Application of the differential transformation method to bifurcation and chaotic analysis of an AFM probe tip

Abstract

The AFM (atomic force microscope) has become a popular and useful instrument for measuring intermolecular forces with atomic resolution, that can be applied in electronics, biological analysis, and studying materials, semiconductors etc. This paper conducts a systematic investigation into the bifurcation and chaotic behavior of the probe tip of an AFM using the differential transformation (DT) method. The validity of the analytical method is confirmed by comparing the DT solutions for the displacement and velocity of the probe tip at various values of the vibrational amplitude with those obtained using the Runge–Kutta (RK) method. The behavior of the probe tip is then characterized utilizing bifurcation diagrams, phase portraits, power spectra, Poincaré maps, and maximum Lyapunov exponent plots. The results indicate that the probe tip behavior is significantly dependent on the magnitude of the vibrational amplitude. Specifically, the tip motion changes first from subharmonic to chaotic motion, then from chaotic to multi-periodic motion, and finally from multi-periodic motion to subharmonic motion with windows of chaotic behavior as the non-dimensional vibrational amplitude is increased from 1.0 to 5.0.