,t)$ is also Itô process, and

$$\begin{aligned} \mathrm {d}f(x_t,t)&=\frac{\partial f}{\partial t}\mathrm {d}t + \frac{\partial f}{\partial x}\mathrm {d}x_t + \frac{1}{2}\frac{\partial ^2 f}{\partial x^2}\sigma (x_t,t)^2\mathrm {d}t \end{aligned}$$

(69a)

$$\begin{aligned}&= \left( \frac{\partial f}{\partial t} + m(x_t,t) \frac{\partial f}{\partial x} + \frac{\sigma (x_t,t) ^2}{2}\frac{\partial ^2 f}{\partial x^2}\right) \mathrm{d}t\nonumber \\&\quad + \sigma (x_t,t) \frac{\partial f}{\partial x}\,\mathrm{d}W_t. \end{aligned}$$

(69b)

Equation (69) is the 1-dimensional Itô’s formula (for more details and proof see [26], Chap. 4).

B General formulation of the stochastic averaging method

In this appendix, the stochastic averaging method [24, 34] is briefly described. For that we consider the following system of differential equations in standard form

$$\begin{aligned} \dot{\mathbf{x}}_t=\mathbf{f}(\mathbf{x}_t,t)+\mathbf{g}(\mathbf{x}_t,t)\varvec{\eta }_t \end{aligned}$$

(70)

where $\mathbf{x}_t \in \mathbb {R}^n$. If the deterministic vector function $\mathbf{f}(\mathbf{x}_t,t)\in \mathbb {R}^n$ and matrix function $\mathbf{g}(\mathbf{x}_t,t)\in \mathbb {R}^n \times \mathbb {R}^n$ satisfy certain requirements [24] and the elements of the vector $\varvec{\eta }_t$ are broadband processes, with zero means, then the slow (or averaged) dynamics of Eq. (70) may be approximated by the following Itô equations

$$\begin{aligned} \mathrm {d} \mathbf{x}_t=\mathbf{m}(\mathbf{x}_t ) \mathrm {d}t+\varvec{\sigma }(\mathbf{x}_t,t) \mathrm {d}{} \mathbf{W}_t, \end{aligned}$$

(71)

where $\mathbf{W}_t\in \mathbb {R}^n$ is a vector of n Wiener processes. The vector $\mathbf{m}$ and the matrix $\varvec{\sigma }$ are called drift vector and diffusion matrix, respectively, and defined by

$$\begin{aligned} \mathbf{m} = T^{\text {av}} \left\{ \mathbf{f}+ \int _{-\infty }^0 \mathbb {E}\left[ \left( \frac{\partial {(\mathbf{g} \varvec{\eta })}}{\partial \mathbf{x} } \right) _t (\mathbf{g} \varvec{\eta })_{t+\tau }\right] \mathrm {d}\tau \right\} , \nonumber \\ \end{aligned}$$

(72)

where $\frac{\partial {(\mathbf{g} \varvec{\eta })}}{\partial \mathbf{x}}$ is the Jacobian matrix of $\mathbf{g} \varvec{\eta }$, and

$$\begin{aligned} \varvec{\sigma } \varvec{\sigma }^T= T^{\text {av}} \left\{ \int _{-\infty }^{+\infty } \mathbb {E}\left[ (\mathbf{g} \varvec{\eta })_{t} (\mathbf{g} \varvec{\eta })_{t+\tau }^T\right] \mathrm {d}\tau \right\} \nonumber \\ \end{aligned}$$

(73)

where $\{.\}^T$ and $\mathbb {E}\left[ \{.\}\right]$ denotes, respectively, the transpose and the expected value of $\{.\}$. $T^{\text {av}}$ is an averaging operator defined as follows

$$\begin{aligned} T^{\text {av}} \left\{ . \right\} = \lim \limits _{T \rightarrow +\infty } \frac{1}{T}\int _{t_0}^{t_0+T}\left\{ .\right\} \mathrm {d}t. \end{aligned}$$

(74)

It should be noted that in the case of a periodic variables with period $T_0$ (which is the case in this paper), the operator $T^{\text {av}}$ becomes a classical Krylov–Bogolyubov time averaging over one period $T_0$, i.e.,

$$\begin{aligned} T^{\text {av}} \left\{ . \right\} = \frac{1}{T_0}\int _{t_0}^{t_0+T_0}\left\{ .\right\} \mathrm {d}t \end{aligned}$$

(75)

and the result is independent of $t_0$.

