The discontinuous limit case of an archetypal oscillator with constant excitation and van der Pol damping: A single equilibrium

TL;DR

The paper analyzes the discontinuous limit case of a constant-excitation oscillator with van der Pol damping, focusing on the regime $a>1$ where a single equilibrium $E_r=(a+1,F(a+1))$ governs the dynamics. It derives a global bifurcation structure with Hopf, grazing, and double-crossing surfaces, and shows that up to three limit cycles can encircle the single equilibrium, using Melnikov analysis for small $\delta$ and rotated vector fields for general $\delta$ to establish a complete global picture. The authors provide rigorous proofs (Theorems $\mathrm{mr1}$ and $\mathrm{mr2}$) and supplement them with numerical examples that illustrate grazing, three-crossing scenarios, and the interactions between surfaces $G$, $DL_1$, and $DL_2$. The results extend prior work on systems with multiple equilibria and reveal a richer set of global dynamics even in the single-equilibrium discontinuous case, with implications for nonsmooth oscillator theory and bifurcation analysis.

Abstract

This paper investigates the global dynamics of the discontinuous limit case of an archetypal oscillator with constant excitation that exhibits a single equilibrium. For parameter regions in which this oscillator possesses two or three equilibria, the global bifurcation diagram and the corresponding phase portraits on the Poincare disc have been presented in [Phys. D, 438 (2022) 133362]. The present work completes the global structure of the discontinuous limit case of an archetypal oscillator with constant excitation. Although the dynamical phenomena are less rich compared to systems with more than one equilibrium, the presence of a single equilibrium gives rise to additional limit cycles surrounding it, thereby enriching the overall dynamics and making the analysis substantially more intricate than in the previously studied cases.

Figures (14)

• Figure 1: A shallow elastic arch and the simplified oscillator.
• Figure 2: The slice $\delta=\delta_*$ of the global bifurcation diagram in $\mathcal{G}$.
• Figure 3: The global phase portraits in $\mathcal{G}$.
• Figure 4: Possible connections of the grazing orbit and the saddles at infinity.
• Figure 5: Dynamical behaviors near $I_x$ and $I_y$.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Lemma 2.1
• proof
• Proposition 2.1
• proof
• Lemma 2.2
• Lemma 3.1
• proof