Conjecture: Non-Devaney chaos for the discrete Ziegler pendulum map with pin friction (Δ = 0)

Establish that the discrete map f: R^4 → R^4 for the generalized Ziegler pendulum with Δ = 0 and pin friction, defined by x_{n+1} = y_n; y_{n+1} = [−(a + b) k_1 x_n + a k_2 z_n + a c sin(x_n) − a μ_O ω_n cos(2 z_n) − a μ_A y_n cos(x_n + 2 z_n)]/(a b); z_{n+1} = ω_n; ω_{n+1} = [a k_1 x_n − a k_2 z_n − a c sin(x_n) + a μ_O ω_n cos(2 z_n) + a μ_A y_n cos(x_n + 2 z_n)]/(a b), with parameters a > 0, b > 0 and real constants k_1, k_2, c, μ_O, μ_A, is not chaotic in the sense of Devaney (i.e., does not have both topological transitivity and a dense set of periodic points).

Background

After showing that for the frictionless discrete map with Δ = 0 the sets of periodic points up to period 3 are not dense, the authors consider a discrete map modified to include torsional friction on the pins. They note a symmetry in the added friction terms and argue that the same reasoning about periodic points should extend to this variant.

Based on these observations, they conjecture that the frictional discrete map is likewise not chaotic in the sense of Devaney, due to having identical fixed points and similar structures of periodic points compared with the frictionless map.