Difference equations: linear equations in the plane, nonlinear equations: Grobman-hartman theorem, invariant manifold theorem, homoclinic tangle. Smale's horsechoe, symbolic dynamics, chaos.

Bibliography:
Sections 2.1, 2.2 and 3.5 of D.K. Arrowsmith and C.M. Place
An introduction to dynamical systems
Cambridge University Press, 1990
For Smale's horseshoe, see also Example 4, Chapter IV, section 4 of
J. Palis, W. de Melo
Introdução aos sistemas dinâmicos
Projeto Euclides, 1978

Definitions of alpha and omega-limit sets. Poincaré:-Bendixson theorem: in a vector field on the plane if the omega-limit set of a point is not empty and if it does not contain an equilibrium, then it is a periodic orbit (called a limit cycle).

Persistence of hyperbolic equilibria and periodic trajectories. Genericity. Statement of Peixoto's theorem: characterisation of generic and persistent vector fields on a compact two-dimensional manifold.

Bibliography:
Chapters 7 and 11 of M.W. Hirsch and S. Smale
Differential Equations, Dynamical Systems and Linear Algebra
Academic Press, 1974
Sections 3. to 3.3 of D.K. Arrowsmith and C.M. Place
An introduction to dynamical systems
Cambridge University Press, 1990

Sufficient conditions for instability of an equilibrium in terms of Lie derivatives (Cetaev's theorem) and of eigenvalues.

Equivalence of vector fields, topological equivalence. Examples, equivalence of linear differential equations.

Other results on the local behaviour of equilibria: Grobman-Hatrman theorem, stable manifold theorem (without proofs).

Bibliography:
Chapter 2 of V.I. Arnold
Équations différentielles ordinaires
Mir, 1974
Sections 2.1, 2.2 of D.K. Arrowsmith and C.M. Place
An introduction to dynamical systems
Cambridge University Press, 1990

Changes of coordinates in ordinary differential equations; definition of the pull-back of a vector field by a diffeomorphism.

Statement and proof of the flow-box theorem: every vector field is the pull-back of the constant vector field v(x)=(1,0, . . . ,0) in a neighbourhood of a point p where v(p) in not zero.

Definition of Lie derivative of a function with respect to a vector field, geometrical interpretation. Theorem: If the Lie derivative of a function with respect to a vector field is identically zero, then the trajectories of the vector field are contained in level sets of the function. In this case the function is called a first integral of the vector field. Example: conservative systes with 1 degree of freedom and their phase portraits.

Liapunov's theorems for the stability of equilibria. First theorem: an equilibrium is Liapunov stable if a positive definite function has nonnegative Lie derivative in a neighbourhood of it, it is asymptotically stable if the Lie derivative is strictly negative in a neighbourhood, excluding the equilibrium. Second theorem: equilibrium is asymptotically stable if all the eigenvalues of the linearisation of the vector field around it have negative real parts.

Bibliography:
Chapter 2 of V.I. Arnold
Équations différentielles ordinaires
Mir, 1974
Chapter 2 of D.K. Arrowsmith and C.M. Place
An introduction to dynamical systems
Cambridge University Press, 1990

Linear differential equations in finite-dimensional space. Definition of the uniform norm in the space of linear maps in R^n; properties: invariance under compositon. Definition of exponential of a linear map in R^n, convergence of the series defining the exponential, properties: if two maps commute, then the exponential of their sum is the product of their exponentials; the derivative with respect to the real variable t of exp(tA) is A exp(tA), hence (t,p) -> exp(tA)p is the flow of the differential equation with vector field Ax. Methods for computing exp(tA) using the Jordan canonical form of A.

Definitions: Liapunov stability, asymptotic stability of an equilibrium point. Unstable equilibrium.

Phase portraits for all linear differential equations in the plane, stability of equilibria.

Bibliography:
Section 2.3 of D.K. Arrowsmith and C.M. Place
Ordinary Differential Equations: a qualitative approach with applications
Chapman and Hall, 1982
Sections 5.3 and 5.4 of M.W. Hirsch and S. Smale
Differential Equations, Dynamical Systems and Linear Algebra
Academic Press, 1974

Abstract definition of a dynamical system, examples: ordinary differential equations; difference equations.

Vector fields and ordinary differential equations: existence and uniqueness of solutions, differentiability with initial conditions and parameters. Definitions of: flow; trajectories; phase portrait; equilibrium points. Examples: vector fields in one-dimensional space; pendulum without friction.

Difference equations, relation with differential equation: discretisation of an o.d.e.; first return map to a transverse section. Definition of orbit of a difference equation. Recurrent behaviour: fixed points, periodic points.

Bibliography:
H.W. Broer, Introduction to Dynamical Systems, pages 1-24 of the book
Broer, Dumortier, van Strien, Takens
Structures in Dynamics - finite dimensional deterministic studies
North Holland, 1991