Session ZC33: NLD Bifurcations and Chaos

Vertically-hinged wing as a noise-driven Duffing oscillator

The interaction between solid structure and fluid flow often results in complex behaviors. Here, we report a novel experimental finding of flow-induced oscillation with the bi-stability of a simple hinged wing. The wing is vertically hinged about a fixed shaft in a wind tunnel, such that it can rotate freely under external forcing. By shifting the shaft position away from the leading edge of the wing, it is found that the system experiences a supercritical bifurcation from a mono-stationary regime to a sequence of bi-stable regimes with distinct oscillation dynamics. The switching events between the two meta-stable states could be either periodic, chaotic or extremely rare, depending on both the shaft position and the wind speed. Detailed theoretical analysis reveals that the bifurcation is determined by the balance between the lift and pitching moments, and the bi-stable dynamics can be modeled by a modified noise-driven Duffing oscillator. This study suggests that a simple solid structure (not limited to the wing investigated here) freely hinged in a turbulent flow can be used as a canonical model for studying bi-stability systems that are ubiquitous in nature and engineering.   

Edge State Switching in Plane Couette Flow

Recent work on plane Couette flow that has streamwise period longer than the minimal unit identifies edge state switching from the lower-branch Nagata steady solution to a time-periodic solution (PO3) with comparable fluctuation amplitude to turbulence. This edge state switching results to a basin boundary metamorphosis, where the formation of the basin boundary also switches from the stable manifold of the time-periodic edge state to the stable manifold of the steady edge state. The switching is due to the creation of the vigorous PO3 at a homoclinic bifurcation. In contrast, time-periodic edge states in transitional wall-bounded shear flows typically originate from a saddle-node bifurcation. Another periodic orbit (PO2) originates from a different homoclinic bifurcation and exhibits period-doubling cascade that leads to a chaotic attractor. This chaotic attractor collides with PO3, and a boundary crisis occurs at a critical Reynolds number. Such bifurcation scenario is consistent with the occurrence of boundary crisis in transitional wall-bounded shear flows. For these kinds of flows, transient turbulence is observed at Reynolds number above the critical value.   

Transition to chaos in a square vortex flow

We present a combined numerical and experimental study of transition to chaos in a quasi two dimenisonal flow composed of approximately square vortices. Such a flow (at low Reynolds numbers) is generated by driving a shallow fluid layer using an electromagnetic force that is nearly sinusoidal in both lateral directions. With increasing forcing strength the flow undergoes a sequence of symmetry breaking bifurcations that lead to steady, periodic, or quasi-periodic temporal dynamics. By constructing first recurrence maps we analyze how the dynamics of the quasi-periodic solution change with increasing Reynolds number. Our analysis reveals several narrow Reynolds number windows where the dynamics are characterized by resonance and frequency locking.We finally show that the flow transitions to chaos via the break-up of the two-torus.   

Learning Chaotic Dynamics through DMD and Neural Networks

Dynamic mode decomposition (DMD) has become a practical model-free time-series analysis and modeling approach due primarily to its ability to provide modal characterizations of complex flows using only linear spectral techniques without recourse to constitutive equations. While modern machine learning methods have been combined with DMD to enhance its descriptive, reconstructive, and predictive accuracy, chaotic time series can still prove challenging to model accurately with DMD, especially for reconstruction and prediction.

We present extensions to DMD using autoencoders coupled with Takens embeddings that allow us to accurately learn and then predict dynamics across a range of chaotic dynamical systems. These include the classic Lorenz-63 system, the multiscale Rossler system, and the Kuramoto-Sivashinsky equations. The success of our approach rests on allowing the autoencoders to embed dynamics in higher dimensional spaces while also making the Takens embeddings adaptive during the training of the autoencoders. Together, we obtain high reconstruction accuracies over test data while also allowing for nontrivial predictive windows. Likewise, we also explore the impact of our autoencoder networks by studying how they change the mutual information shared across different dimensions in the dynamics. Experiments show that the encoding process significantly alters the information between dimensions, helping to explain better the role neural networks play in learning dynamical systems.    

Flexible, Automated Determination of Incommensurable Basis Frequencies Underlying Quasi-Periodic Processes

In quasi-periodic processes, where power spectra consist of discrete “observed” frequencies with at least one pair incommensurable (i.e., having an irrational ratio), determining the underlying “basis” frequencies is of interest in understanding dynamical behavior. Typically, one seeks, by trial and error, a set of basis frequencies (frequently two) whose integer multiples can be summed to reconstruct the observed frequencies. This can be daunting if there are more than a few observed frequencies. Here we describe an automated method that treats basis frequencies as positive reals and reconstruction coefficients as integers and uses a mixed integer nonlinear programming approach to find the combination minimizing the sum of the magnitudes of the integers, which is in some sense the simplest solution. The method is quite efficient even with many observed frequencies, and can accommodate extrinsic information about either observed frequency precision or one or more underlying frequencies. Compared to alternatives, this method provides greater accuracy and gives a posteriori information on accuracy of observed frequencies. We demonstrate the approach for quasi-periodic experimental buoyancy-driven convection data of Fein, Heutmaker & Gollub (1985) and cylinder wake data of Van Atta & Gharib (1987), and computational data of Blanchard et al. (2019).   

A frequency-domain shadowing approach for sensitivity analysis of chaotic systems

We present a frequency-domain approach for the evaluation of sensitivities of time-averaged quantities of chaotic systems with respect to input parameters. These sensitivities cannot be computed using standard adjoint methods because of the exponential growth of the adjoint variables (due to the presence of positive Lyapunov exponents). The proposed method is based on the well-established least-square shadowing (LSS) approach, that formulates the evaluation of sensitivities as an optimisation problem. Existing formulations of LSS (and its variants) are in the time domain. A reformulation of the LSS method in the frequency (Fourier) space using harmonic balancing is presenred. The resulting system is closed using periodicity. The method is tested on the chaotic Kuramoto-Sivashinsky equation, and the results match with those obtained using the time-domain formulation. However, the storage and computing requirements grow rapidly with the size of the system. To mitigate these requirements, we propose a resolvent-based iterative method that needs much less storage. Application to the Kuramoto-Sivashinsky system gave accurate results with low computational cost. Truncating the large frequencies with small energy content from the harmonic balancing operator did not affect the accuracy of the computed sensitivities. Further details can be found in K. Kantarakias and G. Papadakis (2023), J Comp. Physics, vol. 474, 111757, https://doi.org/10.1016/j.jcp.2022.111757   

Control-based exploration of non-linear systems.

Control-based continuation (CBC) is an experimental method used to explore the bifurcation structure of non-linear systems. Feedback control and continuation techniques are used to discover and stabilize a system’s steady states non-invasively, thus allowing them to be observed. To date CBC has been applied to simple mechanical systems such as oscillating pendulums, springs and bending beams. We apply CBC in a spatially extended system for the first time. The system investigated is an air bubble confined in a Hele-Shaw channel filled with silicone oil. The bubble is placed in the centre of a straining flow and is unstable to both translation and deformation. Real-time feedback control is used to achieve bubble steady-states by injecting/withdrawing fluid from the channel, based on the bubble’s position and shape. A bifurcation diagram is constructed mapping the bubble’s deformation to the injection flow rate of the straining flow. Two unstable solution branches are found which are not observable otherwise.