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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.
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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.
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.
We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from the Kostlan-Shub-Smale ensemble. For the related Bargmann-Fock ensemble of real analytic functions we establish an asymptotic result for the average number of empty limit cycles (limit cycles that do not surround other limit cycles) in a large viewing window. Concerning the special setting of limit cycles near a randomly perturbed center focus (where the perturbation has i.i.d. coefficients) we prove that the number of limit cycles situated within a disk of radius less than unity converges almost surely to the number of real zeros of a certain random power series. We also consider infinitesimal perturbations where we obtain precise asymptotics on the global count of limit cycles for a family of models. The proofs of these results use novel combinations of techniques from dynamical systems and random analytic functions.
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.
We seek solutions $uinR^n$ to the semilinear elliptic partial difference equation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to the Laplacian operator on a graph $G$ and $f_s$ is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) {it Nonlinear Elliptic Partial Difference Equations on Graphs} (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and b) {it Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region} wherein we present some of our recent advances concerning symmetry, bifurcation, and automation fo We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newtons method, create informative graphics, and be in the underlying variational structure. We use two modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimension we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithms capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelis of a graph. We highlight interesting symmetry and variational phenomena.
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