# Dynamical systems and chaos

### Details

**Course code**: 11001
**Semester**: 1st
**Credits (ECTS)**: 6 credits
**Teaching hours**:
**Teaching guide**: Teaching Guide (PDF)
**Professors**:

### Contents

**Introduction**. Phase Space, Existence and unicity of trajectories, Liouville theorem, Hamiltonian vs dissipative systems.
**1 dimensional flows**. Bifurcations. Geometric representation, Fixed points, Stability, Linear stability analysis, Potential representation, 1d flows in a circle, Saddle-node bifurcation, Transcritical bifurcation, Pitchfork bifurcation, Normal forms, Bifurcation diagrams, Examples, Structural stability, Imperfect bifurcations and Catastrophes.
**2 dimensional flows. Limit Cycles**. Phase portraits, topological consequences, Fixed points, Linear stability analysis, Conservative systems, Harmonic oscillator, Forced damped oscillators, Pendulum, Index theory, Limit Cycles (examples), Hopf bifurcation, Gradient systems, Lyapunov functions, Poincaré Bendixon theorem, Liénard Systems, Van Der Pol oscillator, Relaxation oscillations, Weakly nonlinear oscillators, Multiple time scale analysis.
**One dimensional maps. Chaos**. Logistic map. Fixed points. Periodic solutions. Chaos. Calculation of Lyapunov exponents. Routes to chaos. Universality. Feigenbaum’s renormalization theory.
**3 dimensional flows**. Lorenz model. Chaos. Strange attractors. Poincaré map. Lorenz map. Lyapunov exponents. Floquet analysis.
**Fractals**. Cantor set. Self-similarity. Dimension of self-similar fractals: Hausdorff, box counting, information and correlation dimensions. Kaplan-Yorke conjecture. Generalized dimensions D_q.
**Entrainment**. Circle map. 1:1 frequency locking. Rational lockings. Arnold tongues. Devil’s staircase.
**Synchronization of oscillators.** Weakly coupled oscillators. Reduction to phase dynamics. Synchronization. Landau-Stuart oscillators. Oscillator death. Kuramoto model. Diversity. Order Parameter. Self-consistent solution.
**Excitability.** Biological motivation. Active rotator. Fizhugh-Nagumo model.
**Non-linear time series analysis**. Poincaré section. Fourier characterization. Embedding methods.
**Delayed systems**. Delay in physical and biological systems. Fixed points. Stability analysis. Mackey-Glass model.