Dynamical system theory for engineers

Course description: Establishing the theoretical basis of linear and nonlinear dynamical systems in both continuous and discrete time. Learning how to anticipate the qualitative behavior of the time-evolution of linear and nonlinear dynamical systems. Introduce the students to some tools from ergodic theory and/or stochastic approximation and their relation to dynamical systems. The course is fairly abstract and mathematical, and does not cover applications (other than some examples used for illustration purposes).

Lectures: Wednesday 8:15 - 11:00 in ELA 1. Teaching is in class (most classes are taught on the blackboard). Do NOT take this course if you have a conflict of schedule and cannot attend the lectures. A few video recordings are available on this channel, but the course format has evolved and these videos do NOT cover only a portion of the course material. They are NOT meant to replace the lectures.

Exercises: Wednesday 11:00 - 12:00 in ELA 1 Teaching Assistant: Paula Mürmann.

• The exercise set will be posted on moodle for the week where the exercises are assigned. 
  

Prerequisites:

• Linear Algebra (MATH-111 or equivalent),
• Calculus (MATH-101, -102 and -103 or equivalent),
• Signals & Systems and/or Signal Processing (COM-202, EE 205 or an equivalent linear circuits/systems/signals class),
• Probability (MATH-232 or equivalent).
  

These are mandatory prerequisites: Do NOT take this course unless you have followed successfully all these classes. Complex analysis (MATH-104 or equivalent), Stochastic processes (COM-300 or equivalent) are very strongly recommended prerequisites. Some chapters of the course will require some notions of general topology and measure theory.

Midterm: Wednesday April 9, 8h15am-10h15am. The midterm (max 30 points) covers all the material seen in class until and including Chapter 5, Section 5.3.4. You may use only the extended class notes summary available on the moodle site that you should print yourself. No device with communication capabilities, thus no calculator, computer, and mobile phones have to be stowed away.

Final exam: Monday June 30, 15h15-18h15. Room CE 1 105. The exam (max 70 points) covers all the material seen in class during the semester and is made of two parts:

1. the first part (20 points) is completely "closed book" : no document is allowed. This part will only have questions calling for a true/false answer with a short justification.
   
2. the second part (50 points) is partially "open book", but you may use only the extended class notes summary uploaded on moodle. No class notes, no exercise set, no solution of exercise sets, nor any other material are allowed. 

Final grade (100%) = Midterm (30%) + Final exam (70%).

Office hour. We will hold an extra office hour for last minute questions on Wednesday June 25 from 1h30pm until 3pm in Room INF 211.

Subjects Covered during Week 1:

0. Introduction to the course

• Objectives, content, logistics
  

Subjects covered during week 2:

2. Linear Systems

• Time domain solution
• Stability: Definition
• Exponential of a diagonalizable matrix
• Stability of a linear system with a diagonalizable state matrix
• Jordan normal form
• Stability of a linear system with a non-diagonalizable state matrix
• General form of the free solution of linear systems
• Stability of linear discrete-time systems

Subjects covered during week 3:

2. Linear systems (continued):

• Classification of the flows of 2-d autonomous continuous-time systems
  
• Phase portraits of 2-dim and 3-dim autonomous systems.
• Stability of linear systems: general case
• Link with Frequency Domain Analysis
• BIBO-stability

Exercises:

• Asymptotic Behavior
• Solution of linear autonomous systems
  

Subjects covered during week 4:

4. Introduction to nonlinear systems

• Introduction to nonlinear systems
• Van der Pol oscillator
• Strange attractors
• Fractals

Exercises

• Jordan normal form
• Stability of linear systems
• Classification of equilibria and sketching of phase portraits

• Large-scale notions of (in)stability
• Boundedness and asymptotic uniform boundedness of solutions
• Lyapunov functions for proving boundedness and asymptotic uniform boundedness of the solutions
• Special class of systems: Hamiltonian systems
  

Exercises:

• Solutions of linear discrete-time systems.
• BIBO Stability
• Observability and controllability

Subjects covered during week 6:

5. Stability of nonlinear systems (continued)

• Small-scale notions of (in)stability
• Stability of a solution
• Criterion for stability of a fixed/equilibrium point
• Sketching the flow in the vicinity of a fixed/equilibrium point in 2 dimensions

5. Stability of Nonlinear Systems (continued)

Exercises 

• Stability of equilibrium/fixed points
  

Midterm on Chapters 1, 2, 3, 4 and 5 (until and including Section 5.3.4). (max 30 points)

5. Stability of Nonlinear Systems (end)

Subjects covered during week 10:

• Correction of midterm
• Local and Global Stability of Non-linear Systems.

• Existence of periodic solutions
• Stability of periodic solutions

Subjects covered during week 12:

8. Introduction to chaos

• Property 1: Irregular and aperiodic trajectories
  
• Property 2: Sensitivity to initial conditions
• Elements from the theory of ergodic dynamical systems: probability space, measurable and measure-preserving transformations, ergodic transformations.
• Lyapunov Exponents.
  

• Bifurcations

Subjects covered during week 13:

8. Introduction to chaos

Exercises:

• Elements from ergodic theory.
  

Program for week 14:

8. Introduction to chaos (End, Only 1 hour 8h15- 9h00)

Exercises (from 9h15)

• One-dimensional maps and chaos.
• Final exam of previous years (note: the material is expanded this year, as the course moved from 4ECTS to 6ECTS).

Additional office hours in case of a (very) last-minute question

• Friday June 17: Room BC 329, from 8h30am until 10am. 

Final exam: Monday June 20 15:15 - 18:15, Room CE 1 106. The exam (max 80 points) covers all the material seen in class during the semester and is made of two parts:

1. the first part (20 points) is completely "closed book" : no document is allowed. This part will only have questions calling for a true/false answer with a short justification.
   
2. the second part (60 points) is partially "open book", but you may use only the extended class notes summary that will be handed in during the exam (it is the same summary as the one allowed for the midterm). No class notes, no exercise set, no solution of exercise sets, nor any other material are allowed. 

Back up links (to stay hidden from students)

• This chapter is pre-recorded and available at the links Chap.1 - Introduction Part.1 and Chap.1 - Introduction Part.2.

Extra Office hours:

• Thursday June 22, 14h00-15h00, Room BC 229
• Friday June 23, 11h00-12h00, Room BC 229