Forecasting the post-bifurcation dynamics of large-dimensional slow-oscillatory systems using critical slowing down and center space reduction

Abstract

This paper presents a method for forecasting Hopf bifurcations and the post-bifurcation dynamics in complex large-dimensional nonlinear systems. This method is based on the phenomenon of critical slowing down and the projection of the system dynamics on the center space. The approach does not require a model of the system because forecasting is based only on time-series measurements of the system response to perturbations (only) in the pre-bifurcation regime. Techniques are introduced to increase forecasting accuracy and to enable the use of data from recoveries after large perturbations. In contrast to other existing methods, the proposed technique solves the problem of insufficient data due to low-frequency system oscillations. The proposed approach is applied to a nonlinear aeroelastic system which is exposed to gust loads. Numerical simulations show that the method accurately predicts the bifurcation point, the bifurcation type, and the post-bifurcation dynamics in both supercritical and subcritical cases despite the fact that it uses only pre-bifurcation regime data and it does not use a model of the system.

Appendix

Consider the following damped single degree of freedom nonlinear system. For clarity, the nonlinearity is assumed to be cubic as

$$\begin{aligned} {\ddot{x}}+c{\dot{x}}+\omega ^2 x=\gamma x^{3}. \end{aligned}$$

(24)

Writing this equation in state space form gives

$$\begin{aligned} \left\{ {{\begin{array}{l} {{\dot{x}}} \\ {{\ddot{x}}} \\ \end{array} }} \right\} =\left[ {{\begin{array}{ll} 0&{} 1 \\ {-c}&{} {-\omega ^{2}} \\ \end{array} }} \right] \left\{ {{\begin{array}{l} x \\ {{\dot{x}}} \\ \end{array} }} \right\} +\left\{ {{\begin{array}{l} 0 \\ {\gamma x^{3}} \\ \end{array} }} \right\} , \end{aligned}$$

(25)

where the states are chosen to be position and velocity.

Next, define the following transformation

$$\begin{aligned} \mathbf{x}=\Phi \mathbf{y}, \end{aligned}$$

(26)

where $\mathbf{x}$ is the state variables vector($\mathbf{x}=[{\begin{array}{ll} x&{} {{\dot{x}}} \\ \end{array} }]^{T}), \mathbf{y}$ is vector containing the complex modal coordinates $\mathbf{y}=[{\begin{array}{ll} {y_1 }&{} {y_2 } \\ \end{array} }]^{T}$, and $\Phi$ is the matrix of complex mode shapes of the system. Matrix $\Phi$ can be written as

$$\begin{aligned} \Phi =\left[ {{\begin{array}{ll} 1&{} 1 \\ {\lambda _1 }&{} {\bar{{\lambda }}_1 } \\ \end{array} }} \right] , \end{aligned}$$

(27)

where $\lambda _1 =(-c-ir)/2$, and $r=\sqrt{4\omega ^{2}-c^{2}}$.

Substituting (26) into (25), solving for $y_1$ and $y_2$, and defining $a=\lambda _1 +i\omega$, one obtains the following expressions which can be used in the normal form theory

$$\begin{aligned}&{\dot{y}}_1 =-i\omega y_1 +a y_1 +\frac{\gamma }{\lambda _1 - \lambda _2 }(y_1 +y_2 )^{3}, \nonumber \\&{\dot{y}}_2 =i\omega y_2 +\bar{{a}}y_2 -\frac{\gamma }{\lambda _1 -\lambda _2 }(y_1 +y_2 )^{3},\nonumber \\&{\dot{a}}=0. \end{aligned}$$

(28)

Next, one may introduce the following nonlinear change of coordinates

$$\begin{aligned} y_1= & {} u_1 +\left( {\frac{-\gamma }{4\omega ^{2}}u_1^3 +\frac{3\gamma }{4\omega ^{2}}u_1 u_2^2 +\frac{\gamma }{8\omega ^{2}}u_2^3 } \right) ,\nonumber \\ y_2= & {} u_2 +\left( {\frac{-\gamma }{4\omega ^{2}}u_2^3 +\frac{3\gamma }{4\omega ^{2}}u_2 u_1^2 +\frac{\gamma }{8\omega ^{2}}u_1^3 } \right) . \end{aligned}$$

(29)

Using normal form theory, one can show that the system can be transformed using Eq. (29) into the following normal form up to $3^{\mathrm{rd}}$ order

$$\begin{aligned} {\dot{u}}_1= & {} -i\omega u_1 +au_1 +\frac{3i\gamma }{2\omega }u_1^2 u_2 ,\nonumber \\ {\dot{u}}_2= & {} i\omega u_2 +\bar{{a}}u_2 -\frac{3i\gamma }{2\omega }u_2^2 u_1. \end{aligned}$$

(30)

The solution of above system of equations is in the following periodic form

$$\begin{aligned} u_1 =\bar{{u}}_2 =u_0 e^{-ct/2} e^{i{\tilde{\omega }} t}, \end{aligned}$$

(31)

where $u_0$ is a complex number, and ${\tilde{\omega }}$ is real number in the following form

$$\begin{aligned} {\tilde{\omega }} =\frac{-r}{2}+\frac{3\gamma }{2\omega }u_0 \bar{{u}}_0. \end{aligned}$$

(32)

Now, using Eqs. (31) and (29), one may observe that $y_1$ and $y_2$ are complex conjugates. Substituting Eq. (31) into Eq. (29) and assuming small nonlinearities, one can obtain an approximation for the modal coordinates in the following form

$$\begin{aligned} y_1= & {} \bar{{y}}_2 \approx \left( {w_1 (t)\cos ({\tilde{\omega }} t)-w_2 (t)\sin ({\tilde{\omega }} t)} \right) \nonumber \\&+\, i \left( {w_1 (t)\sin ({\tilde{\omega }} t)+w_2 (t)\cos ({\tilde{\omega }} t)} \right) , \end{aligned}$$

(33)

where $w_1 (t)$ and $w_2 (t)$ are functions of time containing exponentially decaying functions and the recovery rate of the system. Following the same procedure as in Eq. (33), one finds that both $w_1 (t)$ and $w_2 (t)$ have the same decay rate. The real and imaginary parts of the expression on the right-hand side of Eq. (33) correspond to $q_1$ and $q_2$ used in the text as real valued modal coordinates, i.e.

$$\begin{aligned} q_1= & {} w_1 (t)\cos ({\tilde{\omega }} t)-w_2 (t)\sin ({\tilde{\omega }} t),\nonumber \\ q_2= & {} w_1 (t)\sin ({\tilde{\omega }} t)+w_2 (t)\cos ({\tilde{\omega }} t). \end{aligned}$$

(34)

Note that $q_1$ and $q_2$ are in quadrature and form a regular spiral in the $q_1 -q_2$ plane; the same form as expressed in Eq. (11). Note also that this conclusion came from the fact that the modal bases were chosen to be position and velocity.

The above procedure can be extended to higher-order systems and nonlinearities. More details can be found in refs. [32,33,34].