Olga Bernardi

Ricercatore Universitario

in Fisica Matematica


Ufficio:       337 - terzo piano, corridoio AD - Torre Archimede     

                  via Trieste 63 - 35127 - Padova - Italy


Telefono:   +39 0498271340


Email:        obern@math.unipd.it


Curriculum Vitae et Studiorum


PDF (in italian)


Publications and Preprints

(The papers here available might slightly differ from the published versions)


-- O.B., F. Cardin: On Poincaré-Birkhoff periodic orbits for mechanical Hamiltonian systems on $T^*T^n$. Journal of Mathematical Physics 47, number 7, 072701, 15 pp. (2006). PDF

-- O.B., F. Cardin: Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure and Applied Analysis. Volume 5, number 4, 793-812, (2006). PDF

-- O.B., F. Cardin: Some Global Features of Wave Propagation. Rend. Circ. Mat. Palermo, Serie II, n. 78, 19-29, (2006). PDF

-- O.B. A Symplectic Topology approach to the Poincaré-Birkhoff Theorem and to weak solutions for Hamilton-Jacobi equations. PhD Thesis, (2006). PDF

-- M. Guzzo, O.B., F. Cardin: The experimental localization of Aubry-Mather sets using regularization techniques inspired by viscosity theory. Chaos 17, 033107, 9 pp. (2007). LINK

-- O.B., F. Cardin, A. Siconolfi: Cauchy problems for stationary Hamilton-Jacobi equations under mild regularity assumptions. Journal of Geometric Mechanics (JGM), Volume 1, Number 3, (2009). PDF

-- O.B., F. Cardin, M. Guzzo, L. Zanelli: A PDE approach to finite time indicators in Ergodic Theory. Journal of Nonlinear Mathematical Physics, Vol. 16, no. 2, 195-206, (2009). LINK

-- O.B., F. Cardin: On $C^0$-variational solutions to the Hamilton-Jacobi equation. DCDS-A 31 385-406, (2011). LINK

-- O.B., A. Parmeggiani, L. Zanelli: Mather measures associated with a class of Bloch wave functions. Annales Henri Poincaré 13, no. 8, 1807-1839, (2012). PDF

-- O.B., F. Cardin, M. Guzzo: New estimates for Evans' variational approach to weak KAM theory. Commun. Contemp. Math. 15, no. 2,1250055, 36 pp. (2013). LINK

-- O.B., F. Cardin, M. Guzzo: Convergence to the time average by stochastic regularization. J. Nonlinear Math. Phys. 20 no. 1, 9-27, (2013). LINK

-- A. Abbondandolo, O.B., F. Cardin: Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces. Journal of Dynamics and Differential Equations, on line first 2016. LINK

-- O.B., M. Dalla Riva: Analytic dependence on parameters for Evans' approximated Weak KAM solutions. Discrete and Continuous Dynamical Systems-A 37(9), (2017). PDF

-- O.B. A. Florio: A Conley-type decomposition of the strong chain recurrent set. Ergodic Theory and Dynamical Systems, on line first 2017. PDF

-- O.B. A. Florio: Existence of Lipschitz continuous Lyapunov functions strict outside the strong chain recurrent set. Dynamical Systems: An International Journal, to appear, (2018). PDF

-- O.B. A. Florio, J. Wiseman: The generalized recurrent set, explosions and Lyapunov functions. Submitted (2019). PDF


Events


-- 19 settembre 2017. Séminaire de Systèmes dynamiques, Analyse et Géométrie. Laboratoire de Mathématiques d’Avignon (France). LINK


-- 29 ottobre - 03 novembre 2017. Conference on Hamiltonian Systems. Ascona (Switzerland). LINK


-- 12 febbraio - 16 febbraio 2018. Recent advances in Hamiltonian dynamics and symplectic topology. Padova. LINK


-- 4 ottobre - 6 ottobre 2018. Assemblea scientifica GNFM. Montecatini Terme (PT).


-- 7 novembre 2018. Seminario di Sistemi Dinamici Olomorfi 2018-2019. Centro di Ricerca Matematica Ennio De Giorgi. Pisa.


-- 5 febbraio - 8 febbraio 2019. Workshop “Dynamical Systems: from geometry to mechanics”. University of Rome Tor Vergata.


-- 17 giugno - 21 giugno 2019. Conference “Interactions of Symplectic Topology and Dynamics”. Cortona (AR).


Teaching Publications


-- Temi d’esame senza tema. Esercizi svolti per il corso di Fondamenti di Analisi Matematica 1 per gli studenti di Ingegneria, Edizioni Libreria Progetto Padova, (2011).


Teaching a.a. 2018-2019


-- Mathematical Physics. Corso di Laurea Magistrale in Ingegneria dell’Automazione.


-- Moodle of the course: LINK.


-- Office hours: Every TUESDAY at 13:15 in my office.


