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a/phase_plane_analysis_van_der_pol.py b/phase_plane_analysis_van_der_pol.py new file mode 100644 index 0000000..b276591 --- /dev/null +++ b/phase_plane_analysis_van_der_pol.py @@ -0,0 +1,99 @@ +""" +Simple examples for understanding phase plane analysis, applied to the Van der Pol oscillator +using linearization. + +author: Fabrizio Musacchio +date: Feb 20, 2024 + +For reproducibility: + +conda create -n phase_plane_analysis python=3.10 +conda activate phase_plane_analysis +conda install mamba +mamba install numpy matplotlib scipy + +""" +# %% IMPORT +import numpy as np +import matplotlib.pyplot as plt +from scipy.integrate import solve_ivp +# set global font size for plots: +plt.rcParams.update({'font.size': 14}) +# create a folder "figures" to save the plots (if it does not exist): +import os +if not os.path.exists('figures'): + os.makedirs('figures') +# %% EXAMPLE: VAN DER POL OSCILLATOR +# define the Van der Pol oscillator model: +def van_der_pol(t, z, mu): + x, y = z + dxdt = y + dydt = mu * (1 - x**2) * y - x + return [dxdt, dydt] + +# define the nullclines: +def y_nullcline(x, mu): + return x/(mu*(1-x**2)) +def x_nullcline(y, mu): + return 0*y + +# set time span: +eval_time = 100 +t_iteration = 1000 +t_span = [0, eval_time] +t_eval = np.linspace(*t_span, t_iteration) + +# set initial conditions: +#z0 = [0.5, y_nullcline(0.5, mu)] # [2, 0] +z0 = [4, 0] # [2, 0] + +# set Van der Pol oscillator parameter: +mu = 2.0 # stable: >0, unstable: <0 + +# calculate the vector field: +mgrid_size = 8 +x, y = np.meshgrid(np.linspace(-mgrid_size, mgrid_size, 15), + np.linspace(-mgrid_size, mgrid_size, 15)) +u = y +v = mu * (1 - x**2) * y - x + +# calculating the trajectory for the Van der Pol oscillator: +sol_stable = solve_ivp(van_der_pol, t_span, z0, args=(mu,), t_eval=t_eval) + +# define the x-array for the nullclines: +x_null = np.arange(-mgrid_size,mgrid_size,0.001) + +# plot vector field and trajectory: +plt.figure(figsize=(6, 6)) +plt.clf() +#plt.quiver(x, y, u, v, color='gray') # vector field +#plt.streamplot(x, y, u, v, color='gray', density=2.5) +# plot the streamline plot colored by the speed of the flow: +speed = np.sqrt(u**2 + v**2) +plt.streamplot(x, y, u, v, color=speed, cmap='cool', density=2.0) +plt.plot(x_null, x_nullcline(x_null, mu) , '.', c="darkturquoise", markersize=2) +plt.plot(x_null, y_nullcline(x_null, mu) , '.', c="darkturquoise", markersize=2) +plt.plot(sol_stable.y[0], sol_stable.y[1], 'r-', lw=3, + label=f'Trajectory for $\mu$={mu}\nand $z_0$={z0}') # trajectory +# indicate start point: +plt.plot(sol_stable.y[0][0], sol_stable.y[1][0], 'bo', label='start point', alpha=0.75, markersize=7) +plt.plot(sol_stable.y[0][-1], sol_stable.y[1][-1], 'o', c="yellow", label='end point', alpha=0.75, markersize=7) +# indicate the direction of the trajectory's last point with an arrow: +""" plt.arrow(sol_stable.y[0][-2], sol_stable.y[1][-2], + sol_stable.y[0][-1] - sol_stable.y[0][-2], + sol_stable.y[1][-1] - sol_stable.y[1][-2], + head_width=0.5, head_length=0.5, fc='fuchsia', ec='fuchsia') """ +plt.title('phase plane plot: Van der Pol oscillator') +plt.xlabel('x') +plt.ylabel('y') +plt.legend(loc='lower right') #, bbox_to_anchor=(1, 0.5) +plt.gca().spines['top'].set_visible(False) +plt.gca().spines['right'].set_visible(False) +plt.gca().spines['bottom'].set_visible(False) +plt.gca().spines['left'].set_visible(False) +#plt.xlim(-mgrid_size, mgrid_size) +plt.ylim(-mgrid_size, mgrid_size) +plt.tight_layout() +plt.savefig(f'figures/van_der_pol_oscillator_z_{z0[0]}_{z0[1]}_mu_{mu}.png', dpi=120) +plt.show() +# %% END diff --git a/phase_plane_analysis_van_der_pol_lienard_transformation.py