On the dynamics and integrability of the Ziegler pendulum

TL;DR

The paper analyzes a planar Ziegler pendulum driven by a follower force, with gravity and friction neglected, but with two springs. It shows that when the pivot spring stiffness $k_2=0$, the reduced 3D dynamics admit two-parameter families of periodic orbits, implying local integrability, though the global phase space can remain non-integrable. For $k_2\neq 0$, the system evolves on $\mathbb{T}^2\times\mathbb{R}^2$ with an invariant measure, and numerical experiments reveal coexistence of regular and chaotic dynamics, consistent with KAM theory for reversible systems. The work highlights a mechanism by which integrability can be lost as parameters vary and provides a framework for understanding the transition between regular and chaotic behavior in non-conservative, circulatory systems.

Abstract

We prove that the Ziegler pendulum -- a double pendulum with a follower force -- can be integrable, provided that the stiffness of the elastic spring located at the pivot point of the pendulum is zero and there is no friction in the system. We show that the integrability of the system follows from the existence of two-parameter families of periodic solutions. We explain a mechanism for the transition from integrable dynamics, for which there exist two first integrals and solutions belong to two-dimensional tori in a four-dimensional phase space, to more complicated dynamics. The case in which the stiffnesses of both springs are non-zero is briefly studied numerically. We show that regular dynamics coexists with chaotic dynamics.

Figures (9)

• Figure 1: The Ziegler pendulum.
• Figure 2: Periodic solutions of system \ref{['eq4']} intersecting the plane $\varphi_1 = 0$ at two points for the case in which $F = 0$. Here $m_1 = 1$, $m_2 = 2$, $l_1 = 3$, $l_2 = 1$, $m_3 = 1$, $l_3 = 4$, $k_1 = 0.275$, $\varphi_1(0) = \pi + 1 + 5j$, $j=0,\dots, 4$, $v_1(0) = 0.1$, $v_2(0) = 0.2$.
• Figure 3: Periodic solutions of system \ref{['eq4']} intersecting the plane $\varphi_1 = 0$ at two points for the case in which $m_1 l_1 = m_3 l_3$. Here $F=2$, $m_1 = 1$, $m_2 = 1$, $l_1 = 2$, $l_2 = 1$, $m_3 = 2$, $l_3 = 1$, $k_1 = 1$, $\varphi_1(0) = \pi + 1 + 5j$, $j=0,\dots, 4$, $v_1(0) = 0.1$, $v_2(0) = 0.2$.
• Figure 4: An example of a periodic solution of system \ref{['eq4']} for $F \ne 0$ and $m_1 l_1 \ne m_3 l_3$. Projection onto the $(\varphi_1, v_1)$-plane. Here $F=6$, $m_1 = 1$, $m_2 = 2$, $l_1 = 1$, $l_2 = 4$, $m_3 = 2$, $l_3 = 2.3$, $k_1 = 3$, $\varphi_1(0) = 0$, $v_1(0) = 0.5$, $v_2(0) = 2.25$.
• Figure 5: An example of a periodic solution of system \ref{['eq4']} for $F \ne 0$ and $m_1 l_1 \ne m_3 l_3$. Projection onto the $(\varphi_1, v_2)$-plane. Here $F=6$, $m_1 = 1$, $m_2 = 2$, $l_1 = 1$, $l_2 = 4$, $m_3 = 2$, $l_3 = 2.3$, $k_1 = 3$, $\varphi_1(0) = 0$, $v_1(0) = 0.5$, $v_2(0) = 2.25$.

Theorems & Definitions (23)

• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof