Poincaré Maps and Chaos Classification

Background

Dynamical systems lie at the heart of astrodynamics. Studying the behaviour of these systems is critical for tasks such as designing efficient spacecraft trajectories and predicting the evolution and stability of planetary systems from observations. An important goal of this analysis is to determine if a given system is chaotic [1]. Chaotic motion exhibits a highly sensitive dependence of the future state of the system on its state in the past, and occurs frequently in n-body gravitational systems. This sensitivity makes analysing these systems challenging but has important implications for space science. For instance, this property can be exploited in mission design: placing a spacecraft on an unstable trajectory allows for large changes to the craft’s destination using small amounts of thrust [2]. An understanding of chaos is also critical for identifying stable exoplanetary systems: a chaotic planetary system may appear to evolve regularly for long periods of time prior to a dramatic ejection of one or more of the bodies from the system [3]. In both of these examples the ability to efficiently differentiate between a stable system configuration and a chaotic configuration undergoing transient near-stable motion is critical.

A typical method for analysing dynamical systems involves many integrations of the equations of motions along with their variational equations for a large grid of initial conditions over long time-scales. These integrations can be used to produce `chaos indicators' (e.g the fast Lyapunov indicator [5] or MEGNO [6]) which are designed to reflect the stability of the system. Researchers have proposed various indicators [4-6] with the aim of reducing the integration time needed to predict stability. These methods usually rely on some quickly converging quantity related to an estimator for the Lyapunov Indicator of the system. Figure 1c shows various rates of convergence for the MEGNO indicator starting from different initial conditions on the Henon and Heiles axisymmetric potential.

An alternative method for analysing system stability makes use of Poincaré maps. Poincaré maps are the result of capturing the state of a system every time it crosses a directional hyper-plane in state space. The resulting set of crossings forms a discrete-time dynamical representation of the original system, projected onto a sub-dimensional space with the important property that there is a correspondence between the limits sets of the original system, and those of the discrete map [1]. With an appropriate set of constraints on the system Poincaré maps can form a two-dimensional projection of the original dynamics which experts with domain knowledge can interpret to determine regions of stable and chaotic motion. Figure 3 highlights some characteristics of the appearance of chaotic and stable motion on a Poincaré map for the swinging Atwood's machine [8].

Project Goals

Machine learning tools such as deep learning have been applied to analyse dynamical systems in applications ranging from system stability analysis to system identification [7, 9, 12]. Part of the utility of Poincaré maps lies in their ability to reveal the underlying structure of chaotic motion, such as the chaotic yellow flower pattern traced out in Figure 2. The goal of this project is to apply machine learning techniques to extract the information distilled in these maps. Possible goals of this process are the construction of efficient chaos indicators from Poincaré map orbits, or even better, the ability to classify regions of the state space of a system into 'periodic', 'quasiperiodic' and 'chaotic' groups based on data from Poincaré maps. Another possible goal would be to perform dynamical system identification using data in the form of Poincaré maps, making use of the characteristic patterns that appear in these maps to estimate the parameters of the dynamical system that produced them.

Deep learning architectures which make use of structure in data such as the spatial convolutions of convolutional neural networks (CNNs) and the sequential input structure of recurrent neural networks (RNNs) have risen to popularity with their recent success in a broad range of applications. CNNs are being examined for medical imaging tasks such as segmenting MRI data to highlight tumors [11], and a combination of both paradigms performs well on video segmentation tasks [13]. These architectures appear to be well suited to the goals of this project and will be explored. As a simple example application a CNN could be trained on a dataset similar to Figure 3 to estimate the values of μ and E for a new map from the same system. A more advanced application could incorporate the sequential information contained in the Poincaré maps to make predictions. One approach would be to provide a sequence of Poincaré maps built up as the system continues to pass through the Poincaré section, like in Figure 2, to a convolutional recurrent neural networks to produce a segmentation of the state space of the system into chaotic and stable regions.

References:

[1] - S., H. J., Parker, T. S., & Chua, L. O. (1991). Practical Numerical Algorithms for Chaotic Systems. In Mathematics of Computation (Vol. 56, Issue 193). https://doi.org/10.2307/2008550

[2] - Ross, S. D., Koon, W. S., Lo, M. W., & Marsden, J. E. (2006). Dynamical Systems, the Three-Body Problem and Space Mission Design. ISBN: 978-0-615- 24095-4

[3] - Tamayo, D., Cranmer, M., Hadden, S., Rein, H., Battaglia, P., Obertas, A., Armitage, P. J., Ho, S., Spergel, D. N., Gilbertson, C., Hussain, N., Silburt, A., Jontof-Hutter, D., & Menou, K. (2020). Predicting the long-term stability of compact multiplanet systems. Proceedings of the National Academy of Sciences of the United States of America, 117(31), 18194–18205. https://doi.org/10.1073/pnas.2001258117

[4] - Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J. M. (1980). Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application. Meccanica, 15(1), 21–30. https://doi.org/10.1007/BF02128237

[5] - Froeschlé, C., Lega, E., & Gonczi, R. (1997). Fast Lyapunov indicators. Application to asteroidal motion. In Celestial Mechanics and Dynamical Astronomy (Vol. 67, Issue 1). https://doi.org/10.1023/A:1008276418601

[6] - Cincotta, P. M., & Simó, C. (2000). Simple tools to study global dynamics in non-axisymmetric galactic potentials - I. Astronomy and Astrophysics Supplement Series, 147(2), 205–228. https://doi.org/10.1051/aas:2000108

[7] - Pathak, J., Lu, Z., Hunt, B. R., Girvan, M., & Ott, E. (2017). Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data. Chaos, 27(12). https://doi.org/10.1063/1.5010300

[8] - Tufillaro, N. B., Abbott, T. A., & Griffiths, D. J. (1984). Swinging Atwood’s Machine. American Journal of Physics, 52(10), 895–903. https://doi.org/10.1119/1.13791

[9] - Celletti, A., Gales, C., Rodriguez-Fernandez, V., & Vasile, M. (2022). Classification of regular and chaotic motions in Hamiltonian systems with deep learning. Scientific Reports, 12(1), 1890. https://doi.org/10.1038/s41598-022-05696-9

[10] - Henon, M., Heiles, C., Henon, M., & Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. AJ, 69, 73. https://doi.org/10.1086/109234

[11] - Ranjbarzadeh, R., Bagherian Kasgari, A., Jafarzadeh Ghoushchi, S., Anari, S., Naseri, M., & Bendechache, M. (2021). Brain tumor segmentation based on deep learning and an attention mechanism using MRI multi-modalities brain images. Scientific Reports, 11(1). https://doi.org/10.1038/s41598-021-90428-8

[12] - Brunton, S. L., Proctor, J. L., Kutz, J. N., & Bialek, W. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences of the United States of America, 113(15), 3932–3937. https://doi.org/10.1073/pnas.1517384113

[13] - Wang, W., Zhou, T., Porikli, F., Crandall, D., & Van Gool, L. (2021). A Survey on Deep Learning Technique for Video Segmentation. http://arxiv.org/abs/2107.01153

Outcome