Chaotic dynamics of a continuous and discrete generalized Ziegler pendulum

We present analytical and numerical results on integrability and transition to chaotic motion for a generalized Ziegler pendulum, a double pendulum subject to an angular elastic potential and a follower force. Several variants of the original dynamical system, including the presence of gravity and friction, are considered, in order to analyze whether the integrable cases are preserved or not in presence of further external forces, both potential and non-potential. Particular attention is devoted to the presence of dissipative forces, that are analyzed in two different formulations. Furthermore, a study of the discrete version is performed. The analysis of periodic points, that is presented up to period 3, suggests that the discrete map associated to the dynamical system has not dense sets of periodic points, so that the map would not be chaotic in the sense of Devaney for a choice of the parameters that corresponds to a general case of chaotic motion for the original system.

💡 Research Summary

This paper presents a comprehensive analytical and numerical study on the chaotic dynamics of a generalized Ziegler pendulum, which is a double pendulum subjected to an angular elastic potential and a non-conservative follower force. The authors investigate the conditions for integrability and the transition to chaotic motion, considering several modifications to the original system.

The core model consists of three point masses connected by two massless rods, with torsional springs at the hinges and a follower force of constant magnitude acting along the lower rod. Using the angles φ1 and φ2 as generalized coordinates, the equations of motion are derived via the Lagrangian formalism. Prior results indicating integrability for k2=0 under two specific symmetries (F=0, Hamiltonian case; or Δ=0, non-Hamiltonian case) are recalled. The integrability stems from the cyclicity of the variable φ2 when k2=0.

The analysis first examines the effect of gravity. Introducing a gravitational potential generally destroys the cyclicity of φ2 and thus the integrability. However, the authors demonstrate that an ad-hoc, angle-dependent form of the follower force (F = -Mg cos φ2 / sin φ1) can be chosen to make the generalized force r2 vanish, effectively recovering an integrable-like system. Numerical simulations illustrate a sensitive dependence on parameters, where a tiny change in length l3 can trigger a sharp transition from periodic to chaotic motion, as confirmed by Poincaré sections and the computation of Lyapunov exponents.

Next, the influence of two additional linear springs connecting the fixed point to the ends of the lower rod is studied. Since the potential of these springs depends only on φ1, the cyclicity of φ2 remains intact. Consequently, the two known integrable cases are preserved. A new symmetry condition (k_OB * l1 = k_OC * l3) is identified, corresponding to the physical situation where the two external elastic forces balance each other.

The paper then explores the role of friction in two distinct formulations. The first model incorporates viscous (Stokes) friction acting on each of the three point masses, introduced via a Rayleigh dissipation function. Numerical experiments show that such dissipation tends to suppress chaos, leading the system to settle into stable limit cycles. The second model considers a rotational resistance torque at the hinges. This formulation appears to impose stronger regularity, with chaotic behavior observed only for specific initial conditions.

A significant portion of the work is dedicated to studying a discrete-time version of the system. The continuous equations of motion are discretized using a simple Euler method, resulting in a four-dimensional map. The authors analyze the existence and density of periodic points of this map up to period 3. Their findings suggest that, for a parameter choice corresponding to chaotic motion in the original continuous system, the set of periodic points of the discrete map might not be dense. This implies that the map may not satisfy one of the conditions for chaos in the sense of Devaney (specifically, topological transitivity often associated with a dense set of periodic points), highlighting a potential discrepancy between the chaotic properties of the continuous system and its discrete-time approximation.

In conclusion, the research confirms that gravity typically breaks the integrability of the Ziegler pendulum, though it can be restored with a specific force design. Linear additional springs do not affect the known integrable scenarios. Dissipative forces generally promote regularity and stable limit cycles. Finally, the study raises an important caveat: the chaotic behavior of the continuous system does not necessarily guarantee that an associated discrete map will be chaotic according to rigorous mathematical definitions, underscoring the need for careful interpretation in numerical analyses. Future research directions include investigating other friction models and conducting a more thorough analysis of the periodic point structure in the discrete map.