C Derivation of the expression of the deterministic dynamic bifurcation point

In this appendix, we give the details of the résolution of Eq. (41). First we state $y+\hat{\gamma }^{\text {st}}=X^2$ and then it can be shown that, using Eq. (34), Eq. (41) takes the following form

$$\begin{aligned}&\left( X-X_0\right) \Big (\zeta F_1 X^2+ (\zeta F_1X_0-\alpha _1\omega _1)X\nonumber \\&\quad -\zeta F_1+\zeta F_1 X_0^2-\alpha _1 X_0 \omega _1\Big )=0 \end{aligned}$$

(76)

where $X_0^2=y_0+\hat{\gamma }^{\text {st}}$. Obviously $X=X_0$ and therefore $y=y_0$ is a solution of (76). The second term of the product in the left-hand side of (76) is a second order polynomial equations whose roots $X_1$ and $X_2$ are

$$\begin{aligned} X_{1}&=\frac{1}{2 \zeta F_1}\Bigg (\alpha _1 \omega _1-\zeta F_1 X_0\nonumber \\&\quad -\sqrt{\alpha _1^2 \omega _1^2+2 \alpha _1 \zeta F_1 X_0 \omega _1+\zeta ^2 F_1^2 \left( 4-3 X_0^2\right) }\Bigg ) \end{aligned}$$

(77)

and

$$\begin{aligned} X_2&=\frac{1}{2 \zeta F_1}\Bigg (\alpha _1 \omega _1-\zeta F_1 X_0\nonumber \\&\quad +\sqrt{\alpha _1^2 \omega _1^2+2 \alpha _1 \zeta F_1 X_0 \omega _1+\zeta ^2 F_1^2 \left( 4-3 X_0^2\right) }\Bigg ). \end{aligned}$$

(78)

The initial value $y_0$ is always chosen to be larger than $-\hat{\gamma }^{\text {st}}$ (because the mouth pressure $\gamma$ must be larger than zero). Therefore, X must be larger than zero and, for a realistic set of parameters, only $X_2$ is positive. Consequently, the expression of the deterministic dynamic bifurcation point is given by

$$\begin{aligned} {\hat{y}}^{\text {dyn}}_{\text {det}}=X_2^2-\hat{\gamma }^{\text {st}}. \end{aligned}$$

(79)

D Derivation of the expression of the stochastic dynamic bifurcation point

Using Eqs. (34), (53) takes the following explicit form

$$\begin{aligned} K=\frac{\zeta F_1\sqrt{\hat{\gamma }^{\text {st}} +y}}{2 \omega _1}\left( y-1+\hat{\gamma }^{\text {st}}\right) -\frac{\alpha _1 y}{2}. \end{aligned}$$

(80)

To obtain a solvable cubic form (i.e., without square root), Eq. (80) is transformed into

$$\begin{aligned} \left( K+\frac{\alpha _1 y}{2}\right) ^2= \frac{\zeta ^2 F_1^2\left( \hat{\gamma }^{\text {st}} +y\right) }{4 \omega _1^2} \left( y-1+\hat{\gamma }^{\text {st}}\right) ^2 \end{aligned}$$

(81)

which yields Eq.  (55).

In the remaining of this appendix the Cardano’s method (see, e.g., [32]) is used to solve the latter, i.e., $a_1y^3+a_2y^2+a_3y+a_4=0.$

First, the following parameters are introduced

$$\begin{aligned} p= & {} -\frac{a_2^2}{3a_1^2}+\frac{c_3}{a_1}\quad \text {and}\quad \\ q= & {} \frac{a_2}{27a_1}\left( \frac{2a_2^2}{a_1^2}-\frac{9a_3}{a_1}\right) +\frac{a_4}{a_1}. \end{aligned}$$

The discriminant $\varDelta$ is defined as $\varDelta =-\left( 4 p^3+27 q^2\right) .$ Then:

• 1. If $\varDelta <0$, one root is real and two are complex conjugate.

