Dynamic Map
The Dynamic Map allows exploring the behavior of dynamic systems: the Tent Map and the Logistic Map. These maps are recurrent functions that, depending on the parameter \( r \) and initial conditions, can display a variety of behaviors, including stability, periodicity, and chaos.
Tent Map: This is a type of tent-shaped map that divides the interval [0,1] into two parts. For \( x \) values less than 0.5, the function behaves linearly, increasing with slope \( r \). For \( x \) values greater than or equal to 0.5, the function decreases linearly. This map is useful for studying transitions between ordered and chaotic behaviors.
Logistic Map: This is a classic mathematical model describing population growth with limitations. The logistic map equation is \( f(x) = r x (1 - x) \), where \( x \) represents the current normalized population between 0 and 1, and \( r \) is the growth rate. This map exhibits varied dynamics, including fixed points, periodic cycles, and chaos when \( r \) exceeds certain thresholds.
Cobweb Plot:
1. From \( x_n \), draw a vertical line to the curve \( f(x) \) to find \( f(x_n) \).
2. From \( f(x_n) \), draw a horizontal line to the identity line \( y = x \) to determine \( x_{n+1} \).
3. Repeat this process to visualize subsequent iterations.
Cobweb Plot
Evolution of xn
Bifurcation Diagram
The Bifurcation Diagram is a visual tool that represents how the fixed points and periodic cycles of a dynamic system change as a control parameter, in this case \( r \), varies. This diagram is essential for understanding how small changes in \( r \) can trigger transitions from stable to chaotic behaviors.
By generating the bifurcation diagram for the Tent and Logistic maps, one can observe how, as \( r \) increases, the system may bifurcate into multiple fixed points or cycles, eventually exhibiting chaotic behavior. This phenomenon, known as period-doubling bifurcation, illustrates the sensitivity to initial conditions and the emergent complexity of seemingly simple systems.