Constructing normal forms from experimental observations and time series analysis

A method is proposed for the construction of normal forms from experimental observations which describe the dynamics of a system close to the bifurcation points. The method is applied for the bifurcation of the Fe/2MH(2)SO(4) electrochemical system from a steady state to a chaotic attractor by considering the applied potential and the external ohmic resistance as bifurcation parameters. Steady state and time evolution curves of the response function are recorded. Perturbation experiments, time delay reconstruction of the attractors and calculation of power and Lyapunov spectra are performed. From the above experimental procedure the linear part of the normal form is constructed. The nonlinear part of the normal form is derived only from the knowledge of the linear part. Perturbations of the derived normal form on the bifurcation point are considered through the versal deformation of the normal form and the construction of the versal family of the normal form. The resulting normal form equations reproduce the dynamic characteristics and the bifurcation diagram of the electrochemical system.