Modeling and analysis of complex nonlinear dynamical systems.
Introduction of fundamental mathematical concepts for understanding and analyzing autonomous linear and non linear ordinary differential equations, including qualitative analysis.
Criteria and theorems for the asymptotic stability of equilibria.
Linearization and Hartman-Grobman theorem.
Linear and nonlinear oscillations (limit cycles).
Bifurcations: saddle-node, transcritical, pitchfork and Hopf bifurcations in continuous time, flip and period doubling bifurcations in discrete and continuous time, respectively.
Bifurcation cascades and routes to chaos.
Chaotic attractors and fractals.
Simulation of nonlinear systems. Software tools for the analysis of complex systems: MATLAB.
Analysis and simulation of complex systems in multidisciplinary fields: physical, biological, ecological and economic systems.
1. Continuous time systems
- Linear and nonlinear differential equations
- Vector fields, phase space and differential equations
- Stability of steady 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 Bifurcation
- 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
- 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.