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Hopf and double Hopf bifurcations in a delayed lateral vibration model of footbridges induced by pedestrians

Xuemei Li, Yechi Liu

TL;DR

This work analyzes a delayed lateral footbridge model driven by pedestrian interaction, showing that only Hopf and double Hopf bifurcations arise as parameters vary. It develops a center-manifold reduction for delay differential equations and derives normal-form amplitude equations to characterize both periodic and quasi-periodic vibrations, including the direction and stability of Hopf bifurcations and the structure of the double Hopf normal form. A key contribution is proving, via KAM theory, the existence of robust quasi-periodic invariant 2-tori near double Hopf points for most parameter values in a specified region, indicating complex, yet structured, vibrational regimes beyond simple periodic motion. The results quantify how time delay and pedestrian–structure coupling influence stability and rich dynamics in footbridge lateral vibrations, with implications for design against excessive lateral oscillations.

Abstract

In this paper, we investigate the dynamical behaviors of a delayed lateral vibration model of footbridges proposed based on the facts that pedestrians will reduce their walking speed or stop walking when the response of the footbridge becomes sufficiently large, and that the bridge velocity can not be changed at once when the pedestrians begin to walk on the bridge. By analyzing the distribution of roots of the associated characteristic equation, we find that there are only two types of bifurcations in this model: Hopf bifurcation and double Hopf bifurcation, and give the condition on the stability of the trivial solution. By using the center manifold theorem and bifurcation theory of delayed differential equations, we obtain the dynamical behavior in these bifurcations, specially including the stability of periodic solutions and invariant tori bifurcating from the trivial solution in these bifurcations. Finally, we prove that this model exhibits quasi-periodic vibrations by KAM theorems, besides periodic vibrations.

Hopf and double Hopf bifurcations in a delayed lateral vibration model of footbridges induced by pedestrians

TL;DR

This work analyzes a delayed lateral footbridge model driven by pedestrian interaction, showing that only Hopf and double Hopf bifurcations arise as parameters vary. It develops a center-manifold reduction for delay differential equations and derives normal-form amplitude equations to characterize both periodic and quasi-periodic vibrations, including the direction and stability of Hopf bifurcations and the structure of the double Hopf normal form. A key contribution is proving, via KAM theory, the existence of robust quasi-periodic invariant 2-tori near double Hopf points for most parameter values in a specified region, indicating complex, yet structured, vibrational regimes beyond simple periodic motion. The results quantify how time delay and pedestrian–structure coupling influence stability and rich dynamics in footbridge lateral vibrations, with implications for design against excessive lateral oscillations.

Abstract

In this paper, we investigate the dynamical behaviors of a delayed lateral vibration model of footbridges proposed based on the facts that pedestrians will reduce their walking speed or stop walking when the response of the footbridge becomes sufficiently large, and that the bridge velocity can not be changed at once when the pedestrians begin to walk on the bridge. By analyzing the distribution of roots of the associated characteristic equation, we find that there are only two types of bifurcations in this model: Hopf bifurcation and double Hopf bifurcation, and give the condition on the stability of the trivial solution. By using the center manifold theorem and bifurcation theory of delayed differential equations, we obtain the dynamical behavior in these bifurcations, specially including the stability of periodic solutions and invariant tori bifurcating from the trivial solution in these bifurcations. Finally, we prove that this model exhibits quasi-periodic vibrations by KAM theorems, besides periodic vibrations.

Paper Structure

This paper contains 10 sections, 11 theorems, 145 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Stability and bifurcation values of \ref{['14']}
  3. The distribution of eigenvalues and stability
  4. Decomposition of DDEs with parameters near a bifurcation value
  5. Hopf bifurcation analysis
  6. Case i@: $\tau$ being regarded as the bifurcation parameter
  7. Case iii@: $\alpha_3-\alpha_2$ being regarded as the bifurcation parameter
  8. Double Hopf bifurcation and quasi-periodic solutions
  9. Double Hopf bifurcation analysis
  10. The existence of quasi-periodic solutions

Key Result

Lemma 1

(i). If $\alpha_3>\alpha_2$, then there exists a sequence $\{\tau_j^\pm\}$ with $\tau_{j+1}^\pm>\tau_j^\pm$ such that 23 has a pair of simple purely imaginary roots $\pm{\rm i}\omega_\pm$ when $\tau=\tau_j^\pm$, respectively, where $\omega_\pm$ are defined by 26, (ii). If $C_1:\,\alpha_3=\alpha_2$, then there exists a sequence $\{\tau_j\}$ with $\tau_{j+1}>\tau_j$ such that 23 has a pair of simpl

Figures (4)

  • Figure 1: The decomposition of the $(\alpha_2,\alpha_3)$-parameter plane on the stability of the zero solution of \ref{['22']}. The curves $C_1:\alpha_3=\alpha_2$ and $C_2:\frac{\alpha_2}{\alpha_3}=\cos\left(\pi(1-\sqrt{\frac{\alpha_3^2-\alpha_2^2}{4+\alpha_3^2-\alpha_2^2}})\right)$ divide the first quadrant of the $(\alpha_2,\alpha_3)$-parameter plane into three regions: $D_1,D_2$ and $D_3$
  • Figure 2: The Hopf bifurcation curves $\tau=\tau_k^+(\alpha_3)$ and $\tau=\tau_l^-(\alpha_3)$ of \ref{['22']} for (a) $\alpha_2=0.5$ and (b) $\alpha_2=1$. The intersection points of $\tau_k^+$ and $\tau_l^-$ are possibly double Hopf bifurcation points where the characteristic equation \ref{['23']} has two pairs of simple imaginary roots $\pm{\rm i}\omega_+$ and $\pm{\rm i}\omega_-$.
  • Figure 3: Bifurcation diagram of \ref{['413']} regarding $\sigma_1$ and $\sigma_2$ as bifurcation parameters.
  • Figure 4: The parametric bifurcation portrait of \ref{['416']}.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Lemma 4
  • Theorem 2
  • Theorem 3
  • Lemma 5
  • Theorem 4
  • Theorem 5
  • ...and 3 more