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Abstract: A Glass network is a system of first order ODEs with discontinuous right hand side coming from step function terms. The "ON/OFF" switching dynamics from the step functions makes Glass networks effective at modelling switching behaviour typical of gene and neural networks. They also have potential application as models of true random number generators (TRNGs) in electronic circuits. As random number generators, it is desirable for networks to behave as irregularly as possible to thwart potential hacking attempts. Thus, a measure of irregularity is necessary for analysis of proposed circuit designs. The cybersecurity industry wants bit sequences generated by the circuit to have positive entropy. The nature of the discontinuities allows for Glass networks to be transformed into discrete time dynamical systems, where discrete maps represent transitions through boxes in phase space, where all possible box transitions are represented using a directed graph called the transition Graph (TG). Dynamics on the TG naturally allows for the network dynamics to be represented by shift spaces with an alphabet of symbols representing boxes. For shift spaces, entropy is used to gauge dynamical irregularity. As a result it is a perfect measure for the application to TRNGs. Previously it was shown that the entropy of the TG acts as an upper bound for the entropy of the actual dynamics realized by the network. By considering more dynamical information from the continuous system we have shown that the TG can be reduced to achieve more accurate entropy upper bounds. We demonstrate this by considering examples and use numerical simulations to gauge the accuracy of our improved upper bounds.

As studied in ​many examples, higher order correction added to the Newtonian potential often provides more realistic and accurate quasi-homogeneous models in astrophysics. Important examples include the Schwarzschild and the Manev potentials. The quasi-homogeneous N-body problem aims to study the interaction between N point particles under a prescribed potential. The classical (Newtonian) Hill’s lunar problem aims to improve the solution accuracy of the lunar motion obtained by solving the two-body (Earth-Moon) system. Hill's lunar equation under the Newtonian or homogeneous potentials has been derived from the Hamiltonian of the three-body problem in a uniform rotating coordinate system with angular speed $\omega$, by using symplectic scaling and heuristic arguments on various physical quantities. In this talk, we first introduce a new variational method characterizing relative equilibria with minimal energy. This enables us to classify the dynamic in terms of global existence and singularity for all possible ranges of the parameters. Then we derive Hill’s lunar problem with quasi-homogenous potential, and finally, we implement the same ideas to demonstrate the existence of ``black hole effect" for a certain range of the parameters: below and at some energy threshold, invariant sets (in the phase space) with non-zero Lebesgue measure that either contain global solutions or solutions with singularity are constructed.

The primitive equation is an important model for large scale fluid model including oceans and atmosphere. While solutions to the viscous model enjoy global regularity, inviscid solutions may develop singularities in finite time. In this talk, I will review the methods to show blowup, and share more recent progress on the qualitative properties of the singularity formation. Most notably, I will provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations. This is a joint work with C. Collot and Q. Lin.

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