Course Details

The scope of this course is to analyze the temporal response of physical and chemical systems described by systems of non-linear ordinary differential equations. The course topics are:

1. Description of dynamical system with ordinary differential equations. Steady states (fixed points), orbits, trajectories, periodic orbits and limit cycles
2. Systems of linear ordinary differential equations. Eigenvalues and eigenvectors, similar systems and canonical forms. Autonomous linear systems in two dimensions
3. Stability of non-linear dynamical systems. Linear stability analysis. Analysis of the Duffing oscillator. Volterra-Lotka model.
4. Elementary bifurcations. Center manifold theory. Theory of normal forms. Saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation, Hopf bifurcation. Application in enzyme catalysis, Lorentz model and neurophysiology
5. Coupled oscillators. Type of coupling. Oscillators networks. Application to Bonhoeffer – van der Pol oscillators.
6. Methods, libraries and packages for the analysis of non-linear dynamical systems.
7. Chaos and transition to chaos. Discrete maps and logistic equation. The Rossler oscillator.