Interpretations of Dynamical Laws

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Examples of dynamical systems include: - A pendulum swinging back and forth - Water flowing from a faucet - The decisions made by rational players in negotiation scenarios - Population changes over time

Chaos theory effectively models dynamical systems that are extremely sensitive to initial conditions.

Notable examples of such systems include: - Weather patterns - Two pendulums swinging in sync from the same initial position

Chaos theory categorizes dynamical systems into two types: - Deterministic systems: where identical inputs consistently produce the same outputs. - Non-deterministic systems: where identical inputs can yield different outputs due to randomness.

This article will focus on one of the most renowned equations in deterministic chaos — The Logistic Map.

The Logistic Map

Prepare to witness how complexity emerges from a seemingly basic equation. Notably, "map" here can be understood as a function.

Consider the straightforward quadratic equation: y = ax - ax²

In its factored form, it can be expressed as: y = ax(1 - x)

Let’s rewrite this non-linear quadratic equation as follows:

Graphically, this equation illustrates a downward-opening parabola:

To create this graph, we determined the coordinates of the maximum point.

Two Levels of Understanding the Equation and its Graph

We will draw a vital connection between the quadratic nature of xn+1 = ?xn(1 - xn) (eq. 1) and y = ax(1 - x) (eq. 2).

The key difference lies in their applications. Eq. 1 is formatted to represent an iterative process, linking the present population xn to the future population xn+1.

It’s crucial to understand that n signifies discrete time steps, representing the various states of the dynamical system.

To investigate the iterative relationship between the current state and future states over discrete steps n, we will create a final state diagram.

Final State Diagrams

To construct a final state diagram for eqn. 1, follow these steps: - Fix the value of ? - Select the initial condition x0 (xn = x0) - Iterate and plot the results for 40 iterations

The resulting graph is termed a final state diagram. I will utilize Matlab to execute this for four different values of ?, keeping x0 = 0.7 constant.

Graph for ?=0.5

As n approaches infinity, xn approaches zero, known as the trivial solution. The convergence point is referred to as a fixed point attractor. The behavior leading up to these fixed points is termed transience. A fixed point attractor remains constant irrespective of initial conditions—if x0 varies (as long as it's not zero), the fixed point remains unchanged.

Graph for ?=2.8

In this case, as n increases indefinitely, xn approaches x*, a value between 0.6 and 0.7. This behavior is termed a period 1 cycle.

Graph for ?=3.3

Here, we observe a period 2 cycle, marked by two repeating fixed points.

Graph for ?=3.5

Lastly, we identify a period 4 cycle, characterized by four repeating fixed points.

Do you notice a trend? As ? rises, the system exhibits period doubling. By employing a computer, we can illustrate the following results:

Observe how each increment of ? by smaller and smaller amounts results in a doubling of the period. These increments can be visualized as intervals where ? converges geometrically toward ??.

To analyze where the ratio of differences in successive intervals converges, we evaluate the limit as n approaches infinity:

Keep the value of ? in mind, as we will return to its relevance.

Despite uncovering fascinating results, one question remains unanswered:

What Happens After ???

Thus far, we've observed that as ? increases, the period doubles. However, the notable aspect is that a smaller increment in ? than the previous interval led to ? converging geometrically to ?? = 3.569946... as it approached infinity. This characteristic introduces uncertainty about the behavior beyond ??.

We will employ an iterative approach again, plotting a graph that illustrates: - xn, the set of attractors for each ?, after removing transience - ? on the x-axis - xn on the y-axis

Our aim is to assess how xn varies by adjusting the ? parameter. From 0 to just below 3 in ?, we anticipate stable period 1 behavior. Between 3 and 3.569946..., we expect period doubling. The resulting graph is known as the bifurcation diagram. Bifurcation refers to the phenomenon of splitting into two, which can be visually represented as a fork.

Utilizing Matlab, we obtain:

Surprised? You should be! This is perhaps the most iconic representation in chaos theory, showcasing the simplest nonlinear equation in deterministic chaos. Let’s explore the intervals:

• [0,1): No growth
• [1,3): Period 1 cycle
• [3,3.499...): Period 2 cycle
• [3.499,3.54409...): Period 4
• [3.54409...,3.5644...): Period 8
• [3.5644..., -...): Period 16
• ...
• [-..., 3.569946...): Period ?
• Beyond 3.569946..., periodic behavior ceases, giving way to chaos.

Look closely at the diagram. Do you notice the white area amidst the chaos? This represents a transition from chaotic to periodic behavior. Extending the x-axis would reveal a repeating pattern of transitions from periodic to chaotic and back again.

What might happen if we zoom into the chaotic regions of the diagram?

These discrete points exhibit traits reminiscent of fractals. With additional iterations, the overlapping chaotic zone would increasingly resemble the Mandelbrot set.

The Mandelbrot set is two-dimensional, encompassing both complex and real dimensions. The real dimension (1D) manifests within the bifurcation diagram.

What Does All of This Mean?

Remember ?? This constant holds great significance in mathematics, ranking alongside ? and e. It’s known as the Feigenbaum constant, which denotes the ratio observed in a bifurcation diagram. Its remarkable aspect is its presence not only in this specific nonlinear map (the logistic map) but across all one-dimensional maps with a single quadratic maximum. Consequently, this constant is believed to be transcendental.

Summing Up

The logistic map represents a straightforward yet entirely deterministic equation that can exhibit chaos through iteration, depending on the value of ?.

Iteration is a simple process (made even easier with computers) that can extend infinitely. As demonstrated, iterating a basic equation can lead to complex behaviors that were previously unobserved.

We noted that for a wide range of initial conditions, the system settles into a steady state at attractors/fixed points. Another key observation is the sensitivity to initial conditions, which means that while the future becomes predictable once initial conditions are established, it may appear chaotic. These characteristics make the logistic map a fascinating equation within chaos theory!

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