Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations

To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.