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Modeling of Polymerization Processes
Published in E. Robert Becker, Carmo J. Pereira, Computer-Aided Design of Catalysts, 2020
Mixed monofunctional initiators are often used in industrial free-radical polymerization processes. Let us consider a continuous-flow-stirred-tank reactor in which styrene is solution polymerized with a binary mixture of monofunctional initiators having different thermal stabilities. Here the two monofunctional initiators used are tert-butyl perbenzoate (initiator A, slow initiator) and benzoyl peroxide (initiator B, fast initiator). The effect of initiator feed composition on the steady-state behavior of the reactor is illustrated in Figure 19. yAf is the mole fraction of tert-butyl perbenzoate in the feedstream. The analysis of steady-state and periodic bifurcation branches has been carried out with the numerical package AUTO [58]. This software package is designed for the semi-interactive determination of bifurcation branches and their display in graphical form. Starting from a known or calculated steady-state value, AUTO can trace the associated steady-state branch, determine any steady-state or Hopf bifurcation points, and follow the branches emanating from those bifurcation points. On branches of periodic orbits, it can identify period-doubling bifurcation points and follow branches emanating from them. Therefore, one can obtain a complete picture of both steady-state and periodic bifurcation characteristics of the dynamic systems.
Population dynamics
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Period doubling bifurcation leads to a new behaviour with twice the period of the original system. Period halving bifurcation leads to a new behaviour with half the period of the original system. Period halving leads from chaos to order, whereas period doubling leads from order to chaos.
Fast-slow bursting behaviors of hydroelectric governing system with double periodic excitations
Published in Journal of the Chinese Institute of Engineers, 2021
Sengkong Khov, Diyi Chen, Kimleng Kheav, Haojuan Wei
From Figure 7(a) and Figure 7(b), the bifurcation diagrams of and show that there are three-region bifurcation behaviors. For , the system enters into chaos via the sort of behavior which is called period-doubling bifurcation. For , the scattered point values of and slowly converge to round up at zero, meaning that the system becomes stable. On the other hand, the time waveforms and the phase plane trajectories of the system variables with the certain excitation frequency (i.e., and ), are given in Figures 8 and Figure 9 to better depict the fast-slow bursting behaviors of the HEGS.
The dynamic regimes of the unsymmetric bistable laminate
Published in Mechanics of Advanced Materials and Structures, 2022
Ting Dong, Wei Zhang, Mingming Dong
Resembling the amplitude-frequency response curves, the amplitude-force response curves for the amplitude sweep also indicate peak-to-peak amplitudes delivering comprehensive dynamic regimes. The amplitude-force response curves for Ω = 21.83 Hz are graphically presented in Figure 9. The single-well vibrations over the region from 12.88 to 15.53 N, the 11-period dynamic snap-through over the regions from 15.59 to 16.11 N and 20.23 to 20.87 N, the 7-period dynamic snap-through over the region from 16.23 to 16.75 N, the chaotic snap-through over the regions from 16.88 to 17.26 N, 19.2 to 19.33 N, 21.13 to 21.52 N and 24.48 to 25.77 N, the 4-period dynamic snap-through over the region from 19.71 to 20.1 N, the 3-period dynamic snap-through over the region from 21.9 to 23.19 N and the intermittency dynamic snap-through over the region from 23.32 to 24.35 N appear to be the more precise dynamic regimes. Attributed to the step size of the control parameter, the single-well region is not elaborated precisely, which can be locally magnified shown in Figure 10. The single-well vibrations for pre-snap-through experiences the period-doubling bifurcation exhibiting the 1-period vibration, the 2-period vibration, the 4-period vibration, the 8-period vibration through the chaotic vibration. The period-doubling bifurcation is a typical path to chaos. The 11-period dynamic snap-through per excitation amplitude is given by 11 points forming 11 coexisting amplitude-force response curves over the corresponding cross-well regions. Analogously, the amplitude-force response curves simultaneously possess seven branches over the cross-well region corresponding to the 7-period dynamic snap-through. Continue to change the excitation amplitude F, and the dynamic regimes exhibit 5 branches, 4 branches, 9 branches and 3 branches successively suggesting the multi-frequency dynamic responses. Besides, there exist chaotic and intermittent cross-well regions between multiple-branch regions. Nevertheless, the response amplitudes for the chaotic vibration are elaborated by a cluster of points per excitation amplitude and therefore cannot compose definite amplitude-force response curves. Attributed to aperiodic and disordered characteristics, chaotic and intermittent cross-well responses are characterized by black areas consisting of points, whereat the points of the former are more dense.