Course on Complex Dynamic Systems

Prof. Chiara Mocenni

General Information

MSc on Computer nd Automation Engineering, MSc on Management Engineering, MSc on Mathematics

Academic Year: 2013-2014

Number of hours: 48

Course duration: October 1st - December 19th

Teacher contacts: Office N. 232 (II floor), email: mocenni at, tel. +39 0577 234850 - 1015

Office hours: Wednsday, 9.00-11.00 and by appointment. Anyway, you can stop to my office by anytime.


1. Continuous time systems
- Linear and nonlinear differential equations
- Vector fields, phase space and differential equations
- Stability of stady states
- Linearization of nonlinear systems
- Oscillating solutions of nonlinear systems
- Simulations and examples (labs.)

2. Discrete time systems
- Linear and nonlinear maps
- Stability of the fixed points of maps
- The logistic map
- Iterations of maps (labs)

3. Bifurcations
- Saddle-Node bifurcation
- Transcritical bifurcation
- Pitchfork bifurcation
- Hopf Bbifurcation
- Flip bifurcation
- Period doubling bifurcation
- Simulations and examples (labs)

4. Deterministic chaos
- Definitions and examples
- Unpredictability and determinism
- Chaos paths
- Poincare' sections
- Strange attractors
- The Lorenz system
- Numerical solutions of chaotic systems, logistic map, Lorenz system, Rossler systems.

5. Introduction to fractals and spatial autorganization

6. Distributed systems
- Definitions and examples
- Reaction-diffusion equations
- Turing bifurcation
- Spatio-temporal chaos

7. Applications
- Ecological systems: Simple and modified Lotka-Volterra equations for predator-prey mechanisms and species competition
- Population dynamics and economic systems: application of the logistic equation
- Biological and physiological systems: glicolysis, circadian rhythms, models of neurons

The course provides a strong laboratory practice where students will learn how to simulate and analyse nonlinear dynamic systems. The laboratories will be held any Thursday from 10.00 to 13.00 in Lab. Room 124 (I floor).

Course timetable:
1. Tuesday 1st, 11-13 AM. Introduction to linear nonlinear first order dynamical systems. Solutions of first order systems. Phase plane and qualitative motion on the line. Strogatz: Chapter 2. Paragraphs: 2.0, 2.1, 2.2.

2. Thursday 3rd, 10-13 AM. Vector field in first order systems. Existence and uniqueness of solutions. Steady states (equilibria) and dynamics close to the equilibria. Lecture notes and Strogatz: Chapter 2. Paragraphs 2.5, 2.6.

3. Tuesday 8th, 11-13 AM. Examples of nonlinear first order systems. Introduction to linear II order systems. Solutions and classification (to be continued). Strogatz: Chapter 2. Paragraphs 2.1, 2.4, 2.5. Chapter 5. Paragraph 5.1.

4. Thursday 10th, 10-13 AM. Classification of II order linear systems (download scanned notes).
Numerical solution of linear first order systems (download Matlab files:

5. Tuesday 15th, 11-13 AM. Phase space portrait and nullclines in II order systems ( download scanned notes ). Strogatz: Chapter 6. Paragraphs 6.1,6.2, 6.3.
Introduction to Discrete time linear systems ( download scanned notes ). Strogatz: Chapter 10. Paragraph 10.1 (to be continued).

6. Thursday 17th, 10-13 AM. Numerical solution of linear first order systems (download Matlab files:
Numerical solution of linear second order systems (download Matlab files:
Numerical solution of linear maps discrete_LIN.m)

7. Tuesday 22nd, 11-13 AM. Linearization of II order nonlinear systems. Strogatz: Chapter 6. Hartman-Grobman Theorem on homeomorphism between nonlinear and linear dynamical systems. Strogatz: Paragraph 6.3. ( download scanned notes )
Bifurcations in I order systems. Strogatz: Chapter 3. Hopf Bifurcations in II order systems. Strogatz: Paragraph 8.2. Limit cycles. Strogatz: Paragraphs 7.0, 7.1. ( download scanned notes )

8. Thursday 24th, 10-13 AM. Exercise 1 explains how to find the supercritical Hopf bifurcations of a II order system. The system is also analyzed in the Strogatz book (Exercise 7.3.2 at page 205) ( download text and the matlab files).
Exercise 2 explains how to find supercritical and subcritical Hopf bifurcations ( download text and matlab files)
Exercise 3 iexplains how to find supercritical Hopf bifurcation ( download text and matlab files).

