Lecture 1 (Oct 6th)
Introduction. Discrete dynamical systems. Autonomous and non
autonomous case. Cycles. Simple examples, both 1D and 2D. Relation
between dimensionality and order of the system.
Lecture 2 (Oct 7th)
Solution formula for the linear 1D case: homogeneous and
non-homogeneous case. Example of amortization. Exercises. Equilibrium
points. Examples of equilibrium points.
Lecture 3 (Oct 13th)
Definitions of stability, asymptotic stability, global
stability. Conditions for stability for a linear 1D fisrt-order
equation. Cobweb diagram. Linearization criterion for the 1D case.
Cobweb diagram with GeoGebra.
Lecture 4 (Oct 14th)
Stability criterion: the case of derivative equal to 1 or
-1. Examples. Cobweb Theorem of Economics. Models with expected price.
Lecture 5 (Oct 20th)
Linear discrete dynamical systems. Conditions for
equilibrium. Determinant of a matrix. Eigenvalues of a
matrix. Spectral radius. Stability criterion for linear systems.
Lecture 6 (Oct 21st)
Jury's criterion in dimensions 2 and 3. Examples. Jacobian
matrix. Linearization theorem and idea of the proof. Exercises from
the written tests.
Lecture 7 (Oct 27th)
Linear higher-order discrete dynamical equations.
Explicit solutions. Examples. The non-homogeneous case.
Lecture 8 (Oct 28th)
Second order cobweb model. Gambler's ruin. Examples.
Lecture 10 (Nov 4th)
A trade model between two countries. Criterion of stability for
positive matrices. Discrete logistic map and its bifurcation diagram.
Lecture 11 (Nov 10th)
Exercises from the written test of February 9th, 2024. Ordinary differential equations. Equilibria. General solutions for the first-order 1D linear non-homogeneous case.
Lecture 12 (Nov 11th)
Stability of equilibria. Asymptotic stability. Exponential
stability. General solution for linear k-th ODEs with constant
coefficients. Stability criterion. Solution of the harmonic
oscillator.