Session K06: Nonlinear Dynamics: General (8:45am - 9:30am CST)

Data-driven prediction of multistable systems from sparse measurements

We develop a data-driven method for predicting the asymptotic behavior of nonlinear dynamical systems from sparse measurements. The systems of interest are described by partial differential equations (PDEs). As is usually the case in experiments, we assume that, at any given time, the state of the system can only be measured at a few sparse locations. To make accurate predictions, we formulate and solve a metric-learning optimization problem which promotes sparsity. The resulting metric determines the optimal points where measurements have to be made. The sparse measurements are then used in a clustering algorithm to predict the asymptotic state of the system. We illustrate the application of our method on a reaction-diffusion equation and show that it makes correct predictions more than 85\% of the time.   

Mathematical model of gender bias in physics based on Galtung's triangle of violence

Johan Galtung proposed his theory of a triangle of violence in 1969 to explain how unseen violence hidden in the structures and culture of society can lead to the suppression of whole groups of people. In addition to direct violence, the other two points of the triangle are ``Structural violence'' and ``cultural violence''; indirect methods that dominant groups use to cause harm. This study uses an iterative map based on the well known Lotka-Volterra equations to model the triangle of violence. The r-value is used to represent structural violence, the interaction term between two populations represents cultural violence, and a stochastic term direct violence. Following an earlier study on the stochastic logistic map, the effective r value is calculated for two different interacting populations, dominant and non-dominant. These can be used to calculate the total population and the percent of the total population for either group. Based on these results, we show that physics remains less diverse than other sciences because we graduate many fewer students on average than other sciences.   

Nonlinear damping of sloshing motion caused by a piece-wise linear contact line model

We consider the sloshing motion in an idealized, two-dimensional, container. We show that the presence of a broken-line piece-wise linear contact line model relating the contact line velocity to the contact angle can be accounted for by a projection method on the eigenmode basis pertaining to each linear piece. We demonstrate that each slope discontinuity results in a loss of total energy and eventually contributes to the progressive nonlinear damping of the sloshing motion.   

Low dimensional chaos in the solar magnetic cycle?

The nature of dynamics underlying the solar-cycle remains unclear. Considerable work has been done to understand if it is stochastic or has the character of deterministic chaos. In this talk, the sunspot series is analyzed using direct dynamical tests for deterministic chaos. A model based on the findings is constructed and the possible role of planetary influence on the solar cycle through a stochastic-resonance like mechanism is explored.   

Bifurcations and Chaos in the Fluid-Structure Interaction Dynamics of a Dipteran Flight Motor

The flapping dynamics of a Dipteran flight motor have been studied numerically by using a discrete forcing type Immersed Boundary Method (IBM) based in-house fluid-structure interaction (FSI) solver at a Reynolds number of 100. A bifurcation study has been performed considering the amplitude of the wing actuation force as the control parameter. At lower values of the bifurcation parameters, the wake structures and the aerodynamic loads are similar to that of a rigid foil under the sinusoidal plunge. Further increment in the bifurcation parameter results in multiple harmonic frequencies in the structural response, which results in unequal speeds between the up and down strokes of the wing. Consequently, an asymmetric flow-field is obtained, which is also reflected in the aerodynamic loads. Eventually, the system response exhibits chaos at even higher values of the bifurcation parameter. Interesting dynamical behaviors such as quasi-periodicity, transient chaos, and intermittent transitions between chaotic and quasi-periodic states have also been observed. This study will help in understanding the physics during the transition in Dipteran flight, which can be crucial in developing the futuristic flapping-wing micro air vehicles.   

Predicting the onset of shock-induced buffet using Dynamic Mode Decomposition

During transonic flight aircraft can experience a shock-induced buffet, an oscillation felt by the pilot and the aircraft structure which poses a significant constraint on the aircraft design. In idealized 2D flows buffet is linked to a Hopf-type bifurcation, although realistic flow configurations additionally contain a range of background flow features. In this talk we show how Koopman analysis and Dynamic Mode Decomposition (DMD) techniques can be used to predict the onset of the buffet by tracking decay of transients in pre-buffet simulations generated by a Reynolds-Averaged Navier--Stokes code on unstructured meshes. DMD algorithms decompose a sequence of snapshots into a sum of modes; to predict the buffet bifurcation we track the primary mode associated with the buffet across the threshold of stability. We demonstrate how the approach performs when applied to time-resolved simulations and simulations without physically-accurate timestepping. The results show that in the idealized time-resolved case the bifurcation could be predicted by tracking the change in time constants, although duration and resolution of input data affect the accuracy of the prediction. An additional challenge in realistic flows is identifying the primary mode among other components of the flow.   

Topological entropy calculation in 2D fluid flows using limited-time tracer particles

Topological entropy quantifies the complexity of 2D chaotic flows by measuring the stretching rate of a material curve. This quantity can be estimated by the trajectories of an ensemble of passive tracers, such as beads. Previous work has required these trajectories to persist for the full duration of experimental interest. However, it is common in experimentally tracked data for trajectories to begin and end at different times, either because trajectories have moved out of the focal plane, left the image domain, or have simply not been tracked properly from frame to frame. In this talk, we extend our previous method---the ensemble-based topological entropy calculation (E-tec)---to accommodate such limited-time trajectories. Through numerical simulation, we show that one can still compute an accurate topological entropy even when no single trajectory persists for the full duration of the time interval of interest.   

Fluid Mixing using Braids on a Lattice

Fluid mixing has many important applications, from predicting the behavior of microfluidic devices on the small scale to mitigating the spread of pollutants in the ocean on a larger scale. We often want to maximize mixing, such as when designing an efficient mixer for industrial use. We consider a specific model of mixing in two dimensions, in which stirring rods move the fluid around. To measure mixing, we track the stretching of material lines in the fluid. The exponential stretching rate of these lines over time quantifies the strength of mixing. We start with stirring rods arranged in a lattice, pairs of which execute either clockwise or counterclockwise switches. Collections of these braid generators which can be executed on the lattice simultaneously constitute a braid operator, and a time ordered set of braid operators constitutes a lattice braid. We have a novel algorithm that uses a topological characterization of material lines to find their evolution under the action of lattice braids. We are interested in what patterns of lattice braids maximize the mixing. We also explore the mixing produced through randomly chosen braids.