//! # Fractal Geometry and Chaos Theory
//!
//! This module provides symbolic tools for exploring concepts in fractal geometry
//! and chaos theory. It includes representations for Iterated Function Systems (IFS)
//! and functions for calculating fractal dimensions and Lyapunov exponents.
use crate::symbolic::calculus::differentiate;
use crate::symbolic::core::Expr;
/// Represents an Iterated Function System (IFS).
#[derive(Debug, Clone, PartialEq, Eq)]
pub struct IteratedFunctionSystem {
    pub functions: Vec<Expr>,
    pub probabilities: Vec<Expr>,
}
/// Calculates the fractal dimension (e.g., box-counting dimension) symbolically.
///
/// This is a highly complex symbolic operation, often defined implicitly or through limits.
/// A full symbolic implementation would require advanced set theory and measure theory.
///
/// # Arguments
/// * `_set` - The symbolic representation of the set for which to calculate the dimension.
///
/// # Returns
/// An `Expr` representing the symbolic fractal dimension.
pub fn fractal_dimension(_set: Expr) -> Expr {
    Expr::Variable("FractalDimension(set)".to_string())
}
/// Calculates the Lyapunov exponent for a 1D chaotic map `x_n+1 = f(x_n)`.
///
/// The Lyapunov exponent `λ` quantifies the rate at which nearby trajectories
/// in a dynamical system diverge. A positive Lyapunov exponent is a key indicator of chaos.
/// Formula: `λ = lim (n->inf) (1/n) * sum(ln(|f'(x_i)|))`.
/// This function provides a symbolic representation of this calculation.
///
/// # Arguments
/// * `map_function` - The symbolic expression for the chaotic map `f(x_n)`.
/// * `initial_x` - The initial value `x_0`.
/// * `n_iterations` - The number of iterations to symbolically sum the derivatives.
///
/// # Returns
/// An `Expr` representing the symbolic Lyapunov exponent.
pub fn lyapunov_exponent(map_function: Expr, initial_x: Expr, n_iterations: usize) -> Expr {
    let mut current_x = initial_x;
    let mut sum_log_derivs = Expr::Constant(0.0);
    for _i in 0..n_iterations {
        let derivative_at_x_i = differentiate(&map_function, &current_x.to_string());
        let log_abs_derivative = Expr::new_log(Expr::new_abs(derivative_at_x_i));
        sum_log_derivs = Expr::new_add(sum_log_derivs, log_abs_derivative);
        current_x = Expr::new_apply(map_function.clone(), current_x);
    }
    Expr::new_div(sum_log_derivs, Expr::Constant(n_iterations as f64))
}
