A Bifurcation Analysis of an Open Loop Internal Combustion Engine

The process of engine mapping in the automotive industry identifies steady-state engine responses by running an engine at a given operating point (speed and load) until its output has settled. While the time simulating this process with a computational model for one set of parameters is relatively short, the cumulative time to map all possible combinations becomes computationally inefficient. This work presents an alternative method for mapping out the steady-state response of an engine in simulation by applying bifurcation theory. The bifurcation approach used in this work allows the engine’s steady-state response to be traced through the model’s state-parameter space under the simultaneous variation of one or more model parameters. To demonstrate this approach, a bifurcation analysis of a simplified nonlinear engine model is presented. Using "throttle position" and "desired load torque signal", the engine’s dynamic response is classified into distinct regions bounded by bifurcation points. These bifurcations are shown to correspond to key physical properties of the open-loop system: fold bifurcations correspond to the minimum throttle angle required for a steady-state engine response; Hopf bifurcations bound a region where self-sustaining oscillations occur. The techniques used in this case study demonstrate the efficiency a bifurcation approach has at highlighting different regions of dynamic behavior in the engine's state-parameter space. Such an approach could speed up the mapping process and enhance the automotive engineer's understanding of an engine's underlying dynamic behavior. The information obtained from the bifurcation analysis could also be used to inform the design of future engine control strategies.