This textbook introduces the language and the techniques of the theory of dynamical systems of finite dimension for an audience of physicists, engineers, and mathematicians at the beginning of graduation. Author addresses geometric, measure, and computational aspects of the theory of dynamical systems. Some freedom is used in the more formal aspects, using only proofs when there is an algorithmic advantage or because a result is simple and powerful.

The first part is an introductory course on dynamical systems theory. It can be taught at the master's level during one semester, not requiring specialized mathematical training. In the second part, the author describes some applications of the theory of dynamical systems. Topics often appear in modern dynamical systems and complexity theories, such as singular perturbation theory, delayed equations, cellular automata, fractal sets, maps of the complex plane, and stochastic iterations of function systems are briefly explored for advanced students. The author also explores applications in mechanics, electromagnetism, celestial mechanics, nonlinear control theory, and macroeconomy. A set of problems consolidating the knowledge of the different subjects, including more elaborated exercises, are provided for all chapters.

Differential Equations as Dynamical Systems.- Stability of fixed points.- Difference equations as dynamical systems.- Classification of fixed points.- Hamiltonian systems.- Numerical Methods.-Strange Attractors and Maps of an Interval.- Stable, Unstable and Centre manifolds.-Dynamics in the Centre Manifold.- Lyapunov Exponents and Oseledets Theorem.- Chaos.- Limit and Recurrent Sets.-Poincare Maps.- The Poincare-Bendixon Theorem.- Bifurcations of Differential Equations.-Singular Pertubations and Ducks.-Strange Attractors in Delay Equations.- Complexity of Strange Attractors.-Intermittency.- Cellular Automata.- Maps of the Complex Plane.- Stochastic Iteration of Function Systems.- Linear Maps on the Torus and Symbolic Dynamics.- Parametric Resonance.- Robot Motion.- Synchronisation of Pendula.- Synchronisation of Clocks.- Chaos in Stormer Problem.-Introduction to Celestial mechanics.- Introduction to non-Liner control Theory.- Appendices.

Rui Dilão is Professor of Mathematical Physics and Dynamical Systems at Instituto Superior Técnico of the Technical University of Lisbon. In 1986, he obtained the Ph.D. in Physics (Mathematical Physics) from the Technical University of Lisbon and, in 1997, the Habilitation from the same university. In the period 1986-1988 he has been Fellow at CERN, where he collaborated in the planning of the Large Hadron Collider. He has been Collaborator of the scientific program associated with the Portuguese satellite PoSAT-1 (1992-93). In 1999, together with two colleagues, he received the LabMed prize for original research work on laboratorial research medicine. He has also experience on general computational techniques and on mathematical techniques in finance and economics. He has also experience in general computational techniques, biophysics and economics.