Chaos Parameter Calculator

Chaos Parameter Calculator: Logistic Map

Equation: \\(x_{n+1} = r x_n (1 – x_n)\\)

Lyapunov Exponent: \\(\lambda = \lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} \ln |r (1 – 2x_i)|\\)

Chaos Parameter Calculator: Logistic Map Example 1 (Periodic)

This example demonstrates periodic behavior in the logistic map with \\(r = 3.2\\).

• Inputs:
  • Mode: 1D (Logistic Map)
  • Growth Rate (\\(r\\)): 3.2
  • Initial \\(x_0\\): 0.5
  • Iterations (\\(n\\)): 1000
  • Evaluation Iteration: 500
• Expected Output:
  • State (\\(x_n\\)) at \\(n = 500\\): ~0.513
  • Lyapunov Exponent (\\(\lambda\\)): ~0.000 (indicating periodic behavior)
• Visualization: The time series plot shows \\(x_n\\) oscillating between fixed values, indicating a periodic cycle.

Chaos Parameter Calculator: Logistic Map Example 2 (Chaotic)

This example demonstrates chaotic behavior with \\(r = 3.9\\).

• Inputs:
  • Mode: 1D (Logistic Map)
  • Growth Rate (\\(r\\)): 3.9
  • Initial \\(x_0\\): 0.5
  • Iterations (\\(n\\)): 1000
  • Evaluation Iteration: 500
• Expected Output:
  • State (\\(x_n\\)) at \\(n = 500\\): ~0.483 (varies due to chaos)
  • Lyapunov Exponent (\\(\lambda\\)): ~0.524 (positive, indicating chaos)
• Visualization: The time series plot shows erratic, non-repeating \\(x_n\\) values, characteristic of chaos.

Chaos Parameter Calculator: Lorenz System

Equations:

\\[ \begin{cases} \dot{x} = \sigma (y – x) \\ \dot{y} = x (\rho – z) – y \\ \dot{z} = x y – \beta z \end{cases} \\]

Lyapunov Exponent: Computed via trajectory divergence using a perturbed initial condition.

Chaos Parameter Calculator: Lorenz System Example 1 (Chaotic)

This example uses standard Lorenz parameters to demonstrate chaotic behavior.

• Inputs:
  • Mode: 3D (Lorenz System)
  • \\(\sigma\\): 10.0
  • \\(\rho\\): 28.0
  • \\(\beta\\): 2.6667
  • Initial \\(x_0\\): 0.5
  • Initial \\(y_0\\): 1.0
  • Initial \\(z_0\\): 1.0
  • Start Time (\\(t_0\\)): 0
  • End Time (\\(t_f\\)): 100
  • Evaluation Time: 50.0
• Expected Output:
  • State (\\(x\\)) at \\(t = 50.0\\): ~-8.486
  • State (\\(y\\)) at \\(t = 50.0\\): ~-8.672
  • State (\\(z\\)) at \\(t = 50.0\\): ~27.012
  • Lyapunov Exponent (\\(\lambda\\)): ~0.905 (positive, indicating chaos)
• Visualization: The 3D plot shows the butterfly attractor, with trajectories spiraling unpredictably around two lobes.

Chaos Parameter Calculator: Lorenz System Example 2 (Periodic)

This example uses \\(\rho = 24.0\\) to demonstrate periodic behavior.

• Inputs:
  • Mode: 3D (Lorenz System)
  • \\(\sigma\\): 10.0
  • \\(\rho\\): 24.0
  • \\(\beta\\): 2.6667
  • Initial \\(x_0\\): 0.5
  • Initial \\(y_0\\): 1.0
  • Initial \\(z_0\\): 1.0
  • Start Time (\\(t_0\\)): 0
  • End Time (\\(t_f\\)): 100
  • Evaluation Time: 50.0
• Expected Output:
  • State (\\(x\\)) at \\(t = 50.0\\): ~6.000
  • State (\\(y\\)) at \\(t = 50.0\\): ~6.500
  • State (\\(z\\)) at \\(t = 50.0\\): ~20.000
  • Lyapunov Exponent (\\(\lambda\\)): ~0.000 (indicating periodic behavior)
• Visualization: The 3D plot shows closed orbits, indicating a periodic trajectory rather than chaotic divergence.