Dynamical Systems 1
A.Y. 2023/2024
Learning objectives
The main learning objective is to provide the fundamentals of elementary theory of dynamical systems, with particular reference to chaos arising from deterministic systems on the one hand, and to ordered dynamics on the other hand.
Expected learning outcomes
Ther students will acquire knowledge and skilsl towards some relevant results of the theory of dynamical systems.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
The following is last year's program: in general, it will remain valid for the next edition as well, but we reserve the right to make changes as we go along.
FIRST PART: CHAOS.
A) Stable and unstable manifolds. Stable manifolds Theorem.
B) Homocline intersections, Chaotic Dynamics and hyperbolic sets.
C) Shadowing Lemma and chaos in the vicinity of homocline intersections
D) Chaotic dynamics for the pendulum with a periodic external force. The Melnikov's method
SECOND PART: PERIODIC ORBITS
A) Continuation of periodic orbits: the Poincarè theorem. The Poincarè map and its linearization and eigenvalues. Floquet multipliers.
B) The Lyapunov center theorem (Periodic orbits near equilibrium points). Applications to the three body problem.
THIRD PART: NORMAL FORM THEORY AND SIEGEL THEOREM
A) Formal theory. Problem of existence of a change of variables reducing and ordinary differential equation to its linear part. Resonances. Poincarè theorem on the existence of a "formal normal form transformation" eliminating the nonlinear part of the equation.
B) Small divisors and Siegel Theorem. Poincare' and Siegel domains. Non-resonance, diophantine conditions. Measure estimates for the small divisors satisfying diophantine conditions. Proof of the Siegel theorem and quadratic scheme
APPENDIX: INFINITE DIMENSIONAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS
A) Dispersion in infinite chain of particles.
B) Periodic orbits in partial differential equations
FIRST PART: CHAOS.
A) Stable and unstable manifolds. Stable manifolds Theorem.
B) Homocline intersections, Chaotic Dynamics and hyperbolic sets.
C) Shadowing Lemma and chaos in the vicinity of homocline intersections
D) Chaotic dynamics for the pendulum with a periodic external force. The Melnikov's method
SECOND PART: PERIODIC ORBITS
A) Continuation of periodic orbits: the Poincarè theorem. The Poincarè map and its linearization and eigenvalues. Floquet multipliers.
B) The Lyapunov center theorem (Periodic orbits near equilibrium points). Applications to the three body problem.
THIRD PART: NORMAL FORM THEORY AND SIEGEL THEOREM
A) Formal theory. Problem of existence of a change of variables reducing and ordinary differential equation to its linear part. Resonances. Poincarè theorem on the existence of a "formal normal form transformation" eliminating the nonlinear part of the equation.
B) Small divisors and Siegel Theorem. Poincare' and Siegel domains. Non-resonance, diophantine conditions. Measure estimates for the small divisors satisfying diophantine conditions. Proof of the Siegel theorem and quadratic scheme
APPENDIX: INFINITE DIMENSIONAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS
A) Dispersion in infinite chain of particles.
B) Periodic orbits in partial differential equations
Prerequisites for admission
Basic knowledge of Analytical Mechanics and Ordinary differential equations.
Teaching methods
Lectures in presence.
Regular attendance is recommended.
Course materials could be made available on Ariel.
Regular attendance is recommended.
Course materials could be made available on Ariel.
Teaching Resources
Consistent with what was mentioned about the program, this section will also undergo updates in the future.
The topics of the course are contained in the lecture notes of Prof. Dario Bambusi. These lecture notes can be found at the web page http://users.mat.unimi.it/users/bambusi/dinamici_esame.html
Other useful monographs are the following ones:
1) V.I. Arnold: Metodi geometrici della teoria delle equazioni
differenziali ordinarie. Roma : Editori Riuniti, 1989.
2) Earl A. Coddington, Norman Levinson: Theory of ordinary
differential equations. New York : McGraw-Hill Book Company,
1955.
3) Zehnder, Eduard: Lectures on dynamical systems. Hamiltonian
vector fields and symplectic capacities. EMS Textbooks in
Mathematics. European Mathematical Society (EMS), Zürich, 2010
4) Moser, Jurgen. Notes on Dynamical Systems. Courant Lecture Notes
Volume: 12; 2005; 256 pp
5) V.I. Arnold, Andre' Avez:Problemes ergodiques de la mecanique
classique. Paris : Gauthier-Villars, 1967.
6) Anatole Katok, Boris Hasselblatt: Introduction to the modern theory
of dynamical systems. Cambridge : Cambridge University Press,
1995.
The topics of the course are contained in the lecture notes of Prof. Dario Bambusi. These lecture notes can be found at the web page http://users.mat.unimi.it/users/bambusi/dinamici_esame.html
Other useful monographs are the following ones:
1) V.I. Arnold: Metodi geometrici della teoria delle equazioni
differenziali ordinarie. Roma : Editori Riuniti, 1989.
2) Earl A. Coddington, Norman Levinson: Theory of ordinary
differential equations. New York : McGraw-Hill Book Company,
1955.
3) Zehnder, Eduard: Lectures on dynamical systems. Hamiltonian
vector fields and symplectic capacities. EMS Textbooks in
Mathematics. European Mathematical Society (EMS), Zürich, 2010
4) Moser, Jurgen. Notes on Dynamical Systems. Courant Lecture Notes
Volume: 12; 2005; 256 pp
5) V.I. Arnold, Andre' Avez:Problemes ergodiques de la mecanique
classique. Paris : Gauthier-Villars, 1967.
6) Anatole Katok, Boris Hasselblatt: Introduction to the modern theory
of dynamical systems. Cambridge : Cambridge University Press,
1995.
Assessment methods and Criteria
The examination consists of an oral test (by appointment), during which you will be asked to explain ideas, definitions, and results (with proofs) from the course syllabus in order to assess your knowledge and understanding of the topics covered.
The examination is considered passed if the oral test is successfully completed. The grade is expressed on a scale of thirty and will be communicated immediately at the end of the oral test.
The examination is considered passed if the oral test is successfully completed. The grade is expressed on a scale of thirty and will be communicated immediately at the end of the oral test.
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Professor(s)
Reception:
Contact me via email
Office 1039, 1st floor, Dipartimento di Matematica, Via Saldini, 50