Assignment

Coupled ODEs: Millennium Bridge

This is a joint assignment with the main course Mathematical Modelling. Follow the official handout posted on the main course Blackboard: Kuramoto_assignment.pdf.

Your main goal is to simulate the Millennium Bridge crowd–bridge synchrony model from the handout. Conceptually, it is Kuramoto-like: pedestrians are phase oscillators, and their coupling is mediated by the bridge motion (Strogatz et al. (2005); Eckhardt et al. (2007)).

Upload your submission to the Mathematical Modelling course (it counts towards both courses). This time, assignments are individual.

Summary of Goals

Below is a brief summary of the goals; use the handout for full details and instructions.

Kuramoto Model (Coupled Oscillators)

Start with the basic all-to-all Kuramoto ODE (Kuramoto (2003); Strogatz (2000)) \[\dot\theta_i = \omega_i + \frac{K}{N}\sum_{j=1}^N \sin(\theta_j-\theta_i),\qquad i=1,\dots,N. \tag{1}\]

From this, you can implement the following tasks:

Plot $r(t)$ (order parameter) versus time. $r_\infty\approx 0$ for $K<K_c$ (up to fluctuations) and $r_\infty\in(0,1)$ for $K>K_c$, with fluctuations typically decreasing like $\mathcal{O}(N^{-1/2})$.
Animate oscillators on the unit circle (optionally plot the centroid $r e^{i\Psi}$).
Estimate $K_c$ numerically by sampling multiple $K$ values and plotting $r_\infty(K)$ with error bars. We suggest averaging over a tail window to reduce fluctuations.
Compare with the theory in Strogatz (2000) (implicit curve for $r_\infty(K)$) and the special case of the Cauchy distribution.

Millennium Bridge Crowd Synchrony

Modeling assumptions:

The bridge is a single lateral vibration mode (damped harmonic oscillator).
Each pedestrian is a phase oscillator whose lateral forcing is periodic in its stepping phase.

Bridge dynamics: \[M\,\ddot X(t)+B\,\dot X(t)+KX(t)=\sum_{i=1}^N G\sin\theta_i(t),\qquad \Omega=\sqrt{K/M}. \tag{2}\]

Pedestrian phase dynamics (see Strogatz et al. (2005)): \[\dot\theta_i(t)=\Omega_i + C\,A(t)\,\sin\big(\Psi(t)-\theta_i(t)+\alpha\big),\qquad i=1,\dots,N. \tag{3}\]

Bridge amplitude/phase definitions: \[X(t)=A(t)\sin\Psi(t),\quad \dot X(t)=A(t)\Omega\cos\Psi(t), \tag{4}\] \[A(t)=\sqrt{X(t)^2+\Big(\dot X(t)/\Omega\Big)^2},\qquad \Psi(t)=\operatorname{atan2}\!\Big(X(t),\dot X(t)/\Omega\Big). \tag{5}\]

Synchronization measure: \[R(t)e^{i\Phi(t)}=\frac{1}{N}\sum_{j=1}^N e^{i\theta_j(t)}. \tag{6}\]

Critical crowd size (simplest case $\alpha=\pi/2$ and symmetric $P(\Omega)$ about the bridge frequency): \[N_c=\frac{2B\Omega}{\pi G C\,P(\Omega)}. \tag{7}\]

Wobbling and synchronization emerge together when $N$ crosses $N_c$ (see also Strogatz et al. (2005)).

Required (Millennium Bridge experiment)

Use scipy.integrate.solve_ivp to run the controlled crowd ramp experiment:

Start with $N=N_0$ pedestrians. Every $\Delta T$ seconds, increase the crowd by $\Delta N$ (add new pedestrians with fresh $\theta_i(0)$ and $\Omega_i$) until reaching $N_{\max}$.
Compute $N_c$ using the formula in the handout (using your chosen parameters/distribution).
Define $t_c=\inf\{t: N(t)\ge N_c\}$.
Produce three stacked plots and draw a vertical dashed line at $t=t_c$ labeled “$N=N_c$”:
- $N(t)$ vs $t$ (crowd size staircase)
- $A(t)$ vs $t$ (bridge wobble amplitude)
- $R(t)$ vs $t$ (degree of synchronization)
Briefly describe what happens before/after $t_c$ and whether the observed onset is close to the theoretical prediction.

Reference code is available in amlab/odes_coupled/bridge.py. Use it for ideas, but do not copy it.

Good luck and enjoy your coding!

References

Eckhardt, Bruno, Edward Ott, Steven H. Strogatz, Daniel M. Abrams, and Alan McRobie. 2007. “Modeling Walker Synchronization on the Millennium Bridge.” Physical Review E 75: 021110.

Kuramoto, Yoshiki. 2003. Chemical Oscillations, Waves, and Turbulence. Dover Publications.

Strogatz, Steven H. 2000. “From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators.” Physica D: Nonlinear Phenomena 143: 1–20.

