Periodic orbits of dynamical systems


Periodic orbits of dynamical systems

Dimitri Breda (University of Udine)
Hinke Osinga (University of Auckland)
Bernd Krauskopf (University of Auckland)

Board contact:
Dimitri Breda

attendance: 4 credits
validation: 2 credits

1st part: 1-5 July 2019, 8hrs (start on July 1st, 2pm “sala riunioni” DMIF, other lectures on the morning of 3-4-5 July, 8:45am “sala riunioni” DMIF)
2nd part: 9-10 September 2019, 6hrs (start on September 9th, 10.30am “sala riunioni” DMIF, then September 10th, 10.30 and 14.30, “aula A021 – ex 48), program below
3rd part: 11-13 September 2019, 6hrs (start on September 11th, 10.30am “sala riunioni” DMIF, then September 12th and 13th, 10.30, same room), program below

Standard on differential equations, mathematical and numerical analysis, linear algebra; teaching language: English.

Certification via seminar/lecture or laboratory activities on a specific argument related to the subject of the course.


It is an advanced course devoted to the study of periodic orbits of continuous-time dynamical systems generated by autonomous ordinary differential equations. The term “advanced” is used not because the specific subject is advanced itself, rather because the concerned arguments hardly find proper room in standard courses at the MSc level. Therefore, the main objective is to provide those interested in dynamical systems with a further step to complete a basic knowledge of the subject of stability beyond the study of equilibria and stationary solutions, the latter being commonly dealt with in traditional courses.


he course focuses on periodic orbits of dynamical systems generated by ordinary differential equations. In the first part we introduce the necessary tools to answer the fundamental question of local stability of these invariants, resorting to the classic theory of Floquet in connection with Poincaré maps. In the second part we address the problem of computing these orbits, as solutions of boundary value problems in the framework of numerical continuation. In the third and last part we deal with the relevant bifurcation analysis under parameter variation, with an eye on specific models and applications.

Teaching methodologies: front theoretical lectures; possible laboratory activities on implementation and use of continuation methods; possible seminars on specific arguments.

Other info: course offered also to students of the Scuola Superiore, open also to motivated MSc students in mathematics, computer science, engineering or related disciplines. Course notes and material provided by the lecturers.

  • Program 2nd part:
  1. 9/9, 10.30: continuation of equilibria as solutions of F(x) = 0.
  2. 10/9, 10.30: boundary-balue problem setup for periodic orbits and their continuation.
  3. 10/9, 14.30: practical examples and AUTO demo.
  • Program 3rd part:
  1. 11/9, 10.30: bifurcations of periodic orbits (Hopf, SN/fold, PD, homoclinic).
  2. 12/9, 10.30: period-doubling route to chaos; chaos characterised by infinitely many saddle periodic orbits (laser example).
  3. 13/9, 10.30: torus bifurcation and resonances; route to chaos (laser example).