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Abstract. Recently some methods have been presented to extract ordinary differential equations (ODE) directly from an experimental time series. Here, we introduce a new method to find an ODE which models both the short time and the long time dynamics. The experimental data are represented in a state space and the corresponding flow vectors are approximated by polynomials of the state vector components. We apply these methods both to simulated data and experimental data from human limb movements, which like many other biological systems can exhibit limit cycle dynamics. In systems with only one oscillator there is excellent agreement between the limit cycling displayed by the experimental system and the reconstructed model, even if the data are very noisy. Furthermore we study systems of two coupled limit cycle oscillators. There, a reconstruction was only successful for data with a sufficently long transient trajectory and relatively low noise level.

The maximum energy exchange of two harmonically coupled nonlinear oscillators is investigated.We calculate the maximum energy exchange close to resonance and show that the corresponding resonance curves have a universal shape and become broader and smaller when the amplitudefrequency coupling becomes large. Since there is a large variety of nonlinear oscillators where the trajectories are nearly homothetic curves in a phase-space representation, we furthermore investigate the special situation where the oscillators are homothetic. We argue that in this case there is a scaling of the maximum energy exchange at resonance. Numerical investigations show that these relations remain valid if the oscillators are slightly damped or perturbed by random noise. MS code no. AF4266 1990 PACS number(s): 03.20.+ i, 46. 10.+ z, 05.45. +b

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