# ECE 515, 8.28

$latex \def\Real{\bathbb{R}}$

1. Course overview (see here).
2. Dynamics:
• Dynamical systems, autonomous and non-autonomous. Controlled dynamical systems.
• Equilibrium points of a dynamical system.
• Linearization of the dynamical system $latex \dot{x}=f(x)$ near an equilibrium point $latex x_*$:
$$\dot{\xi}=\left.\left(\frac{Df}{Dx}\right\vert_{x_*}\right)\xi.$$
• Linearization along a trajectory
• Example: linearization near equilibria and a separatrix trajectory for physical pendulum.

Practice exercise: Consider the dynamical system
$$\dot{x}=x^2+y^2-2;\\ \dot{y}=x^2-y^2.$$

1. Find its equilibrium points
2. Linearize the system near those.

### 2 Responses to ECE 515, 8.28

1. Shuo Wang September 4, 2018 at 6:11 pm #

Dear Prof. Baryshnikov,

Could you please post the lecture notes, I mean, in a more detailed way just like your blackboard-writing, before or after class? I know this request may not be very appropriate considering your limited time, but I have to say, sometimes it is hard to follow your pace since there are so many new notations and they are different from those in the notes we get from ECE copy shop. And I believe this is not only my concerns, half of the class or even more are having the same trouble like me. I would be so appreciated if you could post the detailed lectures. Thank you so much!

Shuo

• yuliy September 5, 2018 at 7:19 pm #

Notes published, here.