Table 7.

Hopf normal form: here, we report discovered governing equations using noisy measurement data, representing dynamics of Hopf bifurcation. We note that RK4-SINDy recovers the Hopf normal form very accurately; on the other hand, Std-SINDy breaks down.

method	discovered model
RK4-SINDy	$\begin{array}{rl}\dot{\mathbf{\text{x}}}(t)& \hspace{0.17em}=1.001\mu \mathbf{\text{x}}(t)-1.001\mathbf{\text{y}}(t)-0.996\mathbf{\text{x}}(t)(\mathbf{\text{x}}{(t)}^2+\mathbf{\text{y}}{(t)}^2)\\ \dot{\mathbf{\text{y}}}(t)& \hspace{0.17em}=1.001\mathbf{\text{x}}(t)+1.010\mu \mathbf{\text{y}}(t)-1.006\mathbf{\text{x}}{(t)}^2\mathbf{\text{y}}(t)-1.004\mathbf{\text{y}}{(t)}^3\end{array}$
Std-SINDy	$\begin{array}{rl}\dot{\mathbf{\text{x}}}(t)& \hspace{0.17em}=-0.961\mathbf{\text{y}}(t)+0.719\mu \mathbf{\text{x}}(t)+0.822\mu \mathbf{\text{y}}(t)-0.735\mathbf{\text{x}}{(t)}^3-1.044\mathbf{\text{x}}{(t)}^2\mathbf{\text{y}}\\ & \hspace{0.17em}-0.686\mathbf{\text{x}}(t)\mathbf{\text{y}}{(t)}^2-0.846\mathbf{\text{y}}{(t)}^3\\ \dot{\mathbf{\text{y}}}(t)& \hspace{0.17em}=0.986\mathbf{\text{x}}(t)+0.899\mu \mathbf{\text{y}}(t)-0.882\mathbf{\text{x}}{(t)}^2\mathbf{\text{y}}(t)-0.904\mathbf{\text{y}}{(t)}^3.\end{array}$