Symmetries and geometrical Ansatze in dynamical systems: Hybrid dynamics, control with moving averages, microflows, and a network robustness problem

Four distinct dynamical systems, spanning the areas of hybrid dynamics, control theory, microfluidics and network theory are analyzed. The common thread running through these four model problems is the use in every case of symmetry and geometrical arguments to obtain insights into the dynamics under study. Next is a description of each of the four investigations. (i) Hybrid dynamics: A two-degree of freedom impact oscillator model is studied. The presence of a mixture of continuous dynamics and discrete rebounds yields a rich dynamical behavior. I prove the existence of a nonzero measure set of orbits that lead to infinite impacts in a finite time. I modify the mathematical model to ensure forward existence and uniqueness of solutions. Existence of hybrid periodic orbits is shown. Finally, I present a geometrical interpretation of the dynamics for visualization of numerical simulations. (ii) Control in systems with moving averages: I formally modify the standard state space linear control framework to allow for the inclusion of moving averages in both the input and output signals. I develop observability and controllability tests for this extended framework. (iii) Stokes flow through microfilters: Pressure driven viscous flow through micromachined filters consisting of perforated thin plates is analyzed. I present an analytical formula for the pressure drop across the microfilter versus flow rate ratio that accounts for filter thickness, pore size, pore separation, pore geometric layout pattern, and wall slip-effects. I show that the small inertial effects correction to the analysis is of order the Reynolds number squared. (iv) Robust complex networks: Current literature emphasizes the robustness tradeoff between scale-free and exponential networks. I prove analytically that the optimal network configuration under a classic measure of robustness is altogether different from both of the above: in all cases, random failure and/or attack, there are no more than three distinct node connectivities in the optimal network.;Finally, in Appendix A, I comprehensively review the current state of research in the development of an inhalable form of insulin, specifically discussing the bioefficiency, pharmacokinetics, reliability and safety technical challenges that must be overcome for inhaled insulin to become a reality.