Probing the phase space of coupled oscillators with Koopman analysis

With the development of probing and computing technology, the study of complex systems has become a necessity in various science and engineering problems, which may be treated efficiently with Koopman operator theory based on observed time series. In the current paper, combined with a singular value decomposition (SVD) of the constructed Hankel matrix, Koopman analysis is applied to a system of coupled oscillators. The spectral properties of the operator and the Koopman modes of a typical orbit reveal interesting invariant structures with periodic, quasiperiodic, or chaotic motion. By checking the amplitude of the principal modes along a straight line in the phase space, cusps of different sizes on the magnitude profiles are identified whenever a qualitative change of dynamics takes place. There seems to be no obstacle to extend the current analysis to high-dimensional nonlinear systems with intricate orbit structures.

Figure 1

The $n$-DOF nonlinear oscillator system. $x_i$ is the displacement of the $i\mathrm{th}$ oscillator, with $k$ being the elastic coefficient and $c$ controlling the nonlinearity.

Figure 2

The images on the Poincaré section Eq. (A2) of four typical orbits of the oscillator system Eq. (A3) with $k=0.4,c=3$.

Figure 3

The magnitude of the Koopman modes in a descending order for the oscillator system Eq. (A3) with $k=0.4,c=3$ and the initial point $x_{01}=0,x_{02}=0,x_{04}=0.056465$.

Figure 4

A comparison of the obtained frequencies by the FFT (blue lines) and the Koopman technique (red lines) for the oscillator system Eq. (A3) with $k=0.4,c=3$ and the initial point $x_{01}=0,x_{02}=0,x_{04}=0.056465$. The first five dominant frequencies obtained by the Koopman method are plotted. The ordinate is the amplitude and the abscissa is the frequency.

Figure 5

The variation of the magnitude of the first two Koopman modes of Eq. (A3) with $k=0.4,c=0,h=0.4$ along a straight line defined by $x_{01}=0,x_{02}=0$, and $x_{04}\in [-0.89,0.89]$. (a) The blue and the red dashed line mark the magnitude of mode 1 and 2; (b) the frequency curves of mode 1 and 2.

Figure 6

The variation of the magnitude of the first two Koopman modes of Eq. (A3) with $k=0.4,h=0.4$ along a straight line defined by $x_{01}=x_{02}=0$ and $x_{04}\in [-0.89,0.89]$. The blue and the red dashed line represent the magnitude of mode 1 and 2 (a, c) with their frequency curves marked by the brown lines (b, d). The parameter $c=1.9$ (a, b) and $c=2$ (c, d).

Figure 7

Typical orbits of Eq. (A3) on the Poincaré section defined by Eq. (A2) with $k=0.4,c=1.9,h=0.4$. The initial points have $x_1,x_2=0,x_{04}\in [-0.89,0.89]$.

Figure 8

Typical orbits of Eq. (A3) on the Poincaré section defined by Eq. (A2) with $k=0.4,h=0.4$: (a) for arrow 2 in Fig. 7, (b) for arrow 2 in Fig. 9.

Figure 9

Typical orbits of Eq. (A3) on the Poincaré section defined by Eq. (A2) with $k=0.4,c=2,h=0.4$. The initial points have $x_1,x_2=0,x_{04}\in [-0.89,0.89]$.

Figure 10

Typical orbits of Eq. (A3) on the Poincaré section defined by Eq. (A2) with $k=0.4,h=0.4$ (a) $c=1.9$, (b) $c=2$. The initial points are $x_1,x_2=0$, $x_{04}\in [-0.8,-0.4]$.

Figure 11

The variation of the magnitude of the first three Koopman modes of Eq. (24) with $k=0.4,c=0,h=0.4$ along a straight line defined by $x_{01}=x_{02}=x_{03}=x_{04}=0$ and $x_{06}\in [-0.89,0.89]$. (a) The red, blue, and black lines mark the magnitude of modes 1, 2, and 3, respectively; (b) the frequency curves of modes 1, 2, and 3.

Figure 12

The projection of a few orbits on the Poincaré section of Eq. (24) with $k=0.4,h=0.4$ along a straight line defined by $x_{01}=x_{02}=x_{03}=x_{04}=0$ and $x_{06}\in [-0.89,0.89]$. (a) The orbits 1, 2, 3 for arrows 1, 2, 3 in Fig. 11; (b) the orbits 1, 2, 3 for arrows 1, 3, 5 in Fig. 13, at (a) $c=0$ and (b) $c=0.6$.

Figure 13

The variation of the magnitude of the first three Koopman modes of Eq. (24) with $k=0.4,c=0.6,h=0.4$ along a straight line defined by $x_{01}=x_{02}=x_{03}=x_{04}=0$, and $x_{06}\in [-0.89,0.89]$. (a) The red, blue, and black lines mark the magnitude of modes 1, 2, and 3, respectively; (b) the frequency curves of modes 1, 2, and 3.

Figure 14

The projection of a few orbits on the Poincaré section of Eq. (24) with $k=0.4,c=0.6,h=0.4$ along a straight line defined by $x_{01}=x_{02}=x_{03}=x_{04}=0$, and $x_{06}\in [-0.89,0.89]$. (a) The orbit for arrow 2 in Fig. 13; (b) the orbit for arrow 4 in Fig. 13; (c) the orbit for arrow 6 in Fig. 13.

Figure 15

The Poincaré return map for system (A3) at $k=0.4,c=0$. The initial points start with $x_1=x_2=0,x_4\in [-0.89,0.89]$ with a step size 0.02 and energy $h=0.4$.

Figure 16

Five typical orbits on the Poincaré section (A2) of the two-DOF coupled oscillators described by Eq. (A3) with $k=0.4,c=3$.

Figure 17

The projection of three orbits on the Poincaré section of Eq. (A3) with $k=0.4,c=0.6,h=0.4,x_{01}=x_{02}=0$, (a) $x_{04}=-0.62$, (b) $x_{04}=-0.63$, (c) $x_{04}=-0.635$.

Figure 18

The magnitude of the Koopman modes ordered descendingly for the oscillator system Eq. (A3) with $k=0.4,c=3$ and the initial point $x_{01}=x_{02}=0$, (a) $x_{04}=-0.62$, (b) $x_{04}=-0.63$, (c) $x_{04}=-0.635$.