One-dimensional maps are mathematical functions that describe how a system changes over time. They're used to study complex systems like weather patterns and population dynamics by simplifying them into a single variable that evolves with each iteration.

These maps are crucial in chaos theory, revealing how small changes in initial conditions can lead to drastically different outcomes. This concept, known as the butterfly effect, has implications for fields like weather forecasting and stock market predictions.

One-Dimensional Maps

One-dimensional maps in dynamical systems

• Mathematical functions mapping single variable from one value to another
  • Denoted as $x_{n+1} = f(x_n)$ where $x_n$ is current state and $x_{n+1}$ is next state
  • Function $f$ determines evolution of system over time
• Used to study behavior of dynamical systems
  • Provide simplified representation of complex systems (weather patterns, population dynamics)
  • Allow analysis of long-term behavior and identification of patterns and structures (attractors, bifurcations)

Iterative processes for chaos theory

• Repeatedly applying function to its own output
  • Output of each iteration becomes input for next iteration
  • Generates sequence of values analyzed to understand system's behavior (convergence, divergence)
• Fundamental to chaos theory
  • Study sensitivity of system to initial conditions (butterfly effect)
  • Reveal presence of chaos where small changes in initial conditions lead to drastically different outcomes (weather forecasting, stock market predictions)

Examples of One-Dimensional Maps

Behavior of simple one-dimensional maps

• Tent map is piecewise linear function
  1. $f(x) = 2x$ for $0 \leq x \leq 0.5$
  2. $f(x) = 2(1-x)$ for $0.5 < x \leq 1$
  • Exhibits chaotic behavior and has Lyapunov exponent of $\ln 2$ indicating sensitivity to initial conditions
• Quadratic map (logistic map) defined as $x_{n+1} = rx_n(1-x_n)$ where $r$ is parameter controlling system behavior
  • For certain $r$ values, exhibits period-doubling bifurcations and chaos (population growth, disease spread)
  • Feigenbaum constants $\delta \approx 4.669$ and $\alpha \approx 2.502$ describe scaling behavior of bifurcations

Maps vs discrete-time dynamical systems

• One-dimensional maps are type of discrete-time dynamical system
  • Describe evolution of system at discrete time steps
  • State at each time step determined by previous state and governing function (difference equations)
• Provide insights into behavior of more complex discrete-time dynamical systems
  • Exhibit fixed points, periodic orbits, and chaotic attractors (ecological models, economic systems)
  • Stability of fixed points and periodic orbits analyzed using cobweb diagrams and Lyapunov exponents