Dynamical Systems

Introduction Basic concepts What is a dynamical system? Dynamical systems viewpoint Asymptotic behaviour Homeomorphisms of the interval Maps of the circle Generalities Lift and degree Degree one circle maps Rotation number Rational rotation number Irrational rotation number Families of circle maps Monotonicity of rotation number The Arnold family Stability, persistence and bifurcations Persistence and bifurcation Chaos and non-invertible maps Chaos Sequence spaces Shift map Subshifts of finite type Properties of A Chaos Sharkovsky's theorem The quadratic family 1 < 4 3 4 Period doubling bifurcation Accumulation of period-doublings Dynamics on K Two dimensional invertible maps The horseshoe map Dynamics on S The Hénon map (Un)stable manifolds and homoclinic points Saddle points Transverse homoclinics imply chaos The Melnikov method