Electric force microscopy (EFM) and Kelvin probe force microscopy (KFM) have been widely used for studying electrical properties of nanometer-scale structures. Since they were invented at the early stage of atomic force microscopy (AFM) history, both techniques have been drastically developed with dynamic-mode AFM, where the cantilever is used as a mechanical oscillator. They are capable of mapping local work function and electronic states of a wide variety of material surfaces on an atomic/molecular scale. Here in this article we review the present status of EFM and KFM mainly based on frequency modulation AFM and discuss the future prospects.

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It has been rigorously proved that nonlinear lattices can support spatially localized and periodic in time vibrational modes called either discrete breathers (DBs) or intrinsic localized modes (ILMs). DB does not radiate its energy because its frequency does not belong to the spectrum of small-amplitude running waves. Discreteness and nonlinearity are often said to be the two major necessary conditions for the existence of DBs. Interatomic interactions are nonlinear and the discovery of DBs in crystals, which are nonlinear lattices at the atomic scale, was just a question of time. The first successful attempt to excite DB in alkali-halide NaI crystal in molecular dynamics simulations dates back to 1997. However, the first report on DBs in pure metals was delayed till 2011. In this review we discuss the reason of this delay, describe the latest results on DBs in pure metals and ordered alloys, and outline the open problems in this area.

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We investigate the indentation process of lead zirconate titanate (PZT) cantilever on a nanoscale level and attempt a high-speed observation of the process using dynamic force microscopy (DFM), taking advantage of the converse piezoelectric property of the cantilever. We confirm that the PZT cantilever has a good converse piezoelectric effect, works as an excellent actuator, and can produce an indentation on the polymer film, making it suitable for nanoscale applications. Moreover, for use as an actuator instead of a displacement sensor, the PZT cantilever was incorporated into the feedback control system in dynamic force microscopy (DFM), and could successfully be operated. We attempted a high-speed observation using DFM, and we successfully acquired the topographic image on a 1× 1µm scan area within 8.5seconds.

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We have used linear response measurements to explore the frequency shift and peak height of the natural frequency resonance of single cantilevers in a nonlinear state versus driver frequency as the bifurcation frequency is approached. Analytical calculations of these properties for a single Duffing resonator are in good agreement in terms of softening of the natural frequency-pump difference frequency and the diverging of its peak amplitude. The deviation of the natural frequency from the analytical curve that is observed for a short cantilever can be ascribed to a large amplitude effect, which, as yet, cannot be explained by simple modifications of the Duffing equation.

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The oscillating behavior of a micro-cantilever probe plays a central role in the atomic force microscope for studying a nanoscale sample. The oscillatory phenomena in the microscope are numerically investigated by exciting the probe with a single frequency. We observe the non-resonant frequency components, which correspond to a frequency of transient beats superimposed on the stable solution, around the natural frequency of the probe, when the probe is close to the sample. The difference between the non-resonant frequency and the natural frequency changes when the tip-sample distance decreases. Furthermore, we investigate the originating point of the non-resonant frequency components as a function of the tip-sample distance. In addition, we perform an actual experiment for observing the frequency components near the resonant frequency.

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Nonlinear dynamics of a model of acoustic metamaterials with local resonators are investigated numerically and theoretically. We focus on dynamics of band edge modes (BEMs) and zone boundary modes (ZBMs) which are on the upper bounds of acoustic bands and optical bands of the phonon dispersion band. It is found that, in a region of weak anharmonicity, higher harmonics of a fundamental mode and static displacement are excited in both BEM and ZBM if the geometrical relation between the main lattice and the local resonators has even-order nonlinearity. Numerical solutions of nonlinear periodic orbits which are continued from vibrations in the small amplitude limit by the shooting method indicate that structure of the periodic orbits of the local resonators depends on the form of nonlinear terms of the geometrical relation. Moreover, the nonlinear periodic orbits become unstable when the amplitude of the periodic orbit becomes larger. Direct numerical simulations show that unstable dynamics occur due to modulational instability. After destabilization of the nonlinear periodic orbits, spatial energy localization is also observed.

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A mechanical apparatus of coupled oscillator chains which are levitated on a track by air has been constructed with designing anew nonlinear springs. By use of the apparatus, excitation of mobile type of intrinsic localized modes has been demonstrated with driving sinusoidally at one end of the chains. The apparatus consists of twenty identical oscillators, nonlinear springs, a long and straight air track with a blower at one end and a driver unit for forcing the chains. The oscillators are connected with neighboring ones through the nonlinear springs and levitated above the air track. The relation of restoring force of the spring to deflection is piecewise linear but approximately cubic. In the apparatus with appropriate tensions, the curve of relation is to be symmetrical with respect to the equilibrium position. One end of the chains is fixed and the other is driven sinusoidally in the direction of the chains at a frequency. It has been observed that, driven with a frequency above the cutoff, localized oscillations can be excited intermittently at the driven end and they are propagated along the chains at a constant speed.

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We model the Fermi-Pasta-Ulam lattice, in which masses move in a two-dimensional plane, and identify different types of intrinsic localized modes (ILMs): longitudinal and transverse. The stability of the ILMs is evaluated by using characteristic multipliers. Longitudinal ILMs tend to be unstable because of the buckling effect of the chain. In contrast, transverse ILMs become stable if the chain is initially stretched. This difference between the longitudinal and the transverse ILMs is revealed by computing existence regions with respect to the angular frequency and the initial extension of the chain. The results show that the longitudinal ILMs tend to be stable in low-frequency and low-extension areas whereas the transverse ILMs become stable upon strongly stretching the chain.

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The present paper considers chaotic synchronization in a network consisting of Bernoulli maps with a time-delay connection. We demonstrate that a connection strength and a map parameter at which chaotic synchronization occurs can be systematically designed, even for cases in which the connection delay, the number of maps, and the detailed topology information are unknown. The primary advantage of the proposed design is that the designed connection strength and map parameter are valid for any connection delay. This result is due to the fact that the stability of the synchronized state is the same as that of a time-invariant linear system having both an uncertain dimension and an uncertain parameter. For such a linear system, it is quite difficult to obtain the necessary and sufficient condition of the stability. However, a simple sufficient condition enables us to provide the design. The analytical results are confirmed through numerical examples.

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Many mathematical models in various scientific fields are represented by complicated differential equations, and such models cannot be solved analytically. Therefore, numerical analysis using computers is essential. However, when a system is more complicated, the analysis time is longer. In this study, we propose a fast numerical analysis method using a field programmable gate array (FPGA). The FPGA is a reconfigurable integrated circuit, and it is good at parallel processing. We developed the calculators of bifurcation diagrams and basins of attraction on an FPGA board. As a result, we constructed calculators that were at most 2.8 times faster and that consumed 95.2% less power than the conventional method using a CPU.

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