EEG signals of healthy individuals and epileptic patients, when treated as time series of evolving dynamical systems, are found to display characteristic differences in the behavior of the unstable periodic orbits UPO, marking the transition from regular periodic variations to self-similar dynamics. The UPO, manifesting as broad resonances in the Fourier power spectra, are quite prominent in their presence in the normal signals and are either absent or considerably weakened with a shift towards lower frequency in the epileptic condition. The weighted average and visibility power computed for the UPO region are found to distinguish epileptic seizure from healthy individuals EEG. Remarkably, the unstable periodic motion for healthy ones is well described by damped harmonic motion, the orbits displaying smooth dynamics. In contrast, the epileptic cases show bistability and piecewise linear motion for the larger orbits, exhibiting large sudden jumps in the velocity referred to the rate of change of the EEG potentials, characteristically different from the healthy cases, highlighting the efficacy of the UPO as biomarkers. For both the regions, 8 14Hz UPO and 40 45Hz resonance, we used data driven analysis to derive the system dynamics in terms of sinusoidal functions, which reveal the presence of higher harmonics, confirming nonlinearity of the underlying system and leading to quantification of the discernible differences between the healthy and epileptic patients. The gamma wave region in the 40 45Hz range, connecting the conscious and the unconscious states of the brain, reveals well structured coherence phenomena, in addition to the prominent resonance, which potentially can be used as a biomarker for the epileptic seizure. The wavelet scalogram analysis for both UPO and 40 45Hz region also clearly differentiates the healthy condition from epileptic seizure, confirming the above dynamical picture, depicting the higher harmonic generation, and intermixing of different modes in these two regions of interest.