Search

We first review ideas of harnessing chaotic attractors to implement robust and flexible logic gates, and recast these concepts in the context of tipping points. The central idea is as follows: The presence of tipping points in complex systems endows it with the capability to switch between very different attractors under small changes in parameter, and this feature can be exploited to obtain reliable logic operations. The binary logic outputs can be efficiently mapped to dynamical attractors bounded in different phase space regions, and the logic inputs can be simply encoded through small changes in a parameter. We then go on to extend our central idea with new implementations of multiple-input logic operations. We show explicitly that the system jumps consistently in response to an external stream of multiple inputs, thus offering robust realizations of more complex multi-input logic gates. We demonstrate that the existence of tipping points offers the advantage that very low-amplitude inputs can yield highly amplified outputs. We also show that noise may play a constructive role, enhancing the reliability of the emergent multi-input logic operations, thus strengthening the concept of generalized Logical Stochastic Resonance. These results indicate that the tipping mechanism can serve to construct general purpose computing devices, which have the added capacity to reconfigure logic functionalities efficiently by small parameter changes.

Conventional neural networks are universal function approximators, but they may need impractically many training data to approximate nonlinear dynamics. Recently introduced Hamiltonian neural networks can efficiently learn and forecast dynamical systems that conserve energy, but they require special inputs called canonical coordinates, which may be hard to infer from data. Here, we prepend a conventional neural network to a Hamiltonian neural network and show that the combination accurately forecasts Hamiltonian dynamics from generalised noncanonical coordinates. Examples include a predator–prey competition model where the canonical coordinates are nonlinear functions of the predator and prey populations, an elastic pendulum characterised by nontrivial coupling of radial and angular motion, a double pendulum each of whose canonical momenta are intricate nonlinear combinations of angular positions and velocities, and real-world video of a compound pendulum clock.

Inclusion of redundancy has been one of the primary techniques to reduce the probability of error and to achieve reliability in computing systems and many other engineered systems. Rather than using one system, a group of identical systems is typically used and the final output is determined based on the collective outputs of the redundant systems. A very common technique to obtain an output from a redundant set of systems is “majority wins” (MW) where the majority result is assumed to be the final output. Recently we have shown that the dynamical coupling (DC) of redundant, identical systems reduces local noise. In this approach, redundant systems actively and dynamically collaborate to reduce their combined noise level. Here we present a comparison between MW and DC and demonstrate that a well-designed, optimized DC has better performance in reducing error. In this paper, we mostly focus on noise in computing systems and study redundancy and MW and DC in this context. However, the results can be applied to other similar applications, such as data sensing, oscillators, etc. DC can be implemented with minimal overhead or extra computational complexity, which makes it a suitable mechanism for organizing redundancy to achieve noise robustness.

Dramatically improved data from observatories like the CoRoT and Kepler spacecraft have recently facilitated nonlinear time series analysis and phenomenological modeling of variable stars, including the search for strange (aka fractal) or chaotic dynamics. We recently argued [Lindner et al., Phys. Rev. Lett. 114 (2015) 054101] that the Kepler data includes "golden" stars, whose luminosities vary quasiperiodically with two frequencies nearly in the golden ratio, and whose secondary frequencies exhibit power-law scaling with exponent near -1.5, suggesting strange nonchaotic dynamics and singular spectra. Here we use a series of phenomenological models to make plausible the connection between golden stars and fractal spectra. We thereby suggest that at least some features of variable star dynamics reflect universal nonlinear phenomena common to even simple systems., 9 pages, 9 figures, accepted for publication in Physica D

A two-state system driven by two inputs has been found to consistently produce a response mirroring a logic function of the two inputs, in an optimal window of moderate noise. This phenomenon is called logical stochastic resonance (LSR). We extend the conventional LSR paradigm to implement higher-level logic architecture or typical digital electronic structures via carefully crafted coupling schemes. Further, we examine the intriguing possibility of obtaining reliable logic outputs from a noise-free bistable system, subject only to periodic forcing, and show that this system also yields a phenomenon analogous to LSR, termed Logical Vibrational Resonance (LVR), in an appropriate window of frequency and amplitude of the periodic forcing. Lastly, this approach is extended to realize morphable logic gates through the Logical Coherence Resonance (LCR) in excitable systems under the influence of noise. The results are verified with suitable circuit experiments, demonstrating the robustness of the LSR, LVR and LCR phenomena. This article is part of the theme issue ‘Vibrational and stochastic resonance in driven nonlinear systems (part 1)’.

The importance and the necessity of nonlinearity in Artificial Intelligence, AI, and deep learning are very well understood. A multi-layer neural network with linear activation function is equivalent to a single layer of neurons. It is nonlinearity of activation functions that adds complexity to each layer, transforming the network to a universal computing machine that can approximate any continuous function. However, nonlinearity and the complexity that it creates have not been investigated enough in AI and modern deep learning systems. NC State University’s Nonlinear Artificial Intelligence Lab focuses on nonlinearity and the complexity that comes with it, and investigates how this can be an engine of artificial intelligence. We peruse our research at different levels with different goals. In this article we explain our approach, and present an overview of our results.

We quantify how incorporating physics into neural network design can significantly improve the learning and forecasting of dynamical systems, even nonlinear systems of many dimensions. We train conventional and Hamiltonian neural networks on increasingly difficult dynamical systems and compute their forecasting errors as the number of training data and number of system dimensions vary. A map-building perspective elucidates the superiority of Hamiltonian neural networks. The results clarify the critical relation among data, dimension, and neural network learning performance.

Certain nonlinear systems can switch between dynamical attractors occupying different regions of phase space, under variation of parameters or initial states. In this work we exploit this feature to obtain reliable logic operations. With logic output 0 or 1 mapped to dynamical attractors bounded in distinct regions of phase space, and logic inputs encoded by a very small bias parameter, we explicitly demonstrate that the system hops consistently in response to an external input stream, operating effectively as a reliable logic gate. This system offers the advantage that very low-amplitude inputs yield highly amplified outputs. Additionally, different dynamical variables in the system yield complementary logic operations in parallel. Further, we show that in certain parameter regions noise aids the reliability of logic operations, and is actually necessary for obtaining consistent outputs. This leads us to a generalization of the concept of Logical Stochastic Resonance to attractors more complex than fixed point states, such as periodic or chaotic attractors. Lastly, the results are verified in electronic circuit experiments, demonstrating the robustness of the phenomena. So we have combined the research directions of Chaos Computing and Logical Stochastic Resonance here, and this approach has the potential to be realized in wide-ranging systems.

We study the usefulness of coupled redundancy as a mechanism for reduction in local noise in coupled map lattices and investigate the role of network topology, coupling strength, and iteration number in this mechanism. Explicit numerical simulations to measure noise reduction in coupled units connected in different topologies such as ring, star, small-world, random, and grid networks have been carried out. We study both symmetric and asymmetric networks. Linear stability analysis is presented to identify an optimal symmetric topology. The effect of rewiring is also investigated, and we find that dynamic links enhance the noise reduction capabilities.