Session K07: Nonlinear Dynamics: Chaos (8:45am - 9:30am CST)

Multi-point augmented Lagrangian optimization for chaotic flows

Equipped with the adjoint method based on optimal control theory, gradient-based optimization can be a powerful tool for various flow problems. However, its utility is strongly limited in chaotic flows, as the objective functionals becomes irregular in time such as can be described by the horseshoe mapping of the chaotic dynamics. Regularization methods to compute a usefully smooth gradient in ergodic limit are not always applicable, and their computational costs can be comparable to that of the optimization problem itself. Hence, the optimization of chaotic flows is often viable only for short time periods. We propose an augmented Lagrangian method that directly tackles the degrading mechanism of a horseshoe mapping. The computational framework is demonstrated for chaotic Kolmogorov flows, which shows its efficacy for optimization problems with strong controllability.   

Mapping the Geography of Extreme Events with Optimal Sampling

We introduce a smart sampling algorithm for prediction and quantification of extreme events in dynamical systems. The algorithm iteratively probes the phase space, judging the dangerousness of each state visited by advancing the ``black-box'' dynamics over a prescribed prediction horizon. To keep the number of black-box evaluations at a minimum, the algorithm chooses its next move meticulously by minimizing a criterion that accounts for the importance of the output relative to the input. The criterion and its gradients can be computed analytically, which allows for the possibility of the phase space being high-dimensional. We show that no more than a few dozen samples are necessary for the algorithm to learn the complete geography of extreme events in the phase space. The resulting ``danger map'' can be used to a) compute precursors for the dangerous regions; b) predict, assess the severity of, and control the extreme events in real time; and c) quantify the statistics for the observable of interest.   

Can a butterfly in Brazil control the climate of Texas?

The butterfly effect is a well-known phenomenon in fluid dynamics. A small perturbation to a chaotic dynamical system, such as turbulent flows or the Earth's atmosphere, can lead to large differences at a later time. Lorenz famously posed the question, does the flap of a butterfly's wings in Brazil set off a tornado in Texas? The answer is now widely accepted to be yes. While a tiny perturbation can change the state of a chaotic system, it is unclear whether it can change the long-time statistics. Statistics of many turbulent flows are known to be stable, insensitive to initial conditions. Ergodic theory provided a foundation for such stability. If the weather is ergodic, then it seems unlikely that the butterfly in Brazil can affect the long-time statistics of weather, also known as the climate of Texas. Can a butterfly in Brazil change the climate of Texas? Here we investigate this question computationally with simple chaotic systems, including the Lorenz attractor. We show that arbitrarily small perturbations can significantly influence the statistics of a stable, ergodic system. We also study what it takes to exercise such influence. With the skill to forecast well into the future, one may change the climate of a system through imperceptibly small effort.   

Predicting Critical Transitions in Multiscale Dynamical Systems Using Reservoir Computing

Critical transitions are widespread in many systems in nature. Often times these transition events are induced by a fast driving signal, and are rare and random. Since such events could lead to significant effects, it is important to develop effective methods to predict signal-induced critical transitions early. We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster chaotic process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional.   

Removing pollutants: Flow capture in a model chaotic flow~~

To better understand and optimize pollutant capture in complex geophysical flows, we study the simpler, but still chaotic, time-dependent double-gyre flow model.~For a range of model parameters, the domain consists of a chaotic region, characterized by rapid mixing, interspersed with non-mixing islands in which trajectories are regular.~Pollutant particles are assumed to be passive scalars that drift with the flow and are captured with 100{\%} efficiency~upon reaching a capture unit. To predict the flow capture capability of unit at a fixed location in the flow, we track the movement of the non-mixing islands through the domain and characterize the fraction of time that each point in the domain is in each flow region (i.e., chaotic or non-mixing).~With this information, we can predict bounds on the ultimate capture capability of a~unit placed at an arbitrary location, and therefore determine where to place capture units for~optimal results. We also study the time-dependence of the capture process and demonstrate that extending the flow capture time scale is not necessarily efficient for better capture result depending on the flow parameters and capture unit location.~   

Symbolic Dynamics Applied to 3D Chaotic Fluid Flows with Poorly Defined Vortex Domains

Some popular models of complex three-dimensional fluid flows are based on 3D volume-preserving maps. These maps exhibit chaotic dynamics with complicated geometric structures. The underlying topological structure of these maps can be described using symbolic dynamics techniques. One such method is homotopic lobe dynamics (HLD), which has previously been applied to a chaotic three-dimensional Hill's spherical vortex. In the past HLD was confined to cases where there was a well defined topological vortex domain. However, many flows do not exhibit this topological feature. Through a series of examples, we demonstrate how HLD can be extended to extract symbolic dynamics for such systems; we specifically consider systems where there exists an invariant circle of fluid trajectories connecting two stagnation points. Using this technique we obtain a lower bound for the stretching rate of material surfaces in the fluid.   