-- Prof. Giancarlo Benettin notes, HERE.


WEEK 1 - ARGUMENTS

Two examples from population dynamics: Malthusian and Verhulst growth models and their explicit solutions. Some recalls on vector fields: Cauchy problem, Cauchy existence and uniqueness Theorem, one example of a C^0 vector field without uniqueness of solutions. Fundamental definitions for a qualitative study of a vector field: flow, orbit, phase-space, phase-portrait, equilibrium. Phase-portrait of Malthusian and Verhulst models, discussion of equilibria’s role. Properties of flows and orbits. Other examples of 1-dim vector fields and related phase-portraits. Attractors and repellers for 1-di m vector fields. Example of 2-dim vector fields and related phase portraits. Phase portraits (by using solutions) of the harmonic oscillator, the gravitational vector field, the free particle, the harmonic repeller. Qualitative discussion of "stability", "instability", "asymptotic stability”. Some exercises on 1-dim and 2-dim vector fields. Allee effect. Dependence on initial data. General theorem about the exponential convergence-divergence of trajectories (only statement, the theorem will be proved the next week). Dependence on initial data in the previous examples: Malthusian model, x’ = 1, the harmonic oscillator, the harmonic repeller. General observations about sensitivity to initial conditions: the magnetic pendulum, the double pendulum. From determinism to chaos...


WEEK 1 - NOTES


Double pendulum & Magnetic pendulum:


https://www.youtube.com/watch?v=vFdZ9t4Y5hQ


https://www.youtube.com/watch?v=ft5gJs0xIXM


https://www.youtube.com/watch?v=d0Z8wLLPNE0


https://www.youtube.com/watch?v=AwT0k09w-jw


WEEK 2 - ARGUMENTS

General theorem about the exponential convergence-divergence of trajectories (proof). Vector fields depending on parameters. Biforcations and biforcation diagrams. 5 exercises of 1-dim vector fields depending on a real parameter and their corresponding biforcation diagrams. Remarks about the role of X'(x) for the quality of equilibria in the 1-dim case. Linearization of a vector field (examples in 1-dim, 2-dim, 3-dim). Linearization of second order differential equations, example: simple pendulum with friction (equilibria and linearization around equilibria). n-dim linear vector fields: the matrix exponential. 1 exercise: global solution of a 2-dim linear vector field, remarks on the role of eigenvectors and eigenvalues. Classification of 2-dim linear vector fields, with the matrix A diagonalizable: node, saddle, star node, center, spiral. Biforcation diagram for 2-dim linear vector fields, with the matrix A diagonalizable. 2 exercises. 


WEEK 2 - NOTES


WEEK 3 - ARGUMENTS

6 exercises about: biforcation diagrams, linear systems, the effect of small linear terms. Hyperbolic and elliptic fixed point, stable, unstable and central set (definitions). The Grobman-Hartman Theorem (only statement). The "First Lyapunov Theorem" (only statement, we will partially prove this theorem in the sequel). Stable, unstable, asimptotically stable equilibrium (definitions). Lie derivative of a function along a vector field. Topological and differential version of Lyapunov Theorem about stability (with proof). A Lyapunov function for the harmonic oscillator with friction. 1 exercize about Lyapunov functions for a 1-dim vector field. Differential version of Lyapunov Theorem about asymptotic stability (with proof). Rabbits-sheep model: the principle of competitive exclusion in biology. Asymptotic stability of the origin for the pendulm with friction.


WEEK 3 - NOTES


WEEK 4 - ARGUMENTS

A Lyapunov function for the harmonic oscillator with friction. 3 exercises about: Lyapunov functions, biforcation diagrams. Phase-portraits of 1-dim conservative systems: general properties. The phase-portrait by using the conservation of energy for the gravitational force near earth, the harmonic oscillator, the harmonic repeller, the Keplerian gravitational force, the pendulum. 3 exercises. Prof. Benettin's talk:

"The tortuous path of determinism in classical mechanics" (slides in Moodle).


WEEK 4 - NOTES


DOMANDE DI AUTOVALUTAZIONE - 25 MARZO


WEEK 5 - ARGUMENTS

4 exercises about: phase portraits of 1-dim dynamical systems, biforcations, phase portraits of conservative systems, formula for the period, initial conditions giving periodic/non periodic orbits. First integrals. Attractivity is impossible for conservative systems (with proof). The limit cycle: definition, properties and a first example. A detailded discussion of a model for the mechanical clock. The discrete dynamics induced by a one dimensional map. Fixed points and role of the linearization. 1 example: f(x) = x^2. The logistic map: analytical discussion, cobweb plot and biforcation diagram. 1 exercize about equilibria and biforcations of a mechanical system. Constrained dynamical systems on surfaces or on curves. Various examples.


