Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse.

In this paper, the effects of asymmetry in an electrical synaptic connection between two neuronal oscillators with a small discrepancy are studied in a 2D Hindmarsh-Rose model. We have found that the introduced model possesses a unique unstable equilibrium point. We equally demonstrate that the asymmetric electrical couplings as well as external stimulus induce the coexistence of bifurcations and multiple firing patterns in the coupled neural oscillators. The coexistence of at least two firing patterns including chaotic and periodic ones for some discrete values of coupling strengths and external stimulus is demonstrated using time series, phase portraits, bifurcation diagrams, maximum Lyapunov exponent graphs, and basins of attraction. The PSpice results with an analog electronic circuit are in good agreement with the results of theoretical analyses. Of most/particular interest, multistability observed in the coupled neuronal model is further controlled based on the linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the periodic coexisting firing pattern. For higher values of the coupling strength, only a chaotic firing pattern survives. To the best of the authors' knowledge, the results of this work represent the first report on the phenomenon of coexistence of multiple firing patterns and its control ever present in a 2D Hindmarsh-Rose model connected to another one through an asymmetric electrical coupling and, thus, deserves dissemination.

On the dynamics of a simplified canonical Chua's oscillator with smooth hyperbolic sine nonlinearity: Hyperchaos, multistability and multistability control.

A simplified hyperchaotic canonical Chua's oscillator (referred as SHCCO hereafter) made of only seven terms and one nonlinear function of type hyperbolic sine is analyzed. The system is found to be self-excited, and bifurcation tools associated with the spectrum of Lyapunov exponents reveal the rich dynamical behaviors of the system including hyperchaos, torus, period-doubling route to chaos, and hysteresis when turning the system control parameters. Wide ranges of hyperchaotic dynamics are highlighted in various two-parameter spaces based on two-parameter Lyapunov diagrams. The analysis of the hysteretic window using a basin of attraction as argument reveals that the SHCCO exhibits three coexisting attractors. Laboratory measurements further confirm the performed numerical investigations and henceforth validate the mathematical model. Of most/particular interest, multistability observed in the SHCCO is further controlled based on a linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the asymmetric pair of coexisting attractors. For higher values of the coupling strength, only a unique symmetric periodic attractor survives.

A Novel Autonomous 5-D Hyperjerk RC Circuit with Hyperbolic Sine Function.

A novel autonomous 5-D hyperjerk RC circuit with hyperbolic sine function is proposed in this paper. Compared to some existing 5-D systems like the 5-D Sprott B system, the 5-D Lorentz, and the Lorentz-like systems, the new system is the simplest 5-D system with complex dynamics reported to date. Its simplicity mainly relies on its nonlinear part which is synthetized using only two semiconductor diodes. The system displays only one equilibrium point and can exhibit both periodic and chaotic dynamical behavior. The complex dynamics of the system is investigated by means of bifurcation analysis. In particular, the striking phenomenon of multistability is revealed showing up to seven coexisting attractors in phase space depending solely on the system's initial state. To the best of author's knowledge, this rich dynamics has not yet been revealed in any 5-D dynamical system in general or particularly in any hyperjerk system. Pspice circuit simulations are performed to verify theoretical/numerical analysis.

Periodicity, chaos, and multiple attractors in a memristor-based Shinriki's circuit.

In this contribution, a novel memristor-based oscillator, obtained from Shinriki's circuit by substituting the nonlinear positive conductance with a first order memristive diode bridge, is introduced. The model is described by a continuous time four-dimensional autonomous system with smooth nonlinearities. The basic dynamical properties of the system are investigated including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponents' spectrum. It is found that in addition to the classical period-doubling and symmetry restoring crisis scenarios reported in the original circuit, the memristor-based oscillator experiences the unusual and striking feature of multiple attractors (i.e., coexistence of a pair of asymmetric periodic attractors with a pair of asymmetric chaotic ones) over a broad range of circuit parameters. Results of theoretical analyses are verified by laboratory experimental measurements.