Abstract: The main focus of this manuscript is to provide a
highly efficient two-point sixth-order family of super-Halley type
methods that do not require any second-order derivative evaluation
for obtaining simple roots of nonlinear equations, numerically. Each
member of the proposed family requires two evaluations of the given
function and two evaluations of the first-order derivative per iteration.
By using Mathematica-9 with its high precision compatibility, a
variety of concrete numerical experiments and relevant results are
extensively treated to confirm t he t heoretical d evelopment. From
their basins of attraction, it has been observed that the proposed
methods have better stability and robustness as compared to the other
sixth-order methods available in the literature.
Abstract: We have built universal central pattern generator (CPG) hardware by interconnecting Hodgkin-Huxley neurons with reciprocally inhibitory synapses. We investigate the dynamics of neuron oscillations as a function of the time delay between current steps applied to individual neurons. We demonstrate stimulus dependent switching between spiking polyrhythms and map the phase portraits of the neuron oscillations to reveal the basins of attraction of the system. We experimentally study the dependence of the attraction basins on the network parameters: The neuron response time and the strength of inhibitory connections.
Abstract: In this paper, different nonlinear dynamics analysis techniques are employed to unveil the rich nonlinear phenomena of the electromagnetic system. In particular, bifurcation diagrams, time responses, phase portraits, Poincare maps, power spectrum analysis, and the construction of basins of attraction are all powerful and effective tools for nonlinear dynamics problems. We also employ the method of Lyapunov exponents to show the occurrence of chaotic motion and to verify those numerical simulation results. Finally, two cases of a chaotic electromagnetic system being effectively controlled by a reference signal or being synchronized to another nonlinear electromagnetic system are presented.