Question

Here is an iterated map that is easily studied with the help of your calculator: Let \(x_{t+1}=f\left(x_{t}\right)\) where \(f(x)=\cos (x) .\) If you choose any value for \(x_{0},\) you can find \(x_{1}, x_{2}, x_{3}, \cdots\) by simply pressing the cosine button on your calculator over and over again. (Be sure the calculator is in radians mode.) (a) Try this for several different choices of \(x_{0}\), finding the first 30 or so values of \(x_{t}\). Describe what happens. (b) You should have found that there seems to be a single fixed attractor. What is it? Explain it, by examining (graphically, for instance) the equation for a fixed point \(f\left(x^{*}\right)=x^{*}\) and applying our test for stability [namely, that a fixed point \(\left.x^{*} \text { is stable if }\left|f^{\prime}\left(x^{*}\right)\right|<1\right].\)

Short answer

(a) The iterated sequences always converge to around 0.739. (b) The fixed attractor is 0.739, verified by \(|f'(x^*)|<1\).

Step-by-step solution

Understanding the Iterated Map

The function given is an iterated map, where the next value in the sequence is the cosine of the current value. Specifically, we have \(x_{t+1} = \cos(x_t)\). This means that for any initial value \(x_0\), you can find a sequence of values \(x_1, x_2, \ldots\) by repeatedly applying the function \(f(x) = \cos(x)\).

Choosing Initial Values

Choose several initial values \(x_0\) such as \(x_0 = 0, \frac{\pi}{4}, \frac{\pi}{2}, 1\), and so on. Make sure the calculator is in radians mode since the cosine function is periodic with a period of \(2\pi\) radians.

Computing Iterations

For each initial value \(x_0\), repeatedly apply the cosine function up to 30 times, i.e., calculate \(x_1 = \cos(x_0)\), \(x_2 = \cos(x_1)\), ..., \(x_{30} = \cos(x_{29})\). Record the values to observe any patterns.

Observing the Behavior

Regardless of the initial value \(x_0\) you choose, the sequence of values \(x_1, x_2, \ldots \) should converge to a particular value. This suggests that there is a stable fixed point acting as an attractor. All values gravitate towards this point after enough iterations.

Identifying the Fixed Attractor

The primary task is to find the fixed point \(x^*\) such that \(f(x^*) = x^*\), or \(\cos(x^*) = x^*\). This can be found numerically or graphically, and it is around \(x^* \approx 0.739\).

Verifying Stability of the Fixed Point

To check the stability of this fixed point, calculate the derivative \(f'(x) = -\sin(x)\) and evaluate \(f'(x^*)\). Since \(\left|f'(x^*)\right| = \left|\sin(x^*)\right| < 1\), the fixed point is stable. This means any initial value will eventually converge to \(x^*\).

Key concepts

Fixed Point

In mathematics, especially in the study of dynamical systems, a fixed point is a value that does not change when a specific transformation is applied. In the context of an iterated map, a fixed point, denoted as \(x^*\), satisfies the equation \(f(x^*) = x^*\). For the given iterated map based on the cosine function, we are looking for a value \(x^*\) where \(\cos(x^*) = x^*\).

Finding this fixed point can often be done both numerically and graphically. For the function \(f(x) = \cos(x)\), graphical representation involves plotting both \(y = x\) and \(y = \cos(x)\) on the same graph. Where these two plots intersect, a fixed point exists. Numerically, the fixed point of this function is approximately \(x^* \approx 0.739\). This particular situation is common in mathematical explorations, illustrating how iterative processes can stabilize at specific values.

Stability Analysis

Stability analysis is an important concept that helps us understand whether a fixed point is likely to retain its behavior under slight perturbations. For fixed points, we investigate their stability to determine if sequences starting near the fixed point will converge to it or not.

The stability of a fixed point \(x^*\) can be tested using the derivative of the iterated function, \(f(x)\). For a fixed point to be stable, the absolute value of the derivative at that point needs to be less than one: \(|f'(x^*)| < 1\). This condition ensures that deviation from \(x^*\) decreases with each iteration, leading to convergence towards the fixed point.

In our example with \(f(x) = \cos(x)\), the derivative is \(f'(x) = -\sin(x)\). Evaluating this at the fixed point \(x^* \approx 0.739\), you find \(|f'(x^*)| = |\sin(0.739)|\) is indeed less than one, confirming that this fixed point is stable. This means that for any starting value in the sequence, it will eventually approach \(x^*\).

Convergence

Convergence in iterative maps refers to the tendency of a sequence to approach a stable fixed point after numerous iterations. When you start with an initial value \(x_0\) in the cosine iterated map, you iteratively apply \(x_{t+1} = \cos(x_t)\) for multiple steps.

In practical terms, you try this process using several initial values like 0, \(\pi/4\), \(\pi/2\), and watch how quickly they converge to approximately \(0.739\), the fixed point. This behavior indicates that the iterated map pulls values towards the fixed point, acting as an attractor.

Understanding convergence helps visualize the long-term behavior of dynamical systems. It reassures us that despite initial differences, the sequences become almost indistinguishable as they develop. In homework exercises similar to this one, observing convergence can boost intuition about iterative processes and their stable properties.