Stability of equilibrium for an inverted two-link mathematical pendulum with critical tracking forces

We investigate nonlinear stability for equilibrium of a pendulum with viscoelastic components. The tracking force is chosen so that the matrix of the linearized part of the perturbed motion has two purely imaginary roots or one zero and one negative root. The other two roots are complex with negative real part. The boundary of the domain of stability is divided into “dangerous” and “safe” (in the sense of Bautin) zones.