In nonlinear dynamical system self-excited vibrations frequently occur where an equilibrium undergoes a Hopf bifurcation and limit-cycle oscillations develop.
The Hopf bifurcation has two types, supercritical (soft) and subcritical (dangerous). The type of the bifurcation depends on whether the nonhyperbolic equilibrium is weakly stable or unstable. The stability of the equilibrium (and thus the type of the Hopf bifurcation) is determined by the sign of the so-called Poincaré-Lyapunov constant.
This talk discusses three short topics centered around Poincaré-Lyapunov constants:
1, We pose and affirmatively answer the question whether the stability of a nonlinear center can be determined from the eigenvalues of the Jacobian matrix AWAY from the equilibrium point.
2, We recognize that the Poincaré-Liapunov constant is a quadratic form in a 10-dimensional space of the coefficients associated with the normal form of a Hopf bifurcation. This real manifold (the "Hopf quadric") separates regions of the parameter space corresponding to supercritical and subcritical bifurcations.
The stationary points of the squared distance function from a parameter point to the Hopf quadric are the real zeros of a univariate algebraic equation. The distance to the quadric is the minimal positive zero of this equation. This distance can be used as a measure of the "criticality" of the bifurcation. Joint work with Alexei Yu. Uteshev.
3, We use the so-called Carleman embedding technique to recast the normal form of a Hopf bifurcation as an infinite-dimensional linear system. We describe the connection between the Poincaré-Lyapunov constants and the linear algebraic properties of the Carleman matrices. This connection provides a new algorithm to compute Poincaré-Lyapunov constants. Joint work with Csanád Hubay.