Linear Stability Analysis of Multiparameter Dynamical Systems via a Numerical-Perturbation Approach

should write all the equations in scalar form and (inelegantly) drawa trivial bifurcation diagram x 0 in one parameter. As a second strategy, also followed by some users, one could 1) expand the (parameter dependent) characteristic polynomial of the Jacobian matrix,leadingtotwoscalarequationsintherealandimaginaryparts oftheeigenvalue,inwhichtherealpartmustbevanished;2)consider these equations as the right-hand members of a fictitious dynamical system, in which two parameters are taken as dummy state variables and the imaginary part of the eigenvalue is taken as a dummy parameter; and 3) look for equilibrium paths for the dummy system, actually representing the bifurcation loci for the real system. If this procedure is employed, some of the bifurcation points found (e.g., Hopf points) could have no meaning for the true system. Although both of these procedures work, they are highly unsatisfactory from a formal point of view, since they are essentially based on tricks; moreover, theyare also unsatisfactory from a practical point of view, since they require handling a large number of data (scalar equations