b/phase_plane_analysis_van_der_pol_lienard_transformation.py new file mode 100644 index 0000000..57cd6fa --- /dev/null +++ b/phase_plane_analysis_van_der_pol_lienard_transformation.py @@ -0,0 +1,102 @@ +""" +Simple examples for understanding phase plane analysis, applied to the Van der Pol oscillator +using Liénard's transformation. + +author: Fabrizio Musacchio +date: Feb 20, 2024 + +For reproducibility: + +conda create -n phase_plane_analysis python=3.10 +conda activate phase_plane_analysis +conda install mamba +mamba install numpy matplotlib scipy + +""" +# %% IMPORT +import numpy as np +import matplotlib.pyplot as plt +from scipy.integrate import solve_ivp +# set global font size for plots: +plt.rcParams.update({'font.size': 14}) +# create a folder "figures" to save the plots (if it does not exist): +import os +if not os.path.exists('figures'): + os.makedirs('figures') +# %% EXAMPLE: VAN DER POL OSCILLATOR +# define the Van der Pol oscillator model: +def van_der_pol(t, z, mu): + x, y = z + dxdt = mu * (x - (x**3)/3 - y) + dydt = (1/mu) * x + return [dxdt, dydt] + +# define the nullclines: +def x_nullcline(x, mu): + return x-(1/3)*(x**3) +"""the y-nullcline is in this case just a vertical line at x=0 for all y; +we well plot it as a function of x for the sake of consistency with the other nullcline: +""" +def y_nullcline(x): + return 0*x + + +# set time span: +eval_time = 100 +t_iteration = 1000 +t_span = [0, eval_time] +t_eval = np.linspace(*t_span, t_iteration) + +# set initial conditions: +z0 = [1.0, 0.0] # [2, 0] + +# set Van der Pol oscillator parameter: +mu = -4.0 # stable: >0, unstable: <0 + +# calculate the vector field: +mgrid_size = 8 +x, y = np.meshgrid(np.linspace(-mgrid_size, mgrid_size, 15), + np.linspace(-mgrid_size, mgrid_size, 15)) +u = mu * (x - (x**3)/3 - y) +v = (1/mu) * x + +# calculating the trajectory for the Van der Pol oscillator: +sol_stable = solve_ivp(van_der_pol, t_span, z0, args=(mu,), t_eval=t_eval) + +# define the x-array for the nullclines: +x_null = np.arange(-mgrid_size,mgrid_size,0.001) + +# plot vector field and trajectory: +plt.figure(figsize=(6, 6)) +plt.clf() +#plt.quiver(x, y, u, v, color='gray') # vector field +#plt.streamplot(x, y, u, v, color='gray', density=2.5) +# plot the streamline plot colored by the speed of the flow: +speed = np.sqrt(u**2 + v**2) +plt.streamplot(x, y, u, v, color=speed, cmap='cool', density=2.0) +plt.plot(x_null, x_nullcline(x_null, mu) , '.', c="darkturquoise", markersize=2) +plt.plot(y_nullcline(x_null), x_null , '.', c="darkturquoise", markersize=2) +plt.plot(sol_stable.y[0], sol_stable.y[1], 'r-', lw=3, + label=f'Trajectory for $\mu$={mu}\nand $z_0$={z0}') # trajectory +# indicate start point: +plt.plot(sol_stable.y[0][0], sol_stable.y[1][0], 'bo', label='start point', alpha=0.75, markersize=7) +plt.plot(sol_stable.y[0][-1], sol_stable.y[1][-1], 'o', c="yellow", label='end point', alpha=0.75, markersize=7) +# indicate the direction of the trajectory's last point with an arrow: +""" plt.arrow(sol_stable.y[0][-2], sol_stable.y[1][-2], + sol_stable.y[0][-1] - sol_stable.y[0][-2], + sol_stable.y[1][-1] - sol_stable.y[1][-2], + head_width=0.5, head_length=0.5, fc='fuchsia', ec='fuchsia') """ +plt.title('phase plane plot: Van der Pol oscillator') +plt.xlabel('x') +plt.ylabel('y') +plt.legend(loc='lower right') #, bbox_to_anchor=(1, 0.5) +plt.gca().spines['top'].set_visible(False) +plt.gca().spines['right'].set_visible(False) +plt.gca().spines['bottom'].set_visible(False) +plt.gca().spines['left'].set_visible(False) +#plt.xlim(-mgrid_size, mgrid_size) +plt.ylim(-mgrid_size, mgrid_size) +plt.tight_layout() +plt.savefig(f'figures/van_der_pol_oscillator_lienard_z_{z0[0]}_{z0[1]}_mu_{mu}.png', dpi=120) +plt.show() +# %% END