Overall objectives:

O1 - To introduce the fundamental concepts of discrete dynamical systems, nonlinear differential equations, and systems of differential equations (linear and nonlinear).
O2 - To explore applications of Mathematics to other scientific fields, such as Physics, Biology, Ecology, Chemistry, Economics, and Engineering, showing that apparently distinct problems can be described by common mathematical structures.
O3 - To develop the capacity for abstraction by establishing connections between problems from different fields, and to strengthen skills in the analysis, solution, and interpretation of results obtained through mathematical models.
O4 - To consolidate and deepen the knowledge acquired throughout undergraduate and master's studies.

Syllabus:

P1 - Discrete dynamical systems in IR: Linear difference equations; non-linear difference equations; qualitative study of solutions; stability of fixed points and cycles; asymptotic behavior; applications to population dynamics and economics.
P2 - Discrete dynamical systems in the plane: Systems of linear difference equations; asymptotic behavior; phase plane analysis; systems of non-linear difference equations; linearization and qualitative study of solutions; stability; applications to models of competition and species evolution.
P3 - Nonlinear differential equations: Review of ordinary differential equations; second-order nonlinear differential equations in the phase plane; phase plane of the pendulum; autonomous equations in the phase plane; conservative systems; linear oscillator; cycles; parameter space of conservative systems; applications.
P4 - Systems of differential equations: Review of linear systems in the plane; phase plane analysis; linearization near equilibrium points for nonlinear systems; phase trajectories of linearized systems; stability analysis; applications to predator-prey models and engineering models.

Literature/Sources:

Kelley W. G. & Peterson A. C. , 2001 , Difference Equations , Academic Press
Verhulst, F. , 1996 , Nonlinear Differential Equations and Dynamical Systems , Springer
D. W. Jordan & P. Smith , 2007 , Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , Oxford University Press
Saber N. Elaydi , 2007 , Discrete Chaos: with Applications in Science and Engineering , CRC Press
Saber N. Elaydi & Jim M. Cushing , 2025 , Discrete Mathematical Models in Population Biology: Ecological, Epidemic, and Evolutionary Dynamics , Springer
Saber N. Elaydi , 2005 , An Introduction to Difference Equations , Springer
P. Glendinning , 1999 , Stability, instability and chaos: an introduction to the theory of nonlinear differential equations , Cambridge University Press

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Teaching: Classes are primarily based on the use of the whiteboard for the presentation and explanation of the material, as well as for solving exercises. A computer and projector are also used to support visualization and facilitate the understanding of concepts. Slides may occasionally be used to present specific topics; in such cases, the material will be made available to students in detail through lecture notes or other supporting documents. Software tools will also be used for numerical computations, graphical representations, and to support the analysis and solution of nonlinear problems. Assessment: Assessment consists of two written tests (one in the middle and one at the end of the semester), each contributing 35% to the final grade, and a project (in the form of a report or article) to be submitted at the end of the semester, accounting for 30% of the final grade. The project aims to encourage students to engage in independent research and to obtain results on their own initiative, with guidance from the instructor whenever necessary. It also promotes the development of scientific writing skills, less emphasized in purely test-based assessment, and serves as preparation for the writing of the thesis.