Strogatz, Steven H., Daniel M. Abrams, Alan McRobie, Bruno Eckhardt, and Edward Ott. 2005. “Crowd Synchrony on the Millennium Bridge.” Nature 438: 43–44.

--- title: "Assignment" subtitle: "Coupled ODEs: Millennium Bridge" format: html --- This is a joint assignment with the main course **Mathematical Modelling**. Follow the official handout posted on the main course Blackboard: [Kuramoto_assignment.pdf](../../Kuramoto_assignment.pdf). Your main goal is to **simulate the Millennium Bridge crowd–bridge synchrony model** from the handout. Conceptually, it is Kuramoto-like: pedestrians are phase oscillators, and their coupling is mediated by the bridge motion (@strogatz2005bridge; @eckhardt2007modeling). Upload your submission to the Mathematical Modelling course (it counts towards both courses). This time, assignments are **individual**. ## Summary of Goals Below is a brief summary of the goals; use the handout for full details and instructions. ### Kuramoto Model (Coupled Oscillators) Start with the basic all-to-all Kuramoto ODE (@kuramoto2003chemical; @strogatz2000kuramoto) $$\dot\theta_i = \omega_i + \frac{K}{N}\sum_{j=1}^N \sin(\theta_j-\theta_i),\qquad i=1,\dots,N.$$ {#eq-ode-coupled-assignment-1} From this, you can implement the following tasks: - Plot $r(t)$ (order parameter) versus time. $r_\infty\approx 0$ for $K<K_c$ (up to fluctuations) and $r_\infty\in(0,1)$ for $K>K_c$, with fluctuations typically decreasing like $\mathcal{O}(N^{-1/2})$. - Animate oscillators on the unit circle (optionally plot the centroid $r e^{i\Psi}$). - Estimate $K_c$ numerically by sampling multiple $K$ values and plotting $r_\infty(K)$ with error bars. We suggest averaging over a tail window to reduce fluctuations. - Compare with the theory in @strogatz2000kuramoto (implicit curve for $r_\infty(K)$) and the special case of the Cauchy distribution. ### Millennium Bridge Crowd Synchrony Modeling assumptions: - The bridge is a single lateral vibration mode (damped harmonic oscillator). - Each pedestrian is a phase oscillator whose lateral forcing is periodic in its stepping phase. Bridge dynamics: $$M\,\ddot X(t)+B\,\dot X(t)+KX(t)=\sum_{i=1}^N G\sin\theta_i(t),\qquad \Omega=\sqrt{K/M}.$$ {#eq-ode-coupled-assignment-2} Pedestrian phase dynamics (see @strogatz2005bridge): $$\dot\theta_i(t)=\Omega_i + C\,A(t)\,\sin\big(\Psi(t)-\theta_i(t)+\alpha\big),\qquad i=1,\dots,N.$$ {#eq-ode-coupled-assignment-3} Bridge amplitude/phase definitions: $$X(t)=A(t)\sin\Psi(t),\quad \dot X(t)=A(t)\Omega\cos\Psi(t),$$ {#eq-ode-coupled-assignment-4} $$A(t)=\sqrt{X(t)^2+\Big(\dot X(t)/\Omega\Big)^2},\qquad \Psi(t)=\operatorname{atan2}\!\Big(X(t),\dot X(t)/\Omega\Big).$$ {#eq-ode-coupled-assignment-5} Synchronization measure: $$R(t)e^{i\Phi(t)}=\frac{1}{N}\sum_{j=1}^N e^{i\theta_j(t)}.$$ {#eq-ode-coupled-assignment-6} Critical crowd size (simplest case $\alpha=\pi/2$ and symmetric $P(\Omega)$ about the bridge frequency): $$N_c=\frac{2B\Omega}{\pi G C\,P(\Omega)}.$$ {#eq-ode-coupled-assignment-7} Wobbling and synchronization emerge together when $N$ crosses $N_c$ (see also @strogatz2005bridge). ## Required (Millennium Bridge experiment) Use `scipy.integrate.solve_ivp` to run the controlled **crowd ramp** experiment: 1. Start with $N=N_0$ pedestrians. Every $\Delta T$ seconds, increase the crowd by $\Delta N$ (add new pedestrians with fresh $\theta_i(0)$ and $\Omega_i$) until reaching $N_{\max}$. 2. Compute $N_c$ using the formula in the handout (using your chosen parameters/distribution). 3. Define $t_c=\inf\{t: N(t)\ge N_c\}$. 4. Produce three stacked plots and draw a vertical dashed line at $t=t_c$ labeled "$N=N_c$": - $N(t)$ vs $t$ (crowd size staircase) - $A(t)$ vs $t$ (bridge wobble amplitude) - $R(t)$ vs $t$ (degree of synchronization) 5. Briefly describe what happens before/after $t_c$ and whether the observed onset is close to the theoretical prediction. Reference code is available in [amlab/odes_coupled/bridge.py](../../amlab/odes_coupled/bridge.py). Use it for ideas, but **do not copy it**. \vspace{1cm} **Good luck and enjoy your coding!**