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


-- 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.


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...


Double pendulum & Magnetic pendulum:






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. 



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.


Ultimo aggiornamento: Giovedì 14 marzo 2019