
This e-book originated from an introductory lecture direction on dynamical platforms given through the writer for complicated scholars in arithmetic and physics at ETH Zurich. the 1st half facilities round risky and chaotic phenomena attributable to the prevalence of homoclinic issues. The life of homoclinic issues complicates the orbit constitution significantly and provides upward push to invariant hyperbolic units within sight. The orbit constitution in such units is analyzed by way of the shadowing lemma, whose facts relies at the contraction precept. This lemma is additionally used to turn out S. Smale's theorem in regards to the embedding of Bernoulli platforms close to homoclinic orbits. The chaotic habit is illustrated within the uncomplicated mechanical version of a periodically perturbed mathematical pendulum. the second one a part of the publication is dedicated to Hamiltonian platforms. The Hamiltonian formalism is constructed within the stylish language of the outside calculus. the concept of V. Arnold and R. Jost indicates that the options of Hamiltonian platforms which own sufficiently many integrals of movement might be written down explicitly and for life. The life proofs of worldwide periodic orbits of Hamiltonian platforms on symplectic manifolds are in keeping with a variational precept for the previous motion practical of classical mechanics. the mandatory instruments from variational calculus are constructed. there's an intimate relation among the periodic orbits of Hamiltonian structures and a category of symplectic invariants referred to as symplectic capacities. From those symplectic invariants one derives brilliant symplectic stress phenomena. this permits a primary glimpse of the short constructing new box of symplectic topology. A ebook of the eu Mathematical Society (EMS). dispensed in the Americas through the yank Mathematical Society.