Bifurcation and convergence of particle swarm optimization dynamics embedded into upper and lower bound

In this paper, we propose two types of dynamics of Particle Swarm Optimization (PSO) embedded into upper and lower bounds, which have more powerful abilities of searching the global optimal solution on their bounded constraints. The new type of dynamics called "nonlinear operator model" and "nonlinear variable transformation model" are inertial models, which are introduced by discretizing the convolution integral form on continuous time. After certifying stability and/or instability including chaos of dynamics by the bifurcation diagrams, the infuenccs of model parameters settings on the global search are investigated to achieve better performance than the orthodox PSO dynamics in copmutational simulations for a few benchmark objective functions.