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Tutorial 1: Linear dynamical systems

Difficulty level

This tutorial covers the behavior of dynamical systems, systems that evolve in time, where the rules by which they evolve in time are described precisely by a differential equation. 

Differential equations are equations that express the rate of change of the state variable 𝑥. One typically describes this rate of change using the derivative of 𝑥 with respect to time (𝑑𝑥/𝑑𝑡) on the left hand side of the differential equation: 𝑑𝑥𝑑𝑡=𝑓(𝑥). A common notational short-hand is to write 𝑥˙ for 𝑑𝑥𝑑𝑡. The dot means "the derivative with respect to time".


Overview of this tutorial:

  • Explore and understand the behavior of such systems where 𝑥 is a single variable
  • Consider cases where 𝐱 is a state vector representing two variables
Topics covered in this lesson
  • One-dimensional differential equations
  • Oscillatory dynamics
  • Deterministic linear dynamics in two dimensions
  • Stream plots

Experience with Python Programming Language.

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