9. Tuesday 5th, 11-13 AM. Preparation to the intermediate test of November 7th.

10. Thursday 7th, 10-13 AM (I intermediate test) Useful files for the exam: (download).

11. Tuesday 12th, 11-13 AM. Analysis of discrete time systems: logistic equation. See Strogatz, chapter 10, paragraphs 10.0, 10.1 and 10.3. Simulation of the logistic equation (download matlab and README files).

12. Thursday 14th, 10-13 AM. Logistic equation. Second iterate (download matlab file). Sensibility to initial conditions (download matlab file). Bifurcation diagram (download matlab file).

13. Tuesday  19th, 11-13 AM. III order systems. The Lorenz system: description of the system, equations and analysis (Lorenz_system.ppt); stability of equilibria (stability_lorenz.m); the model (lorenz.m); simulation of the system (lorenz_movie.m); sensibility to initial conditions ( lorenz_sensitivity.m).

14. Thursday 21st, 10-13 AM. III order systems. The Rossler system: description and analysis (Notes on the Rossler system). Stability of equilibria (stability_rossler.m); simulation (; Poincare' section (

15. Tuesday 26th, 11-13 AM. Notes on the paradigm of deterministic chaos and its applications (download presentation).

16. Thursday 28th, 10-13 AM. Attractors, Feignebaum constant, Lyapunov exponents, Feigenbaum constant (download scanned notes). List of projects (download presentation).

17. Tuesday 3rd, 11-13 AM. Spatio-temporal systems (by Dario Madeo). Download the slides of this lecture (lecture_rds).

18. Thursday 5th, 10-13 AM. Projects development (by Dario Madeo).

19. Tuesday 10th, 11-13 AM. Projects development.

20. Thursday 12th, 10-13 AM. II intermediate test: chaotic discrete time and continuous time systems.

21. Tuesday 17th, 11-13 AM. Project development.

22. Thursday 19th, 10-13 AM. Project presentation and discussion.

Teaching material

Lecture Notes on Complex Dynamic Systems (C. Mocenni) (download pdf)
Introduction to Fractals (download pdf)
Deterministic chaos and applications (download pdf)
Analysis and bifurcations of ecological models (download pdf)
Spatio-temporal dynamic systems (download pdf)
Analysis and Simulation of the Logistic Map (download Matlab files)
Analysis and Simulation of the Lorenz system (download presentation and Matlab files)
Analysis and Simulation of the Rossler system (download document and Matlab files)
Poincaré maps (download Matlab files)
Exercises proposed in previous years (download Matlab files)

Bibliografic references

Steven Strogatz, "Nonlinear Dynamics and Chaos", Westview (1994)
Differential equation toolbox (by S. Strogatz) (web page)

Intermediate tests and final exam

The exam will consist with a practical part in laboratory and an oral exam. There will be the possibility of participate to two intermediate tests, the results of which will contribute to the final evaluation and avoid the particpation to the final paractical part.  The intermediate tests are not mandatory. The first test will take place on November 7th in the laboratory 124 at 10.00AM. The second test will be sheduled in December. The second test will include exercises and questions on the first and the second part of the course. The students that have passed the first intermediate test will be requested to respond only to the questions on the second part of the course. The students that did not pass the first test or want to improve their results are also admitted to participate to the final test. In this case they have to complete all the exercises. Some homeworks may be also requested to the students.

After the first test, the students are allowed to request to the teacher the assignment of a project based on research issues. The projects can be performed by groups of max 3 students. To be accounted for the final evaluation, the projects will need to include original contributions.