Complex Systems

This study explores the realm of chaotic dynamics, Neurochaos Learning (a brain-inspired machine learning paradigm) and Normal numbers, focusing on the introduction of a novel chaotic trajectory termed the Universal Orbit. The study investigates the characteristics and generation of universal orbits within two prominent chaotic maps: the Decimal Shift Map and the Gauss Map. It explores the set of points capable of forming such orbits, revealing connections with normal numbers and continued fractions. Points within the interval (0, 1) can produce universal orbits under specific conditions, highlighting the intricate relationship between machine learning, chaotic dynamics and number theory. While not all points forming universal orbits are normal numbers, the trajectory of a normal number may represent a universal orbit (under certain conditions). When employing the universal orbit for feature extraction in Neurochaos Learning, the firing time feature can be interpreted by establishing an upper bound and examining its trend. Future research aims to identify sets of points producing universal orbits under various chaotic maps, intending to enhance the performance of algorithms like the Neurochaos Learning algorithm. This study contributes to advancing our understanding of chaotic systems and their applications in artificial intelligence.

Inspired by the human brain's structure and function, Artificial Neural Networks (ANN) were developed for data classification. However, existing Neural Networks, including Deep Neural Networks, do not mimic the brain's rich structure. They lack key features such as randomness and neuron heterogeneity, which are inherently chaotic in their firing behavior. Neurochaos Learning (NL), a chaos-based neural network, recently employed one-dimensional chaotic maps like Generalized Lüroth Series (GLS) and Logistic map as neurons. For the first time, we propose a random heterogeneous extension of NL, where various chaotic neurons are randomly placed in the input layer, mimicking the randomness and heterogeneous nature of human brain networks. We evaluated the performance of the newly proposed Random Heterogeneous Neurochaos Learning (RHNL) architectures combined with traditional Machine Learning (ML) methods. On public datasets, RHNL outperformed both homogeneous NL and fixed heterogeneous NL architectures in nearly all classification tasks. RHNL achieved high F1 scores on the Wine dataset (1.0), Bank Note Authentication dataset (0.99), Breast Cancer Wisconsin dataset (0.99), and Free Spoken Digit Dataset (FSDD) (0.98). These RHNL results are among the best in the literature for these datasets. We investigated RHNL performance on image datasets, where it outperformed stand-alone ML classifiers. In low training sample regimes, RHNL was the best among stand-alone ML. Our architecture bridges the gap between existing ANN architectures and the human brain's chaotic, random, and heterogeneous properties. We foresee the development of several novel learning algorithms centered around Random Heterogeneous Neurochaos Learning in the coming days.

Increased levels of greenhouse gases in the atmosphere, especially carbon dioxide, are leading contributors to a significant increase in the global temperature, and the consequent global climatic changes are more noticeable in recent years than in the past. A persistent increased growth of such gases might lead to an irreversible transition or tipping of the Earth’s climatic system to a new dynamical state. A change of regimes in CO2 buildup being correlated to one in global climate patterns, predicting this tipping point becomes crucially important. We propose here an innovative conceptual model, which does just this. Using the idea of rate-induced bifurcations, we show that a sufficiently rapid change in the system parameters beyond a critical value tips the system over to a new dynamical state. Our model when applied to real-world data detects tipping points, enables calculation of tipping rates and predicts their future values, and identifies thresholds beyond which tipping occurs. The model well captures the growth in time of the total global atmospheric fossil-fuel CO2 concentrations, identifying regime shift changes through measurable parameters and enabling prediction of future trends based on past data. Our model shows two distinct routes to tipping. We predict that with the present trend of variation of atmospheric greenhouse gas concentrations, the Earth’s climatic system would move over to a new stable dynamical regime in the year 2022. We determine a limit of 10.62 GtC at the start of 2022 for global CO2 emissions in order to avoid this tipping. The Earth’s climate has seen many changes over the years, affecting the physical environment (be it terrestrial, marine, or the atmosphere). These major changes or regime shifts from one stable dynamical state of the physical environment to another, each of which may persist for several years, produce major shifts in natural ecosystems involving trophic structures, changes in composition, and abundance of species. The climatic system moves over to a new regime once it crosses a climatic tipping point—a threshold crossed irreversibly by the system’s dynamics. Anthropogenic influences brought about in the physical environment invariably contribute in a substantial way to climate change globally as the dynamics of the climatic system is governed by the coupling between the land, the atmosphere, and the oceans. An increase in levels of greenhouse gases in the atmosphere mainly caused by human activities, especially carbon dioxide, has been one of the important contributing factors leading to climate change in the last few decades. We present here a theoretical model that well captures the rate of increase of the total global concentrations of carbon dioxide, the major contributing greenhouse gas in the atmosphere. We then employ the concept of rate-induced bifurcations to demonstrate that it is possible to determine the climatic tipping points from our model. This way, we predict that the climatic system would relocate to a new stable state early in the year 2022. It has been widely accepted that tipping point mechanisms can be used to study climate change. In this paper, we shall introduce and apply a rate-induced tipping model to global fossil-fuel emissions data. Our model shows two distinct routes in which tipping can occur, and the parameters describing these can be calculated from data and are physically measurable. Through the application of this model, we identify crucial tipping points, which lead to climate change and quantify exact boundaries crossed that induce tipping. Control can be exercised over the parameters describing tipping, if desired, such that tipping can be prevented. The methods developed can further be applied to any growth curve that may have undergone rate-induced tipping.

Coauthored with Subhayan Sahu, Shriya Pai, and Naren Manjunath