A simple chaotic snap circuit based on a single transistor is presented with tunable damping. It suggests, at present, the simplest chaotic snap circuit in the sense that it requires only 9 devices, which offer the minimum number of devices for a chaotic snap circuit. It also suggests the first and simplest circuit realization of either a four-dimensional (4D) chaotic system or a 4th-order (snap) chaotic system that demonstrates a maximized attractor dimension (D L) of a parameter set, or of the entire parameter space of the system, at minimized damping. The tendency of an increase in D L until its peak is illustrated by a decrease in damping. It offers the highest attractor dimension in a category of unit-damping snap chaos. As an initial report, a Clapp oscillator is able to exhibit 4D chaos but does not allow snap chaos. The proposed snap circuit embeds two simple mechanisms: (i) a Clapp oscillator as a simple core engine of oscillations avoiding a need for op-amps, and (ii) a single resistor as a remarkably simple realization of adjustable damping for snap chaos. A current-tunable equilibrium exhibits one of the 4 different types, two of which are of an (unstable) spiral saddle equilibrium, whereas the others are of a spiral stable equilibrium. They reveal the first report on either saddle-equilibrium or stable-equilibrium snap chaos based on a single transistor. Multistability and hidden attractors are demonstrated. The simple circuit offers a novel damping-tunable single-transistorbased approach to such rich dynamics of snap flows through various types of self-excited and hidden attractors. INDEX TERMS Hidden attractor, maximized attractor dimension, minimized damping, multistability, simplest chaotic snap circuit, single transistor, stable equilibrium. BANLUE SRISUCHINWONG received the B.Eng. degree (Hons.) from King Mongkut's Institute of Technology Ladkrabang, Bangkok, Thailand, in 1985, and the M.Sc. and Ph.D. degrees in electronics from The University of Manchester,

Accelerometry data has been widely used to measure activity and the circadian rhythm of individuals across the health sciences, in particular with people with advanced dementia. Modern accelerometers can record continuous observations on a single individual for several days at a sampling frequency of the order of one hertz. Such rich and lengthy data sets provide new opportunities for statistical insight, but also pose challenges in selecting from a wide range of possible summary statistics, and how the calculation of such statistics should be optimally tuned and implemented. In this paper, we build on existing approaches, as well as propose new summary statistics, and detail how these should be implemented with high frequency accelerometry data. We test and validate our methods on an observed data set from 26 recordings from individuals with advanced dementia and 14 recordings from individuals without dementia. We study four metrics: Interdaily stability (IS), intradaily variability (IV), the scaling exponent from detrended fluctuation analysis (DFA), and a novel nonparametric estimator which we call the proportion of variance (PoV), which calculates the strength of the circadian rhythm using spectral density estimation. We perform a detailed analysis indicating how the time series should be optimally subsampled to calculate IV, and recommend a subsampling rate of approximately 5 minutes for the dataset that has been studied. In addition, we propose the use of the DFA scaling exponent separately for daytime and nighttime, to further separate effects between individuals. We compare the relationships between all these methods and show that they effectively capture different features of the time series.

A new current-tunable chaotic jerk equation is revealed for an existing minimal chaotic circuit. Such a circuit has employed six minimum numbers of electronic components in the category of its kind. The chaotic jerk attractors on an plane and an plane are illustrated including the currenttunable bifurcations and the largest Lyapunov exponent. A new homoclinic orbit of the circuit is also demonstrated on an ) plane.