Question

The chaotic evolution of the logistic map shows the same sensitivity to initial conditions that we met in the DDP. To illustrate this do the following: (a) Using a growth rate \(r=2.6\) calculate \(x_{t}\) for \(1 \leq t \leq 40\) starting from \(x_{0}=0.4 .\) Repeat but with the inital condition \(x_{0}^{\prime}=0.5\) (the prime is just to distinguish this second solution from the first \(-\) it does not denote differentiation) and then plot \(\log \left|x_{t}^{\prime}-x_{t}\right|\) against \(t .\) Describe the behavior of the difference \(x_{t}^{\prime}-x_{t} .\) (b) Repeat part (a) but with \(r=3.3 .\) In this case, the long term evolution has period 2. Again describe the behavior of the difference \(x_{t}^{\prime}-x_{t} \cdot\) (c) Repeat parts (a) and (b), but with \(r=3.6 .\) In this case, the evolution is chaotic, and we expect the difference to grow exponentially; therefore, it is more interesting to take the two inital values much closer together. To be definite take \(x_{0}=0.4\) and \(x_{0}^{\prime}=0.400001 .\) How does the difference behave?

Short answer

The difference grows larger with increasing chaos; it may stabilize, oscillate, or grow rapidly depending on \(r\).

Step-by-step solution

Logistic Map Equation Definition

The logistic map equation is given by \(x_{t+1} = r x_t (1 - x_t)\). This equation will be used to compute future values based on an initial value \(x_0\) and growth rate \(r\).

Calculate Series for r=2.6 and x0=0.4

For part (a) with \(r=2.6\) and starting value \(x_0=0.4\), calculate \(x_t\) for \(1 \leq t \leq 40\) using the logistic map equation. Repeat for initial value \(x_0' = 0.5\).

Plot and Evaluate Difference for r=2.6

Compute the difference \(x_t' - x_t\) for each \(t\). Plot \(\log|x_t' - x_t|\) against \(t\). For \(r=2.6\), observe if the difference decreases or becomes significant over time.

Repeat Calculations for r=3.3 and x0=0.4

For part (b) set \(r=3.3\). Calculate \(x_t\) and \(x_t'\) with initial conditions \(x_0=0.4\) and \(x_0' = 0.5\) respectively, for \(1 \leq t \leq 40\).

Plot and Describe Difference for r=3.3

Plot \(\log|x_t' - x_t|\) against \(t\) for \(r=3.3\). Describe the behavior of the difference, focusing on periodic evolution (period 2) and any fluctuations.

Repeat Calculations for r=3.6 and Close Initial Values

For part (c) with growth rate \(r=3.6\), perform calculations with initial values \(x_0=0.4\) and \(x_0' = 0.400001\) for \(1 \leq t \leq 40\).

Plot and Analyze Chaotic Behavior for r=3.6

Plot \(\log|x_t' - x_t|\) against \(t\) for \(r=3.6\). For chaotic evolutions, expect the difference to grow exponentially. Evaluate the rate of this divergence over time.

Key concepts

Logistic Map

The logistic map is a simple yet powerful mathematical model used to describe various types of population growth. It is represented by the equation: \(x_{t+1} = r x_t (1 - x_t)\). Here, \(x_t\) is the population size at time \(t\), and \(r\) is the growth rate.
This equation showcases how a population changes over time under different conditions specified by \(r\).
Despite its simplicity, the logistic map can produce a range of behaviors from stable fixed points to totally chaotic sequences, depending on the value of \(r\).

• For small values of \(r\), the population stabilizes at a certain point.
• For intermediate values, it can lead to periodic cycles, where population size oscillates between distinct values.
• As \(r\) increases further, the system becomes chaotic and unpredictable.

The logistic map serves as a classic example of how complex and chaotic dynamics can arise from simple non-linear equations.

Sensitivity to Initial Conditions

One of the hallmark features of chaos theory is the notion of sensitivity to initial conditions. This is exemplified in the logistic map experiment described in your exercise.
Even small changes in initial state \(x_0\) can lead to vastly different outcomes over time.
Take, for instance, calculations with \(x_0 = 0.4\) and \(x_0' = 0.5\). Initially, these values are close, but as time progresses, the small difference can dramatically grow, showing how unpredictable the system can become.
Such sensitivity implies that even tiny errors or uncertainties in measurement can amplify over time, making long-term predictions practically impossible in chaotic systems.
This characteristic is often referred to as the "butterfly effect," suggesting that minor changes can have disproportionately large effects in complex systems.

Periodic Evolution

At specific growth rates, the logistic map delivers periodic evolution, where the system's behavior becomes cyclical.
For example, with \(r=3.3\), the logistic map exhibits a period-2 cycle.
Instead of settling on a single value or behaving unpredictably, the system alternates between two values indefinitely.
This periodicity showcases how systems that appear chaotic at first sight can have underlying frequencies or patterns in their behavior.
Regular cycles like these can occur at various growth rates and showcase the system's tendency to lock into repeating sequences, offering a glimpse of orderly dynamics within chaos.

Exponential Growth

In the chaotic regime of the logistic map, particularly observed at a growth rate like \(r=3.6\), the system begins to exhibit exponential divergence.
This means that tiny differences in initial conditions—such as between \(x_0=0.4\) and \(x_0' = 0.400001\)—result in differences that grow exponentially over time.
Initially insignificant, these differences quickly become large, filling the system's behavior with unpredictability.
This growth rate captures the essence of chaotic dynamics where instead of converging to an equilibrium or a periodic cycle, the system diverges rapidly.
Such behavior is a fundamental component of chaotic systems and one of chaos theory's most intriguing and counterintuitive aspects.