Consider the Jacobian matrix of $$ \begin{bmatrix} f_{1}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t))\\ \vdots \\ f_{N_2}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t)) \end{bmatrix} $$
Let the jacobian matrix be denoted by $\mathbf{J}$ and let all the eigenvalues of $\mathbf{J}$ be negative.
consider the symmetric matrix $$ \mathbf{A} = \mathbf{J} + \mathbf{J}^\intercal $$
What can we say about the eigenvalues of $\mathbf{A}$? Will they all be negative?
Also, consider a non-zero vector $\mathbf{x}$ What can we say about the sign of: $$ \mathbf{x^\intercal A x}$$ Will it always be negative?
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Consider
$$J=\begin{bmatrix}-1 & 5\\0 & -2\end{bmatrix}.$$
We have that
$$A:=J+J^T=\begin{bmatrix}-2 & 5\\5 & -4\end{bmatrix}$$
which is not negative definite.
As a result, it is not possible to provide a general statement that if $J$ is Hurwitz stable, then $J+J^T$ is negative definite. The structure of the matrix here will play an essential role.