Double-scroll System Research Articles

Cupolets in a chaotic neuron model.

This paper reports the first finding of cupolets in a chaotic Hindmarsh-Rose neural model. Cupolets (chaotic, unstable, periodic, orbit-lets) are unstable periodic orbits that have been stabilized through a particular control scheme by applying a binary control sequence. We demonstrate different neural dynamics (periodic or chaotic) of the Hindmarsh-Rose model through a bifurcation diagram where the external input current, I, is the bifurcation parameter. We select a region in the chaotic parameter space and provide the results of numerical simulations. In this chosen parameter space, a control scheme is applied when the trajectory intersects with either of the two control planes. The type of the control is determined by a bit in a binary control sequence. The control is either a small microcontrol (0) or a large macrocontrol (1) that adjusts the future dynamics of the trajectory by a perturbation determined by the coding function r ( x ). We report the discovery of many cupolets with corresponding control sequences and comment on the differences with previously reported cupolets in the double scroll system. We provide some examples of the generated cupolets and conclude by discussing potential implications for biological neurons.

Multiscroll in Bidirectionally Coupled Jerk Oscillators: Theoretical Analysis and PSpice Verification

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.

Design of coupling for synchronization of chaotic oscillators

A general procedure is discussed to formulate a coupling function capable of targeting desired responses such as synchronization, antisynchronization, and amplitude death in identical as well as mismatched chaotic oscillators. The coupling function is derived for unidirectional, mutual, and matrix type coupling. The matrix coupling, particularly, is able to induce synchronization, antisynchronization, and amplitude death simultaneously in different state variables of a response system. The applicability of the coupling is demonstrated in spiking-bursting Hindmarsh-Rose neuron model, Rössler oscillator, Lorenz system, Sprott system, and a double scroll system. We also report a scaling law that defines a process of transition to synchronization.