The basic idea behind iterated maps involves taking an initial value, applying a defined function to it, and then using the result as the new input for the next iteration. This process is repeated indefinitely, generating a sequence of values that represents the system's trajectory through its state space.
Iterated maps can exhibit a variety of behaviors, ranging from stable fixed points and periodic orbits to chaotic and unpredictable trajectories. The study of iterated maps helps researchers understand the underlying dynamics of systems and predict their long-term behavior under different conditions.
The logistic map is a mathematical model that describes the population growth of a species. It is a 2-degree polynomial equation and a discrete-time dynamical system, meaning it evolves over time in discrete steps. The formula for the logistic map is given by:
The parameter r plays a crucial role in determining the behavior of the logistic map. Different values of r can lead to various dynamic behaviors, including stable equilibrium, periodic oscillations, and chaotic patterns.
- For 0 < r < 1 , the population converges to a stable equilibrium.
- For 1 < r < 3 , the population exhibits periodic oscillations.
- Beyond r = 3 , the system can enter chaotic behavior, displaying sensitivity to initial conditions.
def logistic_map(x0, r, num_iterations):
result = [x0]
for _ in range(num_iterations):
x0 = r * x0 * (1 - x0)
result.append(x0)
return result
The Hénon map is a discrete-time, two-dimensional dynamical system that exhibits chaotic behavior.The Hénon map takes a point (xn, yn) in the plane and maps it to a new point. The map is defined by the following recursive equations:
def henon_map(x0, y0, a, b, num_iterations):
result_x = [x0]
result_y = [y0]
for _ in range(num_iterations):
xn = 1 - a * result_x[-1]**2 + result_y[-1]
yn = b * result_x[-1]
result_x.append(xn)
result_y.append(yn)
return result_x, result_y