• 2. If $\varDelta =0$, all roots are real and at least two are equal.

• 3. If $\varDelta >0$, all roots are real and unequal.

A typical example of the discriminant $\varDelta$, plotted as a function of the noise level $\sigma$, is depicted in Fig. 12 for a typical set of parameters. It can be shown that $\varDelta$ can be expressed as a fourth-order polynomial equation with respect to $\ln \sigma$ which has two distinct roots

$$\begin{aligned} \ln \left( \sigma _1\right)&=\frac{2 \alpha _1^3 \omega _1^3-27 \alpha _1 \hat{\gamma }^{\text {st}} \zeta ^2 F_1^2 \omega _1}{54 \zeta ^2 F_1^2 \omega _1 \epsilon }\nonumber \\&\quad -\frac{2 \left( \alpha _1^2 \omega _1^2+3 \zeta ^2 F_1^2\right) ^{3/2}-9 \alpha _1 \zeta ^2 F_1^2 \omega _1}{54 \zeta ^2 F_1^2 \omega _1 \epsilon }\nonumber \\&\quad -\frac{1}{2} \ln \left( \frac{2 \sqrt{2 \pi } \hat{\gamma }^{\text {st}^{3/4}} \omega _1 e^{-\frac{(\hat{\gamma }^{\text {st}} -1) \sqrt{\hat{\gamma }^{\text {st}} } \zeta F_1}{\omega _1 \epsilon }}}{\sqrt{(3 \hat{\gamma }^{\text {st}} +1) \zeta F_1 \omega _1 \epsilon }}\right) \nonumber \\&\quad +\ln \left( x_{y_0}\right) \end{aligned}$$

(82)

and

$$\begin{aligned} \ln \left( \sigma _2\right)&=\frac{9 \alpha _1 (1-3 \hat{\gamma }^{\text {st}} ) +\frac{2 \left( \alpha _1^3 \omega _1^3+\left( \alpha _1^2 \omega _1^2+3 \zeta ^2 F_1^2\right) {}^{3/2}\right) }{\zeta ^2 F_1^2 \omega _1}}{54 \epsilon }\nonumber \\&\quad -\frac{1}{2} \ln \left( \frac{2 \sqrt{2 \pi } \hat{\gamma }^{\text {st}^{3/4}} \omega _1 e^{-\frac{(\hat{\gamma }^{\text {st}} -1) \sqrt{\hat{\gamma }^{\text {st}} } \zeta F_1}{\omega _1 \epsilon }}}{\sqrt{(3 \hat{\gamma }^{\text {st}} +1) \zeta F_1 \omega _1 \epsilon }}\right) \nonumber \\&\quad +\ln \left( x_{y_0}\right) \end{aligned}$$

(83)

and a double root

$$\begin{aligned} \ln \left( \sigma _3\right)&=\frac{\alpha _1 (\hat{\gamma }^{\text {st}} -1 )}{2 \epsilon }\nonumber \\&\quad -\frac{\log \left( \frac{2 \sqrt{2 \pi } \hat{\gamma }^{\text {st}^{3/4} }\omega _1 e^{-\frac{(\hat{\gamma }^{\text {st}} -1) \sqrt{\hat{\gamma }^{\text {st}} } \zeta F_1}{\omega _1 \epsilon }}}{\sqrt{(3 \hat{\gamma }^{\text {st}} +1) \zeta F_1 \omega _1 \epsilon }}\right) }{2}\nonumber \\&\quad + \ln \left( x_{y_0}\right) . \end{aligned}$$

(84)

In the example shown in Fig. 12 one has: $\ln \left( \sigma _1\right) =-53.1$, $\ln \left( \sigma _2\right) =-6.8$ and $\ln \left( \sigma _3\right) =-26.6$.

