Robust Method for Experimental bifurcation Analysis

One of the most important tasks in nonlinear dynamics is the determination of bifurcation sets of various dynamical systems. If the dynamical equations are known, bifurcation points in parameter space can be detected by solving some specific fixed point equations. Using suitable continuation algorithms these bifurcation points can be traced in parameter space in order to compute bifurcation curves or surfaces. A well written introduction into numerical bifurcation analysis and continuation can be found in [Seydel, 1988, 1991]. The fixed point conditions and their numerical solution are typically based on some derivatives of the flow with respect to state variables and parameters [Seydel, 1988, 1991; Parlitz, 1990]. These derivatives have to be approximated if the underlying equations are not known as it is the case for most experiments. Any experimental approach to bifurcation analysis requires a computer controlled experiment including manipulation of control parameters and online measurements of the current system dynamics. Using delay reconstruction one may, at least in principle, derive approximating (black-box) model equations which could then be used as starting point for bifurcation analysis very similar to the theoretical case where the system’s equations are known. An example for this state space based approach was presented recently by Anderson et al. [1999] using simulated experiments. We found that state space based methods rely very much on the accuracy of the estimated derivatives and may be very susceptible with respect to noise. Therefore, we devised an algorithm for experimental bifurcation analysis that avoids modeling in (reconstructed) state space but is based directly on features of the measured signal such as autocorrelation functions, changes of magnitude, etc. In order to demonstrate our approach for experimental bifurcation analysis we have used an electronic implementation of Duffing’s equation