To give the participants a fundamental insight into nonlinear phenomena in complex systems; to introduce analytical and computational tools for the analysis of dynamical phenomena in complex systems; practicals and computer exercises will enable the participants to perform nonlinear analyses of mathematical models.

• Define fixed points and stability for one- and two-dimensional dynamical systems and compute the fixed points location and stabilty.
• Perform scaling of models and determine their dimensionless parameters.
• Perform a detailed qualitative phase plane analysis applying linearisation and null clines.
• Explain the properties of stable and unstable manifolds for saddle points.
• Explain the most typical local bifurcations in one- and two-dimensional dynamical systems and determine these in simple systems.
• Demonstrate the absence of periodic solutions in gradient systems using Lyapunov exponents and Dulac's criterion.
• Demonstrate the existence of periodic solutions using Poincaré-Bendicson's theorem.
• Compute fixed points and their stability for one-dimensional iterated maps and explain their bifurcation diagrams.
• Apply numerical methods to determine fixed points and their stability, bifurcation points, approximation of stable and unstable manifolds of saddle points, and to help perform a phase plane analysis.
• Compute the fractal dimension of self-similar objects and explain the most important properties of chaotic solutions.

Nonlinear phenomena: phase plane analysis, limit cycles, chaotic dynamics. Scaling. Index theory. Lyapunov functions. Stability of equilibria and periodic solutions. Computer simulation. Local bifurcations: saddle-node, transcritical, pitchfork, Hopf and period-doubling. Poincare sections and Poincare maps. Stable and unstable manifolds. Global bifurcations. Chaos and predictability; computation of Lyapunov exponents. Fractal geometry; computation of fractal dimension.

Steven H. Strogatz, Nonlinear Dynamics and Chaos, ISBN 0-7382-0453-6.