Compute and plot the first 7 values of the logistic map with parameter λ := 3.57. Describe theplot and what behavior you expect to observe as n → ∞.

To compute the first 7 values of the logistic map with parameter λ = 3.57, we can use the logistic map equation:

x(n+1) = λ * x(n) * (1 - x(n))

where x(n) represents the value at iteration n.

Let's start with an initial value x(0) = 0.5 and calculate the next values iteratively:

x(1) = 3.57 * 0.5 * (1 - 0.5) = 0.8925 x(2) = 3.57 * 0.8925 * (1 - 0.8925) = 2.0024 x(3) = 3.57 * 2.0024 * (1 - 2.0024) = -7.1507 x(4) = 3.57 * -7.1507 * (1 + 7.1507) = -181.3037 x(5) = 3.57 * -181.3037 * (1 + 181.3037) = -116,874.6827 x(6) = 3.57 * -116,874.6827 * (1 + 116,874.6827) = -4,234,674,682.7 x(7) = 3.57 * -4,234,674,682.7 * (1 + 4,234,674,682.7) = -60,000,000,000,000,000,000

Now, let's plot these values on a graph. The x-axis represents the iteration number (n), and the y-axis represents the value of x(n).

Based on the calculated values, we observe that the logistic map with λ = 3.57 exhibits chaotic behavior. The values quickly diverge and become extremely large and negative. This behavior is characteristic of the logistic map when the parameter λ is greater than 3.57.

As n approaches infinity, we expect the values of x(n) to continue oscillating chaotically between positive and negative values, without converging to a specific value or pattern.