A simple method for detecting chaos in nature

Chaos, or exponential sensitivity to small perturbations, appears everywhere in nature. Moreover, chaos is predicted to play diverse functional roles in living systems. A method for detecting chaos from empirical measurements should therefore be a key component of the biologist's toolkit. But, classic chaos-detection tools are highly sensitive to measurement noise and break down for common edge cases, making it difficult to detect chaos in domains, like biology, where measurements are noisy. However, newer tools promise to overcome these limitations. Here, we combine several such tools into an automated processing pipeline, and show that our pipeline can detect the presence (or absence) of chaos in noisy recordings, even for difficult edge cases. As a first-pass application of our pipeline, we show that heart rate variability is not chaotic as some have proposed, and instead reflects a stochastic process in both health and disease. Our tool is easy-to-use and freely available

Dynamical field inference and supersymmetry

Knowledge on evolving physical fields is of paramount importance in science, technology, and economics. Dynamical field inference (DFI) addresses the problem of reconstructing a stochastically driven, dynamically evolving field from finite data. It relies on information field theory (IFT), the information theory for fields. Here, the relations of DFI, IFT, and the recently developed supersymmetric theory of stochastics (STS) are established in a pedagogical discussion. In IFT, field expectation values can be calculated from the partition function of the full space-time inference problem. The partition function of the inference problem invokes a functional Dirac function to guarantee the dynamics, as well as a field-dependent functional determinant, to establish proper normalization, both impeding the necessary evaluation of the path integral over all field configurations. STS replaces these problematic expressions via the introduction of fermionic ghost and bosonic Lagrange fields, respectively. The action of these fields has a supersymmetry, which means there exists an exchange operation between bosons and fermions that leaves the system invariant. In contrast to this, measurements of the dynamical fields do not adhere to this supersymmetry. The supersymmetry can also be broken spontaneously, in which case the system evolves chaotically. This affects the predictability of the system and thereby make DFI more challenging. We investigate the interplay of measurement constraints with the non-linear chaotic dynamics of a simplified, illustrative system with the help of Feynman diagrams and show that the Fermionic corrections are essential to obtain the correct posterior statistics over system trajectories.Comment: 20 pages, 2 figures, 6 Feynman diagrams, 162 numbered equation

Interactions between vortex tubes and magnetic-flux rings at high kinetic and magnetic Reynolds numbers

The interactions between vortex tubes and magnetic-flux rings in incompressible MHD are investigated at high kinetic and magnetic Reynolds numbers, and over a wide range of the interaction parameter. The latter is a measure of the turnover time of the large-scale fluid motions in units of the magnetic damping time, or of the strength of the Lorentz force in units of the inertial force. The small interaction parameter results, that are related to kinematic turbulent dynamo studies, indicate the evolution of magnetic-rings into flattened spirals wrapped around the vortex tubes. This process is also observed at intermediate interaction parameter values, only now the Lorentz force creates new vortical structures at the magnetic spiral edges, that have a striking solenoid vortex-line structure, and endow the flattened magnetic-spiral surfaces with a curvature. At high interaction parameter values, the decisive physical factor is Lorentz force effects. The latter create two (adjacent to the magnetic-ring) vortex rings, that reconnect with the vortex tube by forming an intriguing, serpentine-like, vortex-line structure, and generate, in turn, two new magnetic rings, adjacent to the initial one. In this regime, the morphologies of the vorticity and magnetic field structures are similar. The effects of these structures on kinetic and magnetic energy spectra, as well as on the direction of energy transfer between flow and magnetic fields are also indicated

A Worked Example of the Bayesian Mechanics of Classical Objects

Bayesian mechanics is a new approach to studying the mathematics and physics of interacting stochastic processes. Here, we provide a worked example of a physical mechanics for classical objects, which derives from a simple application thereof. We summarise the current state of the art of Bayesian mechanics in doing so. We also give a sketch of its connections to classical chaos, owing to a particular $\mathcal{N}=2$ supersymmetry.Comment: 34 pages. Presentation revised and expository material added (v2). Condensed version to appear in The 3rd International Workshop on Active Inferenc

Introduction to Supersymmetric Theory of Stochastics

Many natural and engineered dynamical systems, including all living objects, exhibit signatures of what can be called spontaneous dynamical long-range order (DLRO). This order’s omnipresence has long been recognized by the scientific community, as evidenced by a myriad of related concepts, theoretical and phenomenological frameworks, and experimental phenomena such as turbulence, 1/f noise, dynamical complexity, chaos and the butterfly effect, the Richter scale for earthquakes and the scale-free statistics of other sudden processes, self-organization and pattern formation, self-organized criticality, etc. Although several successful approaches to various realizations of DLRO have been established, the universal theoretical understanding of this phenomenon remained elusive. The possibility of constructing a unified theory of DLRO has emerged recently within the approximation-free supersymmetric theory of stochastics (STS). There, DLRO is the spontaneous breakdown of the topological or de Rahm supersymmetry that all stochastic differential equations (SDEs) possess. This theory may be interesting to researchers with very different backgrounds because the ubiquitous DLRO is a truly interdisciplinary entity. The STS is also an interdisciplinary construction. This theory is based on dynamical systems theory, cohomological field theories, the theory of pseudo-Hermitian operators, and the conventional theory of SDEs. Reviewing the literature on all these mathematical disciplines can be time consuming. As such, a concise and self-contained introduction to the STS, the goal of this paper, may be useful