Analyzing Ikeda Map using Cobweb (Spiderweb) and Bifurcation diagram.

Computational Physics Final Project (Fall 2022, PHY391C)

Authors : Inhwan Kim, Casey Christian, Adam Christensen

Setup

1. Download the repository https://github.com/ihkim94/ikeda.git

2. Download the required packages, numpy, matplotlib, and scipy. If you are using conda, you can download them on your_env_name as,

conda activate your_env_name

conda install numpy matplotlib scipy  

conda activate your_env_name

3. You also have to install ddeint to solve the delay differential equation at here.

Examples

0. Open the main.ipynb file. Run the first shell of the script.

1. Plotting the function of the logistic equation.

logistic_model = LogisticEquation(b=4, c=1)
ic = [0.5]  ## Initial condition
tpts, sol = logistic_model.solve(ic, 0, 20, 10000)


plt.figure(figsize=(20, 10))
plt.title(rf'Logistic function when $\beta = {logistic_model.b}$ and $ c = {logistic_model.c}$ ')
plt.xlabel('Time (s)', fontsize=20)
plt.ylabel('f(x)', fontsize=20)
plt.plot(tpts, sol, color='k')

2. Plotting the function of the ikeda equation using ddeint package.

ic = 2.4 ## initial condition
delay = 1 ## delay time
beta = 8 ## intensity of nonlinear function
tau = 0.08  ## the system always has a response time

def values_before_zero(t):
    return ic

def model(y, t):
    return (beta/tau)*(np.sin(y(t - delay))**2) - y(t)/tau

def solve(t_min, t_max, nt):
    tpts = np.linspace(t_min, t_max, nt)
    sol = ddeint(model, values_before_zero, tpts)
    return tpts, sol

tpts, sol = solve(0, 20, 10000) ## time input

plt.figure(figsize=(10, 5))
plt.title(rf'Ikeda function when $\beta = {beta}$ and $\tau_R={delay}$ and $\tau={tau}$')
plt.xlabel('Time (s)', fontsize=20)
plt.ylabel('f(x)', fontsize=20)
plt.plot(tpts, sol, color='k')

3. Plotting the cobweb diagram for the logistic map.

logistic_model = spiderweb(n = 1000)
logistic_model.plot_logistic(r = 3, ic = 0.5)

4. Plotting the cobweb diagram for the adiabatic ikeda map.

ikeda_adiabatic_model = spiderweb(n = 100)
ikeda_adiabatic_model.plot_adiabatic_ikeda(b = 20, ic = 5, t_min = 0, t_max = 20, tn = 10000)

5. Plotting the bifurcation for the logistic map.

model = analysis(beta_min = 1.1, beta_max = 4, n_beta = 3000, n_step = 5000)
model.logistic_plot(n_sol_show = 1999, ic = 0.6)

6. Plotting the bifurcation diagram for the adiabatic ikdea map.

model = analysis(beta_min = 1.1, beta_max = 10, n_beta = 200000, n_step = 5000)
model.ikeda_adiabatic_plot(n_sol_show = 4999, ic = 1)

Reference

• Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos, L. Larger, C.R. Physique 5 (2004)
• Using Synchronization for Prediction of High-Dimensional Chaotic Dynamics, R. Roy et al., Phys. Rev. Lett. 101, 154102 (2008)
• William Gilpin's lecture note
• Feliciano Guistino's Classical Mechnics lecture note
• https://ipython-books.github.io/121-plotting-the-bifurcation-diagram-of-a-chaotic-dynamical-system/