By the end of this section, you should

1. Understand how non-linear systems may behave much differently than linear systems

2. Trajectories can vary wildly due to small changes in initial values or parameters

3. Understand what phase plane is and how it is created

4. Understand what is meant by "chaos" and how it differs from “randomness”

This section introduces non-linear dynamics and why many real systems can behave in ways that linear models cannot capture. You will see how small changes in initial conditions or model parameters can produce dramatically different trajectories, even when the governing equations look simple. The section starts by building the core vocabulary you need for non-linear systems, including phase space, the phase plane, and how phase portraits help you visualize behavior without solving every equation in closed form. From there, you will explore sensitivity to perturbations through classic examples such as the Lorenz system, then connect that sensitivity to practical uncertainty tools like Monte Carlo simulation. You will also examine the difference between chaos and randomness, and why deterministic systems can still appear unpredictable when they are highly sensitive to initial conditions. Later topics extend the toolkit to discrete-time mappings, including the logistic map, and show how bifurcation diagrams reveal qualitative changes in system behavior as a parameter varies. The section closes by tying these ideas to stability, limit cycles, and the kinds of repeatable patterns that can emerge in non-linear systems. Use the table of contents below to move through the topics in order, or jump directly to the concept you need for analysis, simulation, or interpretation.