Application of delay embedding and recurrence analysis to a noisy nonlinear map for cyclic combustion dynamics of SI engines

Abstract

A noisy nonlinear dynamical (NND) model available for cyclic variations in SI engines is examined using nonlinear time series methods. Heat release time series computed from this model is analyzed by employing bifurcation diagrams, return maps, recurrence plots, and recurrence quantitative analysis techniques. The analysis methodology is at first applied to a noisy Logistic map to optimize the necessary parameters, and later the same is used for the NND map. The presence of noise hides the local patterns of the dynamics and higher periodic orbits, making the bifurcation diagrams look fuzzy for both maps. An increase in the noise level and uncombusted residuals in the NND map advance the onset point of bifurcations toward a higher equivalence ratio. The application of the NND model is extended to represent the SI engine combustion dynamics for a higher amount of internal EGR as it is favored for cycle-resolved control due to its feedback period of one combustion cycle. The combined effect of lean charge and internal EGR leads to a more complex dynamical nature. The addition of a higher amount of internal EGR adds some complex chaotic-like regions in the bifurcation diagrams. For a low level of fluctuations in the NND map, more than four noisy periodic orbits are observed for such operating conditions. The geometry of phase space trajectories for the low-level noisy model data with a high level of internal EGR resembles a chaotic system. Bifurcation diagrams, recurrence plots, and quantitative recurrence analysis indicate an intermittent route to chaos at a highly lean and diluted charge. The presence of deterministic chaotic characteristics in combustion dynamics, along with effort-saving techniques of data acquisition and estimating combustion phasing, depicts the possibilities of practical application in active onboard control of engine systems.