fastplotlib · kushalkolar · Feb 5, 2025 · Feb 5, 2025
@@ -0,0 +1,137 @@
+"""
+Phase portrait of ordinary differential equations
+=================================================
+
+Very basic example of how to plot the trajectories of a few trajectories that solve an ordinary differential equation
+given some constants that help define the starting points
+"""
+
+# test_example = false
+# sphinx_gallery_pygfx_docs = 'screenshot'
+
+import numpy as np
+import scipy
+
+import fastplotlib as fpl
+
+
+def create_and_solve_ode(
+        t_start: float,
+        t_end: float,
+        n_t: int,
+        p: float,
+        q: float,
+        A: np.ndarray,
+        coef: float = 1.0,
+) -> float:
+    """
+    Given an ODE of the form:
+
+    x' = Ax
+
+    where:
+    x' ∈ R^2
+    x ∈ R^2, x is a function of x, i.e. x(t)
+    A ∈ R^2x2
+
+    We can solve this ODE through an eigendecomposition of A:
+
+    A η_i = λ_i η_i
+
+    where λ_i, η_i are the eigenvalues and eigenvectors of A.
+
+    Thus this gives us the following solution for x(t):
+
+    x(t) = p * η_1 * e^(λ_1 * t) + q * η_2 * e^(λ_2 * t)
+
+    where p and q are constants that determine the initial conditions
+
+    References
+    ----------
+
+    https://tutorial.math.lamar.edu/Classes/DE/SolutionsToSystems.aspx
+
+    Parameters
+    ----------
+    t_start: float
+    t_end: float
+    p: float
+    q: float
+    A: np.ndarray
+    coef: float
+
+    Returns
+    -------
+
+    np.ndarray
+
+    """
+
+    matrix = A * np.array([[1, 1], [1 / coef, 1 / coef]])
+
+    (lam1, lam2), evecs = scipy.linalg.eig(matrix)
+    eta1 = evecs[:, 0]
+    eta2 = evecs[:, 1]
+
+    x_t = np.zeros((n_t, 2))
+
+    for i, t in enumerate(np.linspace(t_start, t_end, n_t)):
+        x_t[i] = (p * eta1 * np.exp(lam1 * t)) + (q * eta2 * np.exp(lam2 * t))
+
+    return x_t
+
+
+# constants range
+START, STOP = -2, 2
+
+T_START, T_END = -2, 5
+
+A = np.array([
+    [0, 1],
+    [-1, -1]
+])
+
+x_t_collection = list()
+x_t_vel_collection = list()
+
+for p in np.linspace(START, STOP, 8):
+    for q in np.linspace(START, STOP, 8):
+        n_t = 100
+
+        x_t_vel = np.zeros(n_t)
+
+        x_t = create_and_solve_ode(
+            t_start=T_START,
+            t_end=T_END,
+            n_t=n_t,
+            p=p,
+            q=q,
+            A=A,
+            coef=0.5,
+        )
+
+        x_t_vel[:-1] = np.linalg.norm(x_t[:-1] - x_t[1:], axis=1, ord=2)
+        x_t_vel[-1] = x_t_vel[-2]
+
+        x_t_collection.append(x_t)
+        x_t_vel_collection.append(x_t_vel)
+
+fig = fpl.Figure()
+
+lines = fig[0, 0].add_line_collection(x_t_collection, cmap="viridis", cmap_transform=x_t_vel_collection)
+for (line, vel) in zip(lines, x_t_vel_collection):
+    line.cmap = "viridis"
+
+    # color indicates velocity along trajectory
+    line.cmap.transform = vel
+    print(vel)
+
+fig[0, 0].axes.intersection = (0, 0, 0)
+
+fig.show()
+
+# NOTE: `if __name__ == "__main__"` is NOT how to use fastplotlib interactively
+# please see our docs for using fastplotlib interactively in ipython and jupyter
+if __name__ == "__main__":
+    print(__doc__)
+    fpl.loop.run()