Fig. 12

Discriminant $\varDelta$ of (55) as a function the natural logarithm of the noise level $\ln \sigma$. The set of parameters (45) is used

Fig. 13

The roots $r_k$ ($k=0,1,2$) of Eq. (55) and the the stochastic dynamic bifurcation ${\hat{y}}^{\text {dyn}}_{\text {stoch,a}}$ as functions of the natural logarithm of the noise level $\sigma$. The parameters (45) are used and $x_{y_0}=0.01$

If $\sigma _1<\sigma <\sigma _2$ we have $\varDelta >0$ except at $\sigma =\sigma _3$ for which $\varDelta =0$. In general, when $\varDelta >0$ the three real roots are written using trigonometric functions as follows

$$\begin{aligned} r_k= & {} 2 \sqrt{\frac{-p}{3}} \cos \left( \frac{1}{3}\arccos {\left( \frac{3q}{2p}\sqrt{\frac{3}{-p}}\right) }\right. \nonumber \\&+ \left. \frac{2k\pi }{3}\right) -\frac{a_2}{3a_1}\quad \text {with}\quad k=0,1,2. \end{aligned}$$

(85)

Fig. 13 shows the roots $r_k$ ($k=0,1,2$) as functions of the noise level $\sigma$ using again the parameters (45) and $x_{y_0}=0.01$. The stochastic dynamic bifurcation point, denoted ${\hat{y}}^{\text {dyn}}_{\text {stoch,a}}$, is equal to $r_0$ if $\sigma <\sigma _3$ and to $r_2$ if $\sigma _3<\sigma <\sigma _2$. This choice is justified by means of a numerical resolution which shows that this is the unique positive solution of Eq. (80).

E Static bifurcation diagram

The static bifurcation diagram is the result of the bifurcation analysis of a deterministic dynamical system with constant bifurcation parameter, here Eq. (6) with a constant value of y. It plots, as a function of the considered bifurcation parameter (here y), the possible steady-state regimes (fixed points and periodic motions) indicating their stability.

We use here the averaging procedure to obtain the approximated analytical bifurcation diagram of the one-mode model. For that Eq. (27) is considered without noise and with a constant bifurcation parameter y, i.e.,

$$\begin{aligned} \frac{\mathrm {d}x_t}{\mathrm {d}t}=F(x_t,y), \end{aligned}$$

(86)

where the function $F(x_t,y_t)$ is given by Eq. (20).

The fixed points $x^e$ of (86) are obtained by solving $F(x,y)=0.$ We obtain three solutions: the trivial solution $x^e_1=0$ and two non-trivial solutions, one is negative and one is positive. Only the positive non-trivial solution is retained and denoted $x^e_2$, its expression is

$$\begin{aligned} x^e_2=4\sqrt{\frac{\frac{2 \zeta F_1 (3 \hat{\gamma }^{\text {st}} +3 y-1)}{3 \omega _1 \sqrt{\hat{\gamma }^{\text {st}} +y}}-\frac{4 \alpha _1}{3}}{\frac{\zeta F_1 (\hat{\gamma }^{\text {st}} +y+1)}{\omega _1 (\hat{\gamma }^{\text {st}} +y)^{5/2}}}} \end{aligned}$$

(87)

where the expression of ${\hat{y}}^{\text {st}}$ is given by Eq. (7). In the lossless case with $\alpha _1=0$ and $\hat{\gamma }^{\text {st}}=\frac{1}{3}$, Eq. (87) reduces to $x^e_2=4 \sqrt{2} (3 y+1) \sqrt{\frac{y}{9 y+12}}$.

The trivial solution corresponds to zero equilibrium position of (6), whereas the non-trivial solution characterizes its periodic steady-state regimes.

As we know, the trivial fixed point $x^e_1$ is stable if $y<0$ and unstable if $y>0$. One can be shown that the non-trivial fixed point $x^e_2$ exists only for $y>0$ and is stable.