WEEK 5 - NOTES


WEEK 6 - ARGUMENTS

Ideal constraints (definition and discussion), constrained systems of N points, proof of the formula for the kinetic energy in terms of the Lagrangian coordinates (discussion and first examples). The kinetic energy in terms of the Lagrangian coordinates: various examples. The components of the forces, the conservative case. Lagrange equations (only statement, the proof will be given the next week). Lagrange equations in the conservative case. 1 example: Lagrangian and Lagrange equations for a 2-dim mechanical systems. 5 different exercises in preparation for the exam of the next week.


WEEK 6 - NOTES


WEEK 7 - ARGUMENTS

Detailed proof of Lagrange equations, 3 different exercises about Lagrange equations, equilibria and their stability, first integrals. Normal form of Lagrange equations (details in the mechanical case), proof of the Jacobi (first) integral, equilibria and their stability in the mechanical case. Lagrange equations are invariant in the form under changes of coordinates, 1 example in the case of Newton equations. 


WEEK 7 - NOTES     TESTO PRIMO COMPITINO


WEEK 8 - ARGUMENTS

The angular velocity. Poisson formula (with proof). Example: rotation on a plane. Rigid motions: definition and fundamental formula. Konig theorem for the kinetic energy of a rigid system (body). The inertia matrix. Exercises: inertia matrices for the bar, the ring, the disc and the rectangle (square). Various exercises on mechanical systems with rigid bodies: Lagrangian equations, equilibria and their stability, cyclic coordinates, first integrals.


WEEK 8 - NOTES


WEEK 9&10 - ARGUMENTS

Small oscillations. Deduction of the equation for the characteristic frequencies. Potential depending on the velocity. Generalized potential for the Coriolis force and the Lorentz force. Lagrange-Dirichlet Theorem (with proof). Non-degenerate Hessian Theorem (only statement). Routh reduced Lagrangian (with proof). Detailed examples: Routh reduced Lagrangian and phase-portrait for the central motion and the motion on the 2-torus without forces. Noether Theorem (with proof). Examples: cyclic coordinates, central potential. Conservation of the total quantity of motion and the total augular momentum. Various exercises on equilibria, stability, characteristic frequencies.


WEEK 9&10 - NOTES


WEEK 11 - ARGUMENTS

Miscellanea: 1- Chaos in "simple" dynamical systems: the standard map; 2- Another example of limit cycle: the Van der Pol equation; 3- Potential depending on the velocity: the magnetic stabilization. Detailed discussion of the spherical pendulum: phase-portrait of the reduced 1-dim system and re-construction of the trajectories. The Foucault pendulum: detailed solution of the linearized equations around the south pole. 2 exercises.


WEEK 11 - NOTES


Foucault pendulum:


https://www.youtube.com/watch?v=VPxu1zANe0c


https://www.youtube.com/watch?v=Mn4NLqp-4Fc


WEEK 12 - ARGUMENTS

Functionals: definition and various examples. Gateaux-differentiability, stationary point of a functional. The action functional. The Hamilton Principle (or Least Action Principle), with proof. Remarks on the variational formulation of Lagrange equations. Geodesics: definition, variational formulation. Geodesics on the plane, geodesics on the sphere. 1 exercise on Calculus of Variations: minimal surface of revolution. The Hamiltonian formalism: the Legendre transform, the Legendre transform conjugates Lagrange equations and Hamilton equations (with proof). Various exercises on equilibria, stability, characteristic frequencies.  


WEEK 12 - NOTES


ANNUNCI VARI


-- Hanno superato il secondo compitino-primo appello:

BARIN - 30/30

BENCIOLINI - 30/30

BORILLE - 30/30

BRCINA - 25/30

CANOVA - 30/30

CIMOLATO - 28/30

CRESCENTE - 26/30

CUNI 24/30

FACIN - 18/30

FALCONI - 30/30 e lode

FERRARESE - 24/30

FURLAN - 25/30

GAMBARETTO - 20/30

GASPAROTTO - 24/30

GRIGOLETTO - 22/30

IOB - 24/30

LISCIANDRA - 25/30

MARCHINI - 24/30

MONTE - 29/30

ROZZI - 30/30

SCAVAZZA - 25/30

SOLDA' - 27/30

STEFAN - 23/30

TEDESCO - 24/30

TREVISAN - 19/30


-- Hanno superato il secondo appello:


BORDIN - 20/30

TREVISAN T. - 23/30


-- Terzo appello: Martedì 23 luglio ore 14:30 in aula Ee.

-- Orali: Venerdì 26 luglio ore 10:00 (mio ufficio, Torre Archimede).


-- Quarto appello: Venerdì 13 settembre ore 10:00 in aula P3 (Paolotti).

-- Orali: Martedì 17 settembre ore 10:00 (mio ufficio, Torre Archimede).


Ultimo aggiornamento: Giovedì 04 luglio 2019