Conjecture: Non-dense periodic points for the discrete generalized Ziegler pendulum map (Δ = 0)

Prove that the discrete map f: R^4 → R^4 for the generalized Ziegler pendulum with Δ = 0, defined by the update rules 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 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 b), with parameters a > 0, b > 0 and real constants k_1, k_2, c, does not have a dense set of periodic points in R^4.

Background

The paper introduces a discrete map associated with the generalized Ziegler pendulum by replacing time derivatives with forward iterates on variables (x, y, z, ω) and parameters a, b, k_1, k_2, c, with Δ = 0. The authors analyze fixed points and periodic points up to period 3 for this map and prove that none of these sets are dense in R^4.

Building on these partial results, the authors formally state a conjecture that the map lacks a dense set of periodic points for arbitrary periods, which would imply the map is not chaotic in the sense of Devaney despite the continuous system exhibiting chaotic behavior for general parameter choices.