Homeomorphisms, diffeomorphisms generated by differential equations. Phase space. Periodic points, Lyapunov stability. Lyapunov function Method. Van der Pol oscillator, Lorenz equations. Conjugation and equivalence of dynamical systems, flow tube theorem. Hyperbolic points of nonlinear systems. Circle maps. Rotation number. Denjoy theorem. Asymptotic behavior. Limit sets, non-wandering set. Planar flows. Lotka–Volterra Model, gradient flow. Index theory and examples. Poincaré–Bendixson theorem. Bendixson criterion. Elliptic points. Local analysis: local stable and unstable manifolds. Grossman Hartman's theorem. Calculation of stable and unstable manifolds. Examples, dynamical systems of biological interest. Smale's horseshoe. Symbolic dynamics. Shift and subshifts of finite type. Homoclinic intersections. Smale theorem on homoclinic points and examples. The notion of hyperbolic set. Anosov systems. Topological transitivity and minimality. Characterization theorems. Examples of topologically transitive systems. Weyl's theorem. Birkhoff's theorem and applications. Topological mixing. Bowen's shadow theorem. Examples. Cardinality of the set of periodic trajectories. Markov partitions. Markov partitions for the Arnol'd cat, transfer matrix, trace and powers. Topological entropy. Definition and properties.