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Register a new userA timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics
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We discuss how matrix-free/timestepper algorithms can efficiently be used with dynamic non-Newtonian fluid mechanics simulators in performing systematic stability/bifurcation analysis. The timestepper approach to bifurcation analysis of large scale systems is applied to the plane Poiseuille flow of an Oldroyd-B fluid with non-monotonic slip at the wall, in order to further investigate a mechanism of extrusion instability based on the combination of viscoelasticity and nonmonotonic slip. Due to the nonmonotonicity of the slip equation the resulting steady-state flow curve is nonmonotonic and unstable steady-states appear in the negative-slope regime. It has been known that self-sustained oscillations of the pressure gradient are obtained when an unstable steady-state is perturbed [Fyrillas et al., Polymer Eng. Sci. 39 (1999) 2498-2504]. Treating the simulator of a distributed parameter model describing the dynamics of the above flow as an input-output black-box timestepper of the state variables, stable and unstable branches of both equilibrium and periodic oscillating solutions are computed and their stability is examined. It is shown for the first time how equilibrium solutions lose stability to oscillating ones through a subcritical Hopf bifurcation point which generates a branch of unstable limit cycles and how the stable periodic solutions lose their stability through a critical point which marks the onset of the unstable limit cycles. This implicates the coexistence of stable equilibria with stable and unstable periodic solutions in a narrow range of volumetric flow rates.
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Microbial electrolysis cells (MECs) are a promising new technology for producing hydrogen cheaply, efficiently, and sustainably. However, to scale up this technology, we need a better understanding of the processes in the devices. In this effort, we present a differential-algebraic equation (DAE) model of a microbial electrolysis cell with an algebraic constraint on current. We then perform sensitivity and bifurcation analysis for the DAE system. The model can be applied either to batch-cycle MECs or to continuous-flow MECs. We conduct differential-algebraic sensitivity analysis after fitting simulations to current density data for a batch-cycle MEC. The sensitivity analysis suggests which parameters have the greatest influence on the current density at particular times during the experiment. In particular, growth and consumption parameters for exoelectrogenic bacteria have a strong effect prior to the peak current density. An alternative strategy to maximizing peak current density is maintaining a long term stable equilibrium with non-zero current density in a continuous-flow MEC. We characterize the minimum dilution rate required for a stable nonzero current equilibrium and demonstrate transcritical bifurcations in the dilution rate parameter that exchange stability between several curves of equilibria. Specifically, increasing the dilution rate transitions the system through three regimes where the stable equilibrium exhibits (i) competitive exclusion by methanogens, (ii) coexistence, and (iii) competitive exclusion by exolectrogens. Positive long term current production is only feasible in the final two regimes. These results suggest how to modify system parameters to increase peak current density in a batch-cycle MEC or to increase the long term current density equilibrium value in a continuous-flow MEC.
The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on ErdH{o}s--Renyi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.
A special type of rotary-wing Unmanned Aerial Vehicles (UAV), called Quadcopter have prevailed to the civilian use for the past decade. They have gained significant amount of attention within the UAV community for their redundancy and ease of control, despite the fact that they fall under an under-actuated system category. They come in a variety of configurations. The + and x configurations were introduced first. Literature pertinent to these two configurations is vast. However, in this paper, we define 6 additional possible configurations for a Quadcopter that can be built under either + or x setup. These configurations can be achieved by changing the angle that the axis of rotation for rotors make with the main body, i.e., fuselage. This would also change the location of the COM with respect to the propellers which can add to the overall stability. A comprehensive dynamic model for all these configurations is developed for the first time. The overall stability for these configurations are addressed. In particular, it is shown that one configuration can lead to the most statically-stable platform by adopting damping motion in Roll/Pitch/Yaw, which is described for the first time to the best of our knowledge.
We establish the existence of a bifurcation from an attractive random equilibrium to shear-induced chaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent. This addresses an open problem posed by Kevin Lin and Lai-Sang Young, extending results by Qiudong Wang and Lai-Sang Young on periodically kicked limit cycles to the stochastic context.
Using geometric inversion with respect to the origin we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the Riemann sphere. We study its basic properties, and apply it to the study of the Hopf-Takens bifurcation at infinity.
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