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This paper addresses the effects of offset terms on the dynamics of a modified Chua’s oscillator. The mathematical model is derived using Kirchhoff’s laws. The model is analyzed with the help of the maximal Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The investigations show that the offset terms break the symmetry of the system, generating more complex nonlinear phenomena like coexisting asymmetric bifurcations, coexisting asymmetric attractors, asymmetric double-scroll chaotic attractors and asymmetric attraction basins. Also, a hidden attractor (period-1 limit cycle) is found when varying the initial conditions. More interestingly, this latter attractor coexists with all other self-excited ones. A microcontroller-based implementation of the circuit is carried out to verify the numerical investigations.

The simplest megastable chaotic system is built by employing a piecewise-linear damping function which is periodic over the spatial domain. The unforced oscillator generates an infinite number of nested limit cycles with constant distances whose strength of attraction decreases gradually as moving to outer ones. The attractors and the basins of attraction of the proposed system are almost compatible with those of the system with sinusoidal damping. However, the nonzero Lyapunov Exponent of the latter is consistently below that of the former. A comparative bifurcation analysis is carried out for periodically forced systems, showing the chaotic behavior of coexisting attractors in specific values of parameters. Changing the bifurcation parameter results in expansion, contraction, merging, and separation of the coexisting attractors, make it challenging to find the corresponding basins. Three symmetric pairs of attractors are observed; each one consists of two symmetric attractors (with respect to the origin) with almost the same values of the corresponding Lyapunov Exponent.

A megastable oscillator with various types of attractors is proposed. The oscillator shows interesting dynamics like cloud, kite, and arrow-like attractors. As we know, such a megastable oscillator with the dynamic shapes was not previously reported. There is an infinite number of arrow-like attractors in this oscillator. Investigating equilibrium points of the oscillator and their stabilities shows that two of the attractors, the largest cloud-like and smallest kite-like, are self-excited, while the others are hidden. The basin of attraction for the oscillator is studied for various attractors. Then, a forcing term is added to the oscillator, and its dynamics are investigated. The bifurcations and Lyapunov exponents of each type of attractor are investigated by changing the coefficient of the sinusoidal term $K$. Each of the attractors shows various periodic, quasi-periodic, and chaotic dynamics for various $K$. Also, the basin of attraction of the forced oscillator is investigated.

It is known that the unidirectional coupling between a double-scroll system (drive) in the oscillation regime with another system (response) in the steady-state, gives rise to more complex dynamics such as multiscroll attractors [Dana et al., 2008]. This paper shows the way to move from n-scroll to 2n-scroll in a bidirectional coupling scheme between two jerk oscillators. The coupling considered can be interpreted as a perturbation to each oscillator’s amplitude through a signal proportional to the amplitude of the other. Indeed, starting from a situation of two oscillators which can generate double-scroll dynamics individually, we perform a bidirectional coupling of both oscillators which results in more complex dynamics such as multistability and 4-scroll chaotic attractor in the coupled system (not observed in an isolated oscillator). The underlined coupling scheme provides an efficient method for generating higher-order multiscroll attractors. The route to the 4-scroll chaotic attractor is explained with the help of bifurcation diagrams, plots of several phase portraits, and basins of attractions. Indirect validation of the theoretical analysis is carried out by using the PSpice simulation software.

The study of chaotic systems with special properties addresses one of the ongoing research topics. In this paper, we introduce a new class of third-order chaotic systems possessing a cyclic symmetry obeying unidirectional or mutual coupling. These new models are built by considering nonlinearities with three, four, or five segments and are distinguished from each other by the topological structure of the associated chaotic attractor. The projection of the novel chaotic attractors on certain planes reveals their multiscroll or multiwing character. For illustration, one of these models is studied in detail using analytical and numerical methods. The mechanism giving rise to the establishment of the chaotic regime starting from the state of equilibrium under the variation of a parameter is described using bifurcation diagrams, phase portraits, basins of attraction as well as the maximum Lyapunov exponent. Several zones of parameters are identified where the model experiences two or more attractors (i.e. multistability). In order to demonstrate the practical feasibility of the proposed circulant models, a series of experimental measurements are carried out using a physical implementation with an Arduino microcontroller.

Analysis of leg muscle activation and gait variability during locomotion is an important area of research in physiological and sport sciences. In this paper, we analyzed the coupling between the alterations of leg muscle activation and gait variability in single-task and dual-task walking. Since leg muscle activation in the form of electromyogram (EMG) signals and gait variability in the form of stride interval time series have complex structures, fractal theory and approximate entropy were used to evaluate their correlation at various walking conditions. Sixty subjects walked at their preferred speed for 10 min under the single-task condition and for 90s under the cognitive dual-task condition, and we evaluated the variations of the fractal dimension and approximate entropy of EMG signals and stride interval time series. According to the results, dual-task walking caused reductions in the complexity of EMG signals and stride interval time series than single-task walking. This technique can be used to evaluate the correlation between other organs during different locomotion.

Analysis of brain activity for patients with brain disorders is an important research area. Mild cognitive impairment (MCI) is a condition in which patients have more memory or thinking problems compared to healthy people of the same age. In this work, we studied the alterations in brain activity among control subjects and patients with MCI. Three complexity techniques, namely sample entropy, approximate entropy, and fractal dimension, were employed to study electroencephalogram (EEG) signals recorded from 102 control (healthy) subjects, and seven subjects with MCI in a comfortable position, on a bed, with their eyes closed. The results showed that the EEG signals of patients with MCI show greater complexity than the EEG signals of healthy subjects. This analysis method can be applied to compare brain activity among healthy subjects and patients with other brain diseases.