Normal forms for high co-dimension bifurcations of nonlinear time-periodic systems with nonsemisimple eigenvalues

The normal forms for time-periodic nonlinear variational equations with arbitrary linear Jordan form undergoing bifurcations of high co-dimension are found. First, the equations are transformed via the Lyapunov-Floquet (L-F) transformation into an equivalent form in which the linear matrix is constant with degenerate nonsemisimple linear eigenvalues while the nonlinear monomials have periodic coefficients. By considering the resulting coupling of the bases of the near identity transformation, the solvability condition for an arbitrary Jordan matrix is then derived. It is shown that time-independent and/or time-dependent nonlinear resonance terms remain in the normal form for various Jordan matrices. Specifically, the normal forms for quadratic and cubic nonlinearities with the following linear Jordan forms are explicitly derived: double zero eigenvalues (co-dimension two bifurcation), triple zero eigenvalues (co-dimension three bifurcation), and two repeated pairs of purely imaginary eigenvalues (co-dimension two bifurcation). A commutative system with cubic nonlinearities and a double inverted pendulum with a periodic follower force are used as illustrative examples.