Tutorial 1: Linear Dynamical Systems

Tutorial 1: Linear dynamical systems¶

Week 2, Day 3: Linear Systems

By Neuromatch Academy

Content Creators: Bing Wen Brunton, Alice Schwarze

Content Reviewers: Norma Kuhn, Karolina Stosio, John Butler, Matthew Krause, Ella Batty, Richard Gao, Michael Waskom

Production editors: Gagana B, Spiros Chavlis

Tutorial Objectives¶

Estimated timing of tutorial: 1 hour

In this tutorial, we will be learning about 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:

A common notational short-hand is to write $\dot{x}$ for $\frac{dx}{dt}$. The dot means “the derivative with respect to time”.

Today, the focus will be on linear dynamics, where $f(x)$ is a linear function of $x$. In Tutorial 1, we will:

• Explore and understand the behavior of such systems where $x$ is a single variable

• Consider cases where $\mathbf{x}$ is a state vector representing two variables.

Setup¶

Section 1: One-dimensional Differential Equations¶

This video serves as an introduction to dynamical systems as the mathematics of things that change in time, including examples of relevant timescales relevant for neuroscience. It covers the definition of a linear system and why we are spending a whole day on linear dynamical systems, and walks through solutions to one-dimensional, deterministic dynamical systems, their behaviors, and stability criteria.

Note that this section is a recap of Tutorials 2 and 3 of our pre-course calculus day.

\begin{equation}
x(t) = x_0\exp(a t)
\end{equation}

To gain further intuition, let’s explore the behavior of such systems with a simple simulation. We can simulate an ordinary differential equation by approximating or modeling time as a discrete list of time steps $t_0, t_1, t_2, \dots$, such that $t_{i+1}=t_i+dt$. We can get the small change $dx$ over a small duration $dt$ of time from the definition of the differential:

So, at each time step $t_i$, we compute a value of $x$, $x(t_i)$, as the sum of the value of $x$ at the previous time step, $x(t_{i-1})$ and a small change $dx=\dot x\,dt$:

This very simple integration scheme, known as forward Euler integration, works well if $dt$ is small and the ordinary differential equation is simple. It can run into issues when the ordinary differential equation is very noisy or when the dynamics include sudden big changes of $x$. Such big jumps can occur, for example, in models of excitable neurons. In such cases, one needs to choose an integration scheme carefully. However, for our simple system, the simple integration scheme should work just fine!

Coding Exercise 1: Forward Euler Integration¶

Referred to as Exercise 1B in video

In this exercise, we will complete a function, integrate_exponential, to compute the solution of the differential equation $\dot{x} = a x$ using forward Euler integration. We will then plot this solution over time.

Interactive Demo 1: Forward Euler Integration¶

1. What happens when you change $a$? Try values where $a<0$ and $a>0$.

2. The $dt$ is the step size of the forward Euler integration. Try $a = -1.5$ and increase $dt$. What happens to the numerical solution when you increase $dt$?

Section 2: Oscillatory Dynamics¶

Estimated timing to here from start of tutorial: 20 min

We will now explore what happens when $a$ is a complex number and has a non-zero imaginary component.

Interactive Demo 2: Oscillatory Dynamics¶

Referred to as exercise 1B in video

In the following demo, you can change the real part and imaginary part of $a$ (so a = real + imaginary i)

1. What values of $a$ produce dynamics that both oscillate and grow?

2. What value of $a$ is needed to produce a stable oscillation of 0.5 Hertz (cycles/time units)?

Section 3: Deterministic Linear Dynamics in Two Dimensions¶

Estimated timing to here from start of tutorial: 33 min

This video serves as an introduction to two-dimensional, deterministic dynamical systems written as a vector-matrix equation. It covers stream plots and how to connect phase portraits with the eigenvalues and eigenvectors of the transition matrix A.

\begin{aligned}
  \dot{x}_1 &= {a}_{11} x_1 \\
  \dot{x}_2 &= {a}_{22} x_2
\end{aligned}

\begin{aligned}
  \dot{x}_1 &= {a}_{11} x_1 + {a}_{12} x_2 \\
  \dot{x}_2 &= {a}_{21} x_1 + {a}_{22} x_2
\end{aligned}

Coding Exercise 3: Sample trajectories in 2 dimensions¶

Referred to in video as step 1 and 2 of exercise 1C

We want to simulate some trajectories of a given system and plot how $x_1$ and $x_2$ evolve in time. We will begin with this example system:

We will use an integrator from scipy, so we won’t have to solve the system ourselves. We have a helper function, plot_trajectory, that plots these trajectories given a system function. In this exercise, we will write the system function for a linear system with two variables.

Interactive Demo 3A: Varying A¶

We will now use the function we created in the last exercise to plot trajectories with different options for A. What kinds of qualitatively different dynamics do you observe?

Hint: Keep an eye on the x-axis and y-axis!

Interactive Demo 3B: Varying Initial Conditions¶

We will now vary the initial conditions for a given $\mathbf{A}$:

What kinds of qualitatively different dynamics do you observe? Hint: Keep an eye on the x-axis and y-axis!

Section 4: Stream Plots¶

Estimated timing to here from start of tutorial: 45 min

It’s a bit tedious to plot trajectories one initial condition at a time!

Fortunately, to get an overview of how a grid of initial conditions affect trajectories of a system, we can use a stream plot.

We can think of a initial condition ${\bf x}_0=(x_{1_0},x_{2_0})$ as coordinates for a position in a space. For a 2x2 matrix $\bf A$, a stream plot computes at each position $\bf x$ a small arrow that indicates $\bf Ax$ and then connects the small arrows to form stream lines. Remember from the beginning of this tutorial that $\dot {\bf x} = \bf Ax$ is the rate of change of $\bf x$. So the stream lines indicate how a system changes. If you are interested in a particular initial condition ${\bf x}_0$, just find the corresponding position in the stream plot. The stream line that goes through that point in the stream plot indicates ${\bf x}(t)$.

Think! 4: Interpreting Eigenvalues and Eigenvectors¶

Using some helper functions, we show the stream plots for each option of A that you examined in the earlier interactive demo. We included the eigenvectors of $\bf A$ as a red line (1st eigenvalue) and a blue line (2nd eigenvalue) in the stream plots.

What is special about the direction in which the principal eigenvector points? And how does the stability of the system relate to the corresponding eigenvalues? (Hint: Remember from your introduction to linear algebra that, for matrices with real eigenvalues, the eigenvectors indicate the lines on which $\bf Ax$ is parallel to $\bf x$ and real eigenvalues indicate the factor by which $\bf Ax$ is stretched or shrunk compared to $\bf x$.)

Summary¶

Estimated timing of tutorial: 1 hour

In this tutorial, we learned:

• How to simulate the trajectory of a dynamical system specified by a differential equation $\dot{x} = f(x)$ using a forward Euler integration scheme.

• The behavior of a one-dimensional linear dynamical system $\dot{x} = a x$ is determined by $a$, which may be a complex valued number. Knowing $a$, we know about the stability and oscillatory dynamics of the system.

• The dynamics of high-dimensional linear dynamical systems $\dot{\mathbf{x}} = \mathbf{A} \mathbf{x}$ can be understood using the same intuitions, where we can summarize the behavior of the trajectories using the eigenvalues and eigenvectors of $\mathbf{A}$.