@article{12259,
  abstract     = {Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions. },
  author       = {Choueiri, George H and Suri, Balachandra and Merrin, Jack and Serbyn, Maksym and Hof, Björn and Budanur, Nazmi B},
  issn         = {1089-7682},
  journal      = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
  keywords     = {Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics},
  number       = {9},
  publisher    = {AIP Publishing},
  title        = {{Crises and chaotic scattering in hydrodynamic pilot-wave experiments}},
  doi          = {10.1063/5.0102904},
  volume       = {32},
  year         = {2022},
}

@article{7563,
  abstract     = {We introduce “state space persistence analysis” for deducing the symbolic dynamics of time series data obtained from high-dimensional chaotic attractors. To this end, we adapt a topological data analysis technique known as persistent homology for the characterization of state space projections of chaotic trajectories and periodic orbits. By comparing the shapes along a chaotic trajectory to those of the periodic orbits, state space persistence analysis quantifies the shape similarity of chaotic trajectory segments and periodic orbits. We demonstrate the method by applying it to the three-dimensional Rössler system and a 30-dimensional discretization of the Kuramoto–Sivashinsky partial differential equation in (1+1) dimensions.
One way of studying chaotic attractors systematically is through their symbolic dynamics, in which one partitions the state space into qualitatively different regions and assigns a symbol to each such region.1–3 This yields a “coarse-grained” state space of the system, which can then be reduced to a Markov chain encoding all possible transitions between the states of the system. While it is possible to obtain the symbolic dynamics of low-dimensional chaotic systems with standard tools such as Poincaré maps, when applied to high-dimensional systems such as turbulent flows, these tools alone are not sufficient to determine symbolic dynamics.4,5 In this paper, we develop “state space persistence analysis” and demonstrate that it can be utilized to infer the symbolic dynamics in very high-dimensional settings.},
  author       = {Yalniz, Gökhan and Budanur, Nazmi B},
  issn         = {1089-7682},
  journal      = {Chaos},
  number       = {3},
  publisher    = {AIP Publishing},
  title        = {{Inferring symbolic dynamics of chaotic flows from persistence}},
  doi          = {10.1063/1.5122969},
  volume       = {30},
  year         = {2020},
}

@article{5878,
  abstract     = {We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the formation of a global attractor. In this final regime, the droplet’s angular momentum spontaneously changes its sign as observed in the experiments of Perrard et al.},
  author       = {Budanur, Nazmi B and Fleury, Marc},
  issn         = {1089-7682},
  journal      = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
  number       = {1},
  publisher    = {AIP Publishing},
  title        = {{State space geometry of the chaotic pilot-wave hydrodynamics}},
  doi          = {10.1063/1.5058279},
  volume       = {29},
  year         = {2019},
}