Chaos Theory and Fractals

Chaos theory explores systems that are highly sensitive to initial conditions—a concept popularly known as the “Butterfly Effect.” Fractals are complex geometric shapes that exhibit self-similarity across different scales.

Sensitive Dependence on Initial Conditions

In a chaotic system, small differences in initial state lead to vastly different outcomes over time. This makes long-term prediction impossible, even for deterministic systems.

def logistic_map(x, r):
  return r * x * (1 - x)

r = 3.9 # Chaotic regime
x1 = 0.5
x2 = 0.500001
dt = []

print("Comparison of two nearly identical starting points:")
for i in range(10):
  x1 = logistic_map(x1, r)
  x2 = logistic_map(x2, r)
  print(f"Iteration {i}: x1={x1:.4f}, x2={x2:.4f}, diff={abs(x1-x2):.6f}")

Bifurcation

As a system parameter (like $r$ in the logistic map) changes, the qualitative behavior of the system can change abruptly at certain values, called bifurcation points.

Fractals and Hausdorff Dimension

Fractals are objects with non-integer dimensions. The Hausdorff dimension measures how a set fills space as it is scaled. For the Cantor set or the Koch snowflake, this dimension is a fraction.

import numpy as np

def mandelbrot(c, max_iter=100):
  z = 0
  for n in range(max_iter):
      if abs(z) > 2:
          return n
      z = z*z + c
  return max_iter

# Testing a few points
points = [complex(0, 0), complex(1, 1), complex(-1, 0)]
for p in points:
  res = mandelbrot(p)
  status = "Inside" if res == 100 else f"Escaped at iter {res}"
  print(f"Point {p}: {status}")

Strange Attractors

A strange attractor is an attractor that exhibits chaotic behavior. The most famous example is the Lorenz Attractor, which arises from a simplified model of atmospheric convection.