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

Laboratory

The course provides a strong laboratory practice where students will learn how to simulate and analyse nonlinear dynamical systems.