Course on Complex Dynamic Systems
Prof. Chiara Mocenni
Degrees: MSc on Computer nd Automation Engineering, MSc on
Management Engineering, MSc on Mathematics
Number of hours:
Course duration: October
1st - December 19th
Teacher contacts: Office N. 232
(II floor), email: mocenni at dii.unisi.it,
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)
- 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,
5. Introduction to fractals and spatial
6. Distributed systems
- Definitions and examples
- Reaction-diffusion equations
- Turing bifurcation
- Spatio-temporal chaos
- Ecological systems: Simple and modified Lotka-Volterra equations for
predator-prey mechanisms and species competition
- Population dynamics and economic systems: application of the logistic
- 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).
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. 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,
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.
4. Thursday 10th, 10-13 AM. Classification of II order linear systems (download
Numerical solution of linear first order systems (download Matlab files: O1_linear.zip).
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
6. Thursday 17th, 10-13 AM. Numerical solution of linear first order systems
(download Matlab files: O1NL.zip).
Numerical solution of linear second order systems (download Matlab files: O2LIN.zip).
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
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
10. Thursday 7th, 10-13 AM (I intermediate test) Useful files for the exam:
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
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 (
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);
Poincare' section (Rossler_Poincare.zip).
15. Tuesday 26th, 11-13 AM. Notes on the paradigm of deterministic chaos and
its applications (download
16. Thursday 28th, 10-13 AM. Attractors, Feignebaum constant, Lyapunov
exponents, Feigenbaum constant (download
scanned notes). List of projects (download
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.
Lecture Notes on Complex Dynamic Systems (C. Mocenni) (download
Introduction to Fractals (download
Deterministic chaos and applications
Analysis and bifurcations of ecological models (download
Spatio-temporal dynamic systems (download
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)
(download Matlab files)
Exercises proposed in previous years
(download Matlab files)
Steven Strogatz, "Nonlinear Dynamics and Chaos", Westview (1994)
Differential equation toolbox (by S. Strogatz)
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.