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A system plus environment conservative model is used to characterize thenonlinear dynamics when the time averaged energy for the system particle startsto decay. The system particle dynamics is regular for low values of the $N$environment oscillators and becomes chaotic in the interval $13\le N\le15$,where the system time averaged energy starts to decay. To characterize thenonlinear motion we estimate the Lyapunov exponent (LE), determine the powerspectrum and the Kaplan-Yorke dimension. For much larger values of $N$ theenergy of the system particle is completely transferred to the environment andthe corresponding LEs decrease. Numerical evidences show the connection betweenthe variations of the {\it amplitude} of the particles energy time oscillationwith the time averaged energy decay and trapped trajectories.

Chaotic motion at the emergence of the time averaged energy decay