DYNAMICAL BEHAVIOR OF AN INNOVATION DIFFUSION MODEL WITH INTRA-SPECIFIC COMPETITION BETWEEN COMPETING ADOPTERS

In this paper,we proposed an innovation diffusion model with three compartments to investigate the diffusion of an innovation(product)in a particular region.The model exhibits two equilibria,namely,the adopter-free and an interior equilibrium.The existence and local stability of the adopter-free and interior equilibria are explored in terms of the effective Basic Influence Number(BIN)R_(A).It is investigated that the adopter free steady-state is stable if R_(A)<1.By consideringτ(the adoption experience of the adopters)as the bifurcation parameter,we have been able to obtain the critical value ofτresponsible for the periodic solutions due to Hopf bifurcation.The direction and stability analysis of bifurcating periodic solutions has been performed by using the arguments of normal form theory and the center manifold theorem.Exhaustive numerical simulations in the support of analytical results have been presented.

STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage （time delay） is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period （time delay, T） passes through a critical value. Applying the normal form theory and the center manifold argument, we de- rive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Influence of alumina/MWCNT hybrid nanoparticle additives on tribological properties of lubricants in turning operations

A hybrid lubricant with improved thermal and tribological properties was developed by blending multiwalled carbon nanotubes(MWCNTs) with alumina-based nanoparticles into cutting fluid at fixed volumetric proportions(10:90). The hybrid cutting fluid was prepared in different volumetric concentrations(0.25, 0.75, and 1.25 vol%), and the tribological properties and contact angles were measured using pin-on-disc tribometry and goniometry, respectively. The study showed a reduction in wear and friction coefficient with increasing nanoparticle concentration. The cutting fluid performance was investigated using minimum quantity lubrication(MQL) in the turning of AISI 304 stainless steel. Regression models were developed for measuring the temperature and tool flank wear in terms of cutting speed, feed, depth of the cut, and nanoparticle concentration using response surface methodology. The developed hybrid nanolubricants significantly reduced the tool flank wear and nodal temperature by 11% and 27.36%, respectively, as compared to alumina-based lubricants.

Bifurcation behaviors analysis of a plankton model with multiple delays

A mathematical model describing the dynamics of toxin producing phytoplankton- zooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multiple delays; one due to gestation period in the growth of phytoplankton population and second due to the delay in toxin liberated by TPP. It is established that a sequence of Hopf bifurcations occurs at the interior equilibrium as the delay increases through its critical value. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined using the theory of normal form and center manifold. Meanwhile, effect of toxin on the stability of delayed plankton system is also established numerically. Finally, numerical simulations are carried out to support and supplement the analytical findings.