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Lectures on Dynamical Systems
EMS Textbooks in Mathematics

Eduard Zehnder (ETH Zurich, Switzerland)

Lectures on Dynamical Systems

Hamiltonian Vector Fields and Symplectic Capacities

ISBN print 978-3-03719-081-4, ISBN online 978-3-03719-581-9
DOI 10.4171/081
May 2010, 363 pages, hardcover, 16.5 x 23.5 cm.
48.00 Euro

This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at the ETH Zurich.

The first part centres around unstable and chaotic phenomena caused by the occurrence of homoclinic points. The existence of homoclinic points complicates the orbit structure considerably and gives rise to invariant hyperbolic sets nearby. The orbit structure in such sets is analyzed by means of the shadowing lemma, whose proof is based on the contraction principle. This lemma is also used to prove S. Smale’s theorem about the embedding of Bernoulli systems near homoclinic orbits. The chaotic behavior is illustrated in the simple mechanical model of a periodically perturbed mathematical pendulum.

The second part of the book is devoted to Hamiltonian systems. The Hamiltonian formalism is developed in the elegant language of the exterior calculus. The theorem of V. Arnold and R. Jost shows that the solutions of Hamiltonian systems which possess sufficiently many integrals of motion can be written down explicitly and for all times. The existence proofs of global periodic orbits of Hamiltonian systems on symplectic manifolds are based on a variational principle for the old action functional of classical mechanics. The necessary tools from variational calculus are developed. There is an intimate relation between the periodic orbits of Hamiltonian systems and a class of symplectic invariants called symplectic capacities. From these symplectic invariants one derives surprising symplectic rigidity phenomena. This allows a first glimpse of the fast developing new field of symplectic topology.

Keywords: Dynamical systems, ergodicity, transitivity, structural stability, stable and unstable manifolds, hyperbolic sets, shadowing lemma, homoclinic and heteroclinic orbits, Bernoulli systems, chaos, limit sets, Lyapunov function, gradient systems, Morse inequalities, symplectic maps, Hamiltonian vector fields, Hamiltonian formalism, integrable systems, symplectic capacities, Hofer–Zehnder capacity, action functional, mini-max principles, periodic motions on prescribed energy surfaces

Further Information

Review in Zentralblatt MATH 1195.37001

Review in MR 2664490

MAA Reviews