Georgia Institute of Technology College of Sciences

In 1994, Dumortier, Roussarie and Rousseau launched a program aiming at proving the finiteness part of Hilbert’s 16th problem for the quadratic system. For the program, 121 graphics need to be proved to have finite cyclicity. In this presentation, I will show that 4 families of HH-graphics with a triple nilpotent singularity of saddle or elliptic type have finite cyclicity. Finishing the proof of the cyclicity of these 4 families of HH-graphics represents one important step towards the proof of the finiteness part of Hilbert’s 16th problem for quadratic systems. This is a joint work with Professor Christiane Rousseau and Professor Huaiping Zhu.

We present a KAM-like theorem for the existence of quasi-periodic tori with a prescribed Diophantine rotation for a discrete family of dynamical system. The theorem is stated in an a posteriori format, so it can be used to validate numerical computations. The method of proof provides an efficient algorithm for computing quasi-periodic tori. We also present implementations of the algorithm, illustrating them throught several examples and observing different mechanisms of breakdown of qp invariant tori. This is a joint work with Alex Haro.

(Joint work with A. Avila and S. Jitomirskaya). An analytic, complex, one-frequency cocycle is given by a pair $(\alpha,A)$ where $A(x)$ is an analytic and 1-periodic function that maps from the torus $\mathbb(R) / \mathbb(Z)$ to the complex $d\times d$ matrices and $\alpha \in [0,1]$ is a frequency. The pair is interpreted as the map $(\alpha,A)\,:\, (x,v) \mapsto (x+\alpha), A(x) v$. Associated to the iterates of this map are (averaged) Lyapunov exponents $L_k(\alpha,A)$ and an Osceledets filtration. We prove joint-continuity in $(\alpha,A)$ of the Lyapunov exponents at irrational frequencies $\alpha$, give a criterion for domination and prove that for a dense open subset of cocycles, the Osceledets filtration comes from a dominated splitting which is an analogue to the Bochi-Viana Theorem.

Synchronization of coupled oscillators, such as grandfather clocks or metronomes, has been much studied using the approximation of strong damping in which case the dynamics of each reduces to a phase on a limit cycle. This gives rise to the famous Kuramoto model. In contrast, when the oscillators are Hamiltonian both the amplitude and phase of each oscillator are dynamically important. A model in which all-to-all coupling is assumed, called the Hamiltonian Mean Field (HMF) model, was introduced by Ruffo and his colleagues. As for the Kuramoto model, there is a coupling strength threshold above which an incoherent state loses stability and the oscillators synchronize. We study the case when the moments of inertia and coupling strengths of the oscillators are heterogeneous. We show that finite size fluctuations can greatly modify the synchronization threshold by inducing correlations between the momentum and parameters of the rotors. For unimodal parameter distributions, we find an analytical expression for the modified critical coupling strength in terms of statistical properties of the parameter distributions and confirm our results with numerical simulations. We find numerically that these effects disappear for strongly bimodal parameter distributions. *This work is in collaboration with Juan G. Restrepo.