Recently I’ve been programming simulations of continuous dynamical systems at work, and discrete dynamical systems as part of many solutions to the Advent of Code 2020. In this post I’ll first compare continuous vs discrete system simulation methods, and then “grow” a plant by modelling it as a discrete dynamical system.

Dynamical systems

A time-invariant, first-order, continuous dynamical system can be described with a simple equation \( \frac{dx}{dt}=f(x(t)) \), where \( x(t) \) is the state of the system at time \( t \), and \( f \) is some function describing how to get to the next state in terms of the current state \( x(t) \).

How can we simulate a continous-time system with a discrete computer? A computer can simulate a dynamical system evolving over time by applying the function \( f \) over many infinitely small timesteps \( dt \). In other words, at each subsequent timestep \( dt \), we can calculate the next state with \( x_{next}=x_{now} + dx \). From our first equation, we know that \( dx = f(x(t))dt \), and we arrive at the following computation:

\[ x_{next} = x_{now} + f(x_{now})dt \]

It is difficult to compute \( x_{next} \) accurately because \( dt \) is infintesimally small, and computers cannot directly compute boundless quantities (read: Forward Euler Method).

In discrete dynamical systems however, the evolution of the state \( x \) does not depend on time: \( x_{next} \) is a direct function of the current state \( x_{now} \).

\[ x_{next} = f(x_{now}) \]

Lindenmayer systems are interesting discrete dynamical systems to study because they demonstrate how a simple state mutation function applied many times can generate beautifully complex results.

Growing a fractal plant

A particularly interesting Lindenmayer system is the fractal plant. A fractal plant is grown by repeatedly applying a set of “growth” rules that describe how to get from the current plant state to the next. Here’s the rules I used:

• X –> F+[[X]-X]-F[-FX]+X
• F –> FF

The plant’s state is represented as an ordered list of symbols that evolve according to the rules above. Each symbol can be interpreted as an instruction for a 2D renderer, resulting in a plant-like image.

• F –> “draw forward”
• - –> “turn right”
• + –> “turn left”
• X –> “ignore”
• [ –> “start a new branch”
• ] –> “finish the current branch”

Here’s a plant with 6 iterations starting with a state of X.

You can view the source code on GitHub.