Strong Subcritical Dynamics in the Thermal Asymptotic Suction Boundary Layer

The thermal asymptotic suction boundary layer problem occurs when a heated plate drives thermal convection in a fluid subject to a uniform suction perpendicular to the plate itself. The background state is semi-infinite in $z$ with uniform suction velocity $V_z < 0$ and a temperature profile that relaxes to $T_\infty$. This setup becomes linearly unstable at a Rayleigh number $\simeq 20$; it is also known to have a subcritical bifurcation with steady states traced to Rayleigh numbers of $\sim 8$ (Zammert, Fischer, & Eckhardt 2016). In this work, we show that even in the weakly unstable regime, this system is highly chaotic owing to the free energy in the suction flow as well as that of the unstable temperature gradient. Using direct numerical simulations in the infinite half-plane, we follow the subcritical instability below the threshold for linear stability and show that it has highly complex dynamics. We comment on the role played by exact coherent states in unconstrained simulations and the projection of our solutions onto low-dimensional subspaces.   

Automatic-differentiated shadowing methods for optimization and data assimilation of chaotic acoustics

In an acoustic cavity with a heat source, a feedback loop between the pressure waves and the heat released by a source can, under resonant conditions, result in loud oscillations. These undesirable nonlinear oscillations, known as thermoacoustic instabilities, can be chaotic. While effective sensitivity-analysis-based control of instabilities is well-established for eigenvalues, control of chaotic oscillations presents a unique challenge due to the extreme sensitivity of chaotic systems (the butterfly effect). We present a computational analysis and applications to tackle the optimization and data assimilation of the chaotic thermoacoustic oscillations. In particular, we illustrate i) sensitivity analysis ii) parameter optimization and iii) data assimilation, on a chaotic thermoacoustic model. We present a discrete shadowing algorithm, based on unifying the tangent and adjoint versions of the Non-Intrusive Least Squares Shadowing algorithm. We compute the sensitivity of the ergodic averages of the acoustic energy and the Rayleigh index, and use the computed sensitivities for optimizing the heat release rate. Finally, we also develop a shadowing-based algorithm for data assimilation, which improves the predictability of hyperbolic chaos, possibly with a constant time delay.   

Transport of Flexible Filaments in Cellular Flows

The transport and dispersal of suspended objects by flows depends on the interplay between the object's morphology and internal mechanics with the flow at various length scales. We study the transport of flexible filaments in a time independent and spatially periodic cellular flow, focusing on the regime where the size of the structure is comparable with the size of background flow cells. Using experiments and numerical simulations, we show that this regime has surprisingly rich dynamics. Several transport states are identified: ballistic states, trapping states, Brownian walks, and Lévy walks. In particular, we identify a transition from Brownian walks to Lévy walks as filament length is decreased. The positional dynamics of filaments is also shown to be chaotic, even in the limit of rigid filaments. The emergence of chaos and different types of random walks is attributed to the nonlocal interactions between the filaments and the flow. Our results open up new possibilities for the dynamic sorting of filaments according to their length and flexibility.   

De-entanglement of complex flows with time delay coordinates

Time delay coordinates are an important tool to study and model real-world systems. It is well-known that these coordinates can be used to reconstruct an attractor from limited measurements. In this talk, we examine how moderate to long time-delay embeddings de-entangle attractors into simple periodic elements. Many systems, such as chaotic dynamical systems and moderate-Re turbulent flows, exhibit recurrent behavior. During transient phases the flow closely mirrors the motion of periodic orbits before eventually becoming turbulent again. These unstable periodic orbits represent exact solutions of the Navier-Stokes equations and form the skeleton of chaotic dynamics. Here, we use unstable periodic orbit solutions to reveal the de-entanglement of attractors through long time delay embeddings, which may have important implications for the characterization and modeling of complex dynamical systems. This work is demonstrated on several chaotic systems and plane Couette flow of moderate Reynolds number Re=400.   

Energy Exchange In Coupled Systems

We present an approach to identify energy exchange in nonlinear coupled systems and to investigate how this exchange depends on the system control parameters. To illustrate this approach, we evaluate the energy coupling of the bi-dimensional spring pendulum, a paradigm to study nonlinear coupled systems and a model for several systems. The dynamics of a spring pendulum varies according to its total energy and one control parameter that accounts for its physical characteristics. This variation is presented in a sequence of Poincar\'{e} sections of the system which show a characteristic order-chaos-order transition as we change its energy and control parameter. We identify the spring and pendulum like motions and an analytical expression for the coupling energy between them. With this expression we obtain the energy exchange during individual trajectories and identify regions in the parameter space that correspond to strong and weak coupling.