IB课程数学探究选题咨询：混沌理论、三角学或概率论方向

Hey Alia, awesome choice leaning into chaos theory for your IB Math exploration—it’s equal parts mathematically rigorous and visually engaging! Below are targeted project ideas across chaos theory, trigonometry, and probability, each with clear mathematical angles to build your investigation around:

• The Logistic Map: Bifurcations & Lyapunov Exponents
  Dive into the classic logistic equation xₙ₊₁ = r*xₙ*(1 - xₙ) where r is the growth rate parameter. Calculate iterations for different r values, plot bifurcation diagrams to show how the system shifts from periodic to chaotic behavior, and compute Lyapunov exponents to quantify chaos (a positive exponent confirms the system is chaotic). You can code simple iterations in Python or do manual calculations for small r values to verify patterns.
• Mandelbrot Set: Complex Number Iterations & Boundary Analysis
  Explore the math behind the Mandelbrot set, defined by zₙ₊₁ = zₙ² + c (where z and c are complex numbers). Calculate whether points in the complex plane remain bounded after repeated iterations, plot a small region of the set manually, and analyze how the boundary’s fractal dimension relates to iteration behavior. Focus on a specific section (like the "seahorse valley") to dive into local patterns.
• Chaos in the Driven Pendulum
  Model a pendulum with an external driving force using differential equations. Use numerical methods (like Euler’s or Runge-Kutta) to solve the equations, plot phase space portraits, and identify the transition from regular, periodic motion to chaos. Compare how changing the driving frequency or amplitude alters the system’s behavior.

• Fourier Series & Chaotic Signal Decomposition
  Take a chaotic time series (e.g., data from the logistic map or a double pendulum) and use trigonometric Fourier series to decompose it into sine and cosine components. Calculate the coefficients for the series, compare how well the series approximates the original chaotic signal, and analyze whether chaotic signals have distinct spectral patterns compared to periodic ones.
• Trigonometric Models of Chaotic Population Growth
  Modify the standard logistic map to include trigonometric terms (e.g., xₙ₊₁ = r*xₙ*(1 - xₙ) + sin(xₙ)). Investigate how the sine term alters bifurcation behavior, calculate fixed points using trigonometric identities, and compare the Lyapunov exponents of this modified map to the original logistic map.
• Triangular Fractals & Trigonometric Dimension Calculation
  Create a fractal using repeated trigonometric transformations (e.g., dividing an equilateral triangle into smaller triangles using sine/cosine to calculate side lengths). Use trigonometry to derive the fractal’s theoretical dimension, then compute the empirical box-counting dimension and compare the two values.

• Probabilistic Chaos: Randomness vs. Determinism
  Generate a sequence from a chaotic system (like the logistic map with r=4) and compare its statistical properties to a true random number sequence. Calculate measures like mean, variance, autocorrelation, and run-length distributions to see if the chaotic sequence can pass "randomness tests." Discuss whether deterministic chaos can mimic true randomness.
• Fractal Growth & Probabilistic Iterated Function Systems (IFS)
  Use probabilistic IFS to generate fractals like the Barnsley fern. Assign probabilities to each transformation (e.g., 85% chance for the stem, 7% for a left leaf), calculate the expected value of each coordinate transformation, and analyze how changing the probabilities alters the fractal’s shape. You can also use probability to derive the fractal’s dimension.
• Probability in Chaotic Pendulum Predictability
  Run simulations of a chaotic double pendulum with slightly different initial angles. Calculate the probability that the pendulum will end up in a specific region of phase space after a given time, and analyze how this probability changes as time increases (linking directly to the "butterfly effect").

内容的提问来源于stack exchange，提问作者Alia Najwa