Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model

Abstract

Similar to period-doubling bifurcation of fixed points, periodic orbits, it has been found since 1980's that a corresponding doubling bifurcation can also be found in the case of quasiperiodic orbits. Doubling bifurcations of quasiperiodic orbits has an important consequence on the dynamics of the system under consideration. Recently, it has been shown that subsequent doublings of quasiperiodic closed invariant curves leads to the formation of Shilnikov attractors. In this contribution, we illustrate for the first time in a discrete neuron system, the phenomenon of doubling of closed invariant curves. We also show the presence of mode-locked orbits and the geometry of one-dimensional unstable manifolds associated to them resulting in the formation of a resonant closed invariant curve. Moreover, we illustrate the phenomenon of crisis, and multistability in the system.

Dynamical behaviors in discrete memristor-coupled small-world neuronal networks

“…In order to have a deeper understanding of neuronal networks, researchers in various fields have explored the characteristics and dynamical behavior of neuron models from the perspective of different mathematical functions and analyses. [1][2][3][4][5][6][7][8][9][10] Memristors are the fourth basic passive circuit element after resistors, inductors and capacitors. It can be seen from Ref.…”

Influence of the magnetic flux on the dynamics of a self-sustaining system: analytical, numerical and analogical investigations

“…This study helps advance our understanding of complex networks and the process of neuron firing. Muni et al 2022 considered nodal and network behaviours considering electromagnetic flux coupling show the effect of magnetic flux on the phenomenon of bistability and the presence of periodic and chaotic attractor [23][24][25][26]. More recenlty, another authors analyzed the effects of electromagnetic flux on the discrete Chialvo neuron model, and shown that the model can exhibit rich dynamical behaviors such as multistability.…”

“…More recenlty, another authors analyzed the effects of electromagnetic flux on the discrete Chialvo neuron model, and shown that the model can exhibit rich dynamical behaviors such as multistability. The impact of magnetic flux is proved on the bifurcation diagrams, Lyapunov exponent diagram, phase portraits, basins of attraction [24][25][26][27]. Recent studies have shown that magnetic induction can destroy or accelerate enzymatic reactions based on memristors [21,28].…”