Dynamical Systems

Chaos, Ito-Stratonovich dilemma, and topological supersymmetry ArXiv ID: 2512.21539 “View on arXiv” Authors: Igor V. Ovchinnikov Abstract It was recently established that the formalism of the generalized transfer operator (GTO) of dynamical systems (DS) theory, applied to stochastic differential equations (SDEs) of arbitrary form, belongs to the family of cohomological topological field theories (TFT) – a class of models at the intersection of algebraic topology and high-energy physics. This interdisciplinary approach, which can be called the supersymmetric theory of stochastic dynamics (STS), can be seen as an algebraic dual to the traditional set-theoretic framework of the DS theory, with its algebraic structure enabling the extension of some DS theory concepts to stochastic dynamics. Moreover, it reveals the presence of a topological supersymmetry (TS) in the GTOs of all SDEs. It also shows that among the various definitions of chaos, positive “pressure”, defined as the logarithm of the GTO spectral radius, stands out as particularly meaningful from a physical perspective, as it corresponds to the spontaneous breakdown of TS on the TFT side. Via the Goldstone theorem, this definition has a potential to provide the long-sought explanation for the experimental signature of chaotic dynamics known as 1/f noise. Additionally, STS clarifies that among the various existing interpretations of SDEs, only the Stratonovich interpretation yields evolution operators that match the corresponding GTOs and, consequently, have a clear-cut mathematical meaning. Here, we discuss these and other aspects of STS from both the DS theory and TFT perspectives, focusing on links between these two fields and providing mathematical concepts with physical interpretations that may be useful in some contexts. ...

Bayesian framework for characterizing cryptocurrency market dynamics, structural dependency, and volatility using potential field ArXiv ID: 2308.01013 “View on arXiv” Authors: Unknown Abstract Identifying the structural dependence between the cryptocurrencies and predicting market trend are fundamental for effective portfolio management in cryptocurrency trading. In this paper, we present a unified Bayesian framework based on potential field theory and Gaussian Process to characterize the structural dependency of various cryptocurrencies, using historic price information. The following are our significant contributions: (i) Proposed a novel model for cryptocurrency price movements as a trajectory of a dynamical system governed by a time-varying non-linear potential field. (ii) Validated the existence of the non-linear potential function in cryptocurrency market through Lyapunov stability analysis. (iii) Developed a Bayesian framework for inferring the non-linear potential function from observed cryptocurrency prices. (iv) Proposed that attractors and repellers inferred from the potential field are reliable cryptocurrency market indicators, surpassing existing attributes, such as, mean, open price or close price of an observation window, in the literature. (v) Analysis of cryptocurrency market during various Bitcoin crash durations from April 2017 to November 2021, shows that attractors captured the market trend, volatility, and correlation. In addition, attractors aids explainability and visualization. (vi) The structural dependence inferred by the proposed approach was found to be consistent with results obtained using the popular wavelet coherence approach. (vii) The proposed market indicators (attractors and repellers) can be used to improve the prediction performance of state-of-art deep learning price prediction models. As, an example, we show improvement in Litecoin price prediction up to a horizon of 12 days. ...