Chaotic dynamics of a continuous and discrete generalized Ziegler pendulum

Abstract: 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.

• The paper demonstrates that integrability in the generalized Ziegler pendulum arises under precise symmetry and external potential conditions, which are sensitive to minor perturbations.
• It compares continuous and discrete models, revealing that while the continuous system exhibits chaos, its discrete counterpart lacks Devaney chaos due to non-dense periodic points.
• The study highlights how frictional forces induce bifurcations leading to transitions from fixed points to limit cycles and strange attractors with small parameter variations.

Chaotic Dynamics and Integrability in Continuous and Discrete Generalized Ziegler Pendulums

Introduction

The study investigates the complex dynamics of a generalized Ziegler pendulum—a model generalizing the double pendulum subject to both conservative (elastic, gravitational) and non-conservative (follower, friction) forces. The continuous-time system is formulated as a double pendulum with three mass points linked by two massless rods, with elastic potentials at the hinges and a follower force acting at the lower rod. The work comprehensively analyzes integrability, onset of chaos, and how these properties are altered by symmetries, geometry, and dissipative (frictional) effects. Both the continuous and the discrete versions of the system are addressed, with particular attention to the distinction between classical (Liouvillian/Jacobian) integrability and chaos in the sense of Devaney.

Continuous Dynamics: Integrability and Symmetry Breaking

In the absence of non-potential forces (i.e., for vanishing follower force $F$ and for certain mass distributions characterized by $\Delta=0$), the Ziegler pendulum admits integrable regimes. The two prototypical cases are:

• Hamiltonian case ($F=0$): If $k_2=0$, $\varphi_2$ is cyclic and system admits an additional conserved quantity, thus allowing Liouvillian integrability.
• Non-Hamiltonian symmetric case ($\Delta=0$): There remains cyclicity and the system is integrable in the sense of Jacobi.

These integrable classes are robust to small symmetry-breaking perturbations: introducing small $F$ or $\Delta$ only weakly perturbs the integrability—an echo of KAM-type persistence.

However, gravity generically destroys these integrable regimes except in contrived cases with angle-dependent follower force; generically, all symmetry protection is lost and nonlinear dynamics yield chaotic evolution, as evidenced via positive Lyapunov exponents for only slight parameter variation near the transition (e.g., changing $l_3$ by less than $1\%$ switches the dynamics from periodic to chaotic). Isolated islands of regularity persist for specific parameter values, in line with KAM theory expectations.

When linear springs are added (with extensions depending only on lower rod orientation), the cyclicity of $\varphi_2$ is preserved, and all previous integrability results still apply. This shows that chaoticity is highly sensitive to the type (and location) of external potentials, not just their presence.

Geometric and Physical Variants

Variations involving mass redistribution—employing physical rods versus massless connections—do not qualitatively alter the equations of motion. The preserved mathematical structure ensures identical qualitative behavior, both in the presence and absence of chaos. The role of the mass-distribution symmetry is also preserved in all variants; i.e., the distinction between regular and chaotic motion remains controlled by $\Delta$.

Dissipative Dynamics: Effects of Friction

Viscous (Stokes-type) Friction

Introduction of velocity-proportional friction at the masses leads to rich bifurcations in qualitative behavior:

• For a specific symmetry on friction coefficients (analogous to the integrable mass-distribution symmetry), the system evolves to an attractor point (dissipative analog of a fixed point).
• Generic friction coefficient configurations lead to limit cycles, with unconventional approach dynamics (e.g., "jumping" spiral trajectories close to the attractor).

Pin (Torsional) Friction

Friction at the rotational hinges (pins) generates further departure from integrability. Continuous variation in the symmetry-breaking parameter $\Delta$ demonstrates complex transitions from near-periodic to fully chaotic dynamics. For small symmetry-breaking, phase-space portraits exhibit accumulation on pseudo-spherical sets (suggesting a "strange attractor" different from classical Lorenz-type attractors); as symmetry is further broken, the system transitions to complex, highly chaotic regimes. Importantly, the chaotic regime is reached at threshold values for the friction coefficients, as verified by Lyapunov exponent computations, with the maximal exponent peaking at the chaos-onset threshold.

Discrete-Time Dynamics and Devaney-Chaos

A central theoretical investigation is the nature of chaos for an associated discrete map version of the Ziegler system (defined via finite-step approximation to the continuous flow, with canonical variables).

Key results:

• For the discrete map with $\Delta=0$, the set of periodic points (fixed points, 2-cycles, 3-cycles) is always finite and never dense in $\mathbb{R}^4$ for any parameter choice (verified by explicit solution of the periodicity constraints).
• As a result, the map is not chaotic in the sense of Devaney (which requires, among other properties, that periodic points are dense) for parameter choices where the continuous system is chaotic.
• This nontrivial distinction underscores that discrete approximations can fail to inherit the full measure-theoretic/topological complexity of the original continuous chaotic system.

Extending to include friction in the discrete model preserves this structure and the lack of Devaney-chaos, as additional dissipative terms do not change the finiteness of the periodic points' set.

Implications and Future Developments

This work elucidates several crucial points for nonlinear dynamics and engineering design:

• The relationship between integrability and chaos in the Ziegler pendulum is governed by specific symmetries, particularly regarding the distribution of mass and the nature of non-conservative forces.
• The precise type and form of external potentials critically affect the persistence or loss of integrability, highlighting the importance of model details in engineering applications.
• Dissipative effects, particularly their distribution and type, have profound consequences: they can stabilize to fixed points, generate limit cycles, or even produce strange attractors, depending on symmetries.
• The discrete-time analysis suggests that topological chaos (in Devaney's sense) is not inherited from the continuous chaotic system; this cautions against naive extrapolation of results between continuous and time-discretized systems.
• The methodology and findings open the way for deeper bifurcation analysis, more rigorous proofs for the structure of periodic sets, and extensions to control-theoretic interpretations or the application of KAM and Melnikov methods to this and related systems.

Conclusion

The analysis of the generalized Ziegler pendulum, both in continuous and discrete time, underscores the fine balance between integrability and chaos—a balance acutely sensitive to symmetry, force type, and dissipative structure. The result that discrete maps associated with inherently chaotic continuous-time systems may lack Devaney chaos challenges assumptions about the generalization of chaotic properties between domains. The study's rigorous treatment of symmetry-breaking, friction, and external potentials provides a high-resolution road map for both theoretical investigations of nonlinear dynamical systems and for practical design considerations in mechanical systems subject to non-conservative effects.