A Case Study on the Parametric Occurrence of Multiple Steady States

We consider the problem of determining multiple steady states for positive real values in models of biological networks. Investigating the potential for these in models of the mitogen-activated protein kinases (MAPK) network has consumed considerable effort using special insights into the structure of corresponding models. Here we apply combinations of symbolic computation methods for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition. We determine multistationarity of an 11-dimensional MAPK network when numeric values are known for all but potentially one parameter. More precisely, our considered model has 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment, and furthermore positivity conditions on all variables and parameters.Comment: Accepted into ISSAC 2017. This version has additional page showing all 11 CAD trees discussed in Section 2.1.

Vortices in polariton OPO superfluids

This chapter reviews the occurrence of quantised vortices in polariton fluids, primarily when polaritons are driven in the optical parametric oscillator (OPO) regime. We first review the OPO physics, together with both its analytical and numerical modelling, the latter being necessary for the description of finite size systems. Pattern formation is typical in systems driven away from equilibrium. Similarly, we find that uniform OPO solutions can be unstable to the spontaneous formation of quantised vortices. However, metastable vortices can only be injected externally into an otherwise stable symmetric state, and their persistence is due to the OPO superfluid properties. We discuss how the currents charactering an OPO play a crucial role in the occurrence and dynamics of both metastable and spontaneous vortices.Comment: 40 pages, 16 figure

Spontaneous patterns in coherently driven polariton microcavities

We consider a polariton microcavity resonantly driven by two external lasers which simultaneously pump both lower and upper polariton branches at normal incidence. In this setup, we study the occurrence of instabilities of the pump-only solutions towards the spontaneous formation of patterns. Their appearance is a consequence of the spontaneous symmetry breaking of translational and rotational invariance due to interaction induced parametric scattering. We observe the evolution between diverse patterns which can be classified as single-pump, where parametric scattering occurs at the same energy as one of the pumps, and as two-pump, where scattering occurs at a different energy. For two-pump instabilities, stripe and chequerboard patterns become the dominant steady-state solutions because cubic parametric scattering processes are forbidden. This contrasts with the single-pump case, where hexagonal patterns are the most common arrangements. We study the possibility of controlling the evolution between different patterns. Our results are obtained within a linear stability analysis and are confirmed by finite size full numerical calculations.Comment: 15 pages, 9 figure

Bifurcation and chaos in the double well Duffing-van der Pol oscillator: Numerical and analytical studies

The behaviour of a driven double well Duffing-van der Pol (DVP) oscillator for a specific parametric choice ($\mid \alpha \mid =\beta$) is studied. The existence of different attractors in the system parameters ($f-\omega$) domain is examined and a detailed account of various steady states for fixed damping is presented. Transition from quasiperiodic to periodic motion through chaotic oscillations is reported. The intervening chaotic regime is further shown to possess islands of phase-locked states and periodic windows (including period doubling regions), boundary crisis, all the three classes of intermittencies, and transient chaos. We also observe the existence of local-global bifurcation of intermittent catastrophe type and global bifurcation of blue-sky catastrophe type during transition from quasiperiodic to periodic solutions. Using a perturbative periodic solution, an investigation of the various forms of instablities allows one to predict Neimark instablity in the $(f-\omega)$ plane and eventually results in the approximate predictive criteria for the chaotic region.Comment: 15 pages (13 figures), RevTeX, please e-mail Lakshmanan for figures, to appear in Phys. Rev. E. (E-mail: [email protected]

Tidal turbine blade load experiments for oscillatory motion

This paper presents blade root bending moment measurements of a horizontal-axis tidal turbine for planar oscillatory motion, conducted in a stationary water towing tank. By comparing the measurements with quasi-steady reconstructions for both single and multiple frequency oscillatory motion, the bending moment was shown to be sensitive to both frequency and amplitude, as well as to the mean tip-speed ratio. The unsteady loads associated with the separation of the flow and dynamic stall are shown to be of considerably greater importance than those which are already present for attached flow, such as added mass and dynamic inflow. A linear model fit to the unsteady bending moment also indicates that the inertia contribution is relatively small. For cases where attached flow exists over the majority of the load cycle, these reconstruction methods are likely to be sufficient to obtain a reasonable prediction of the root out-of-plane bending moment. However, turbines whose blades are likely to operate near stall are likely to require more complex models for accurate load predictions to mitigate the risk of fatigue failure

Testing for Equilibrium Multiplicity in Dynamic Markov Games

This paper proposes several statistical tests for finite state Markov games to examine the null hypothesis that the data are generated from a single equilibrium. We formulate tests of (i) the conditional choice probabilities, (ii) the steady-state distribution of states and (iii) the conditional distribution of states conditional on an initial state. In a Monte Carlo study we find that the chi-squared test of the steady-state distribution performs well and has high power even with a small number of markets and time periods. We apply the chi-squared test to the empirical application of Ryan (2012) that analyzes dynamics of the U.S. Portland Cement industry and test if his assumption of single equilibrium is supported by the data