Consider ($\epsilon = 0.1$)
\begin{equation}\label{eq:general eq}
\begin{aligned}
\dot{x}_1(t) &= x_1(x_1-0.5)(x_1+0.5)+\epsilon x_2\\
\dot{x}_2(t) &= x_2(x_2-0.5)(x_2+0.5)+\epsilon x_1
\end{aligned}
\end{equation}
Note that:
- If consider the following ODE $$\dot{x}(t) = x(x-0.5)(x+0.5)+\epsilon x,$$ we have three equilibrium points: $\{-0.63,0,0.63\}$. $x=0$ is the stable equilibrium point. The basin of attraction of $x=0$ is $[-0.63,0.63]$.
Go back to the top coupled ODE. Let $$\begin{bmatrix}x_1(0) \\ x_2(0) \end{bmatrix} = \begin{bmatrix}0.63 \\ 0.63 \end{bmatrix}.$$
By doing so (same initial condition of $x_1$ and $x_2$), the top coupled ODE can be reduced to (since both are the same)$$\dot{x}(t) = x(x-0.5)(x+0.5)+\epsilon x,$$ with $x(0) = 0.63$. So $$\begin{bmatrix}x_1(0) \\ x_2(0) \end{bmatrix} = \begin{bmatrix}0.63 \\ 0.63 \end{bmatrix}.$$ is in the basin of attraction of the top coupled ODE.
However, based on the phase plotter, it shows that
$$\begin{bmatrix}x_1(0) \\ x_2(0) \end{bmatrix} = \begin{bmatrix}0.63 \\ 0.63 \end{bmatrix}$$ is not in the basin of attraction of the top coupled ODE.
($\epsilon = 0.1$)
Can anyone please point out where I am wrong?
Best Answer
Hint.
A good stream plot can help a lot!
In red the equilibrium points for $\epsilon = 0.1$.
The corresponding MATHEMATICA script.