Take into account the next dynamical system in $mathbb{R}^n$
$$
dot{x} = -x + Atanh(x)=:f(x)
$$
the place $x = (x_1,…,x_n) in mathbb{R}^n$, $A$ is an actual matrix with spectral radius $rho(A) < 1$, and $tanh(x) := (tanh(x_1),…,tanh(x_n))$. In all simulations that I’ve run to date, $lim_{t rightarrow infty}x(t) =0$ with all preliminary values $x(0)$, so I’m attempting to show this.
The system has a secure equilibrium at $0$, since $f(0)=0$ and all eigenvalues of $nabla f(0) = -I+A$ have destructive actual half. Because the trajectory of $x$ is bounded, to indicate that $x(t)$ converges to zero as $t rightarrow infty$, I have to show that the equation $x = Atanh(x)$ that no answer aside from zero, or its nonzero options correspond to unstable equilibria.