While point in polygon plot points/lines fails (Circle.contains_point; Python; matplotlib; pyplot)

My task: Plot an initial point and calculate new point positions (x, y) while points are in a circle.

My problem: Using a while loop to perform this fails to run the code block as it incorrectly determines the circle doesn't contain the point (fig 1). However, using a for loop it can be seen that multiple points reside in the circle, but still incorrectly identifies that points don't exist in the circle (fig 2).

Additional info: I also can't seem to get lines instead of points to be plotted (e.g. using pyplot.plot(x, y, '-')). If there are more streamlined methods, for instance in plotting the circle/points/lines, or the entire approach I'd be interested to see them. I appreciate any help that can be offered.

After much searching here and elsewhere I've come as far as the following code:

# IMPORT LIBRARIES ####
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.path as mPath
import matplotlib.patches as mPatches


# CREATE CIRCLE ####
# Input circle radius
circleRadius = 2

# Create circle (arguments: centre, radius, face colour (fc), edge colour (ec), alpha)
circle = mPatches.Circle((0, 0), radius = circleRadius, fc = 'm', alpha = 0.3)

# Plot circle
fig1, ax = plt.subplots()
ax.add_patch(circle)
ax.set_xlim(-5, 5)
ax.set_ylim(-5, 5)
ax.set_aspect('equal')


# CREATE POINTS ####
# Input coordinates
x0 = -1
y0 = 0.5
# Assignment for equation
x = x0
y = y0

# Store
curr_pt = [x, y]
counter = 0

# Plot 1st point (x, y)
plt.plot(x, y, '.c')

# Pre-checks
print(circle.contains_point([x, y]))
print(counter)
print(x0, y0)
print(x, y)

# Plot points 
while circle.contains_point([x, y]): # see 1st figure
#for i in range(5):                  # uncomment this line (for) and comment above line (while) see 2nd figure
    xnew = x**2 - y**2 + x0
    ynew = 2 * x * y + y0
    x = xnew
    y = ynew
    # Update current point
    curr_pt = [x, y]
    # Plot point (x, y)
    plt.plot(x, y, '.c')
    counter += 1

# Post-checks
print(circle.contains_point([x, y]))
print(counter)
print(x0, y0)
print(x, y)
print(xnew, ynew) # note this check will output error with while loop as they will not be defined

# Print plot
plt.show()

Output:

# while loop (Fig 1)
False
0
-1 0.5
-1 0.5
False
0
-1 0.5
-1 0.5

Fig 1 (while loop)

# for loop (Fig 2)
False
0
-1 0.5
-1 0.5
False
5
-1 0.5
5.063203536206856 -4.162733919918537
5.063203536206856 -4.162733919918537

Fig 2 (for loop)