I have experience of working with simple systems of ODE's, but recently I found out about Reaction-Diffusion systems.I'm curious if I have a system of differential equations with diffusion terms like below, what are the required mathematical tools to inspect behavior of the system for Pattern formation, Bifurcation, and Chaos? For instance, I know if I linearize an RD system around the stable points of the system in the absence of diffusion, new eigenvalues of Jacobian emerge due to diffusion and those points, and complex eigenvalues lead to Turing pattern formation. $$ \dot{v}_1 = f_1(v_1,v_2,...,v_n) +D_1\nabla^2(v_1)\\ \dot{v}_2 = f_2(v_1,v_2,...,v_n)+D_2\nabla^2(v_2)\\ ...\\ \dot{v}_n = f_n(v_1,v_2,...,v_n)+D_n\nabla^2(v_n)$$
How to find out the necessary conditions for Turing-patterns to form? How do I know if such system has a bifurcation behavior at any point? How do I make sure the system doesn't show any chaotic behavior at any point?