Detection of Hidden Oscillations in Systems Without Equilibrium

Chaotic systems with hidden attractors have been widely studied in recent decades. In this field, systems without equilibrium are of special interest. Since multistable systems can alternate between several hidden attractors, it is difficult to detect their hidden oscillations using known analytical techniques. In this study, we propose to apply recurrence analysis methods as a possible solution to this problem. We consider the new quantity measure, called relative specific volume, which correlates with changes of the attractor. We choose the improved Sprott A system as an example of the multiscroll attractor to verify the proposed approach. We compare the new recurrence measure with other traditional quantitative methods for chaotic systems investigation. It is shown that the relative specific volume possesses the best correlation with the change of attractors among all considered metrics. The proposed approach can be used to study hidden oscillations in experimental data as well.

A Simple Nonautonomous Hidden Chaotic System with a Switchable Stable Node-Focus

This paper presents a simple two-dimensional nonautonomous system, which possesses piecewise linearity constructed by a simple absolute value function. The nonautonomous system has only one switchable equilibrium state with a stable node-focus in the considered control parameter region but can generate periodic, chaotic and coexisting attractors. Therefore, the presented simple two-dimensional nonautonomous system always operates with hidden oscillations, which is not similar to any example reported in the literature. Furthermore, specific hidden dynamical behaviors are numerically disclosed by employing one-dimensional and two-dimensional bifurcation plots, phase plane plots, Poincaré mappings, local attraction basins, and complexity plots. In addition, by utilizing the circuit module of the absolute value function, a multiplierless analog circuit is designed, based on which breadboard experiments are performed to validate the numerically simulated phase plane plots of coexisting attractors.

Infinite Number of Hidden Attractors in Memristor-Based Autonomous Duffing Oscillator

In this paper, the recent and emerging phenomenon of hidden oscillations is observed in a newly implemented memristor-based autonomous Duffing oscillator for the first time. The hidden oscillations are presented and quantified by various statistical measures. The system shows a large number of hidden attractors for a wide range of the system parameters. This study indicates that hidden oscillations can exist not only in piecewise-linear but also in smooth nonlinear circuits and systems. The distribution of Lyapunov exponents and the basin of attraction are explored to understand the nature of the hidden oscillations. We have also discussed the new phenomenon of periodic line invariant. An experimental demonstration is also presented using real time analog circuit.