The dynamics of conformal Hamiltonian flows: dissipativity and conservativity

We study in detail the dynamics of conformal Hamiltonian flows that are defined on a conformal symplectic manifold (this notion was popularized by Vaisman in 1976). We show that they exhibit some conservative and dissipative behaviours. We also build many examples of various dynamics that show simultaneously their difference and resemblance with the contact and symplectic case. Keywords: smooth mappings and diffeomorphisms, attractors of solutions to ordinary differential equations, attractors and repellers of smooth dynamical systems and their topological structure, flows related to symplectic and contact structures, invariant manifolds for ordinary differential equations.