A model of the AFM (atomic force microscope) with certain tip-nanostructure interaction, cantilever elasticity and damping of its oscillations is proposed. Stable and unstable motion of the AFM tip interacting with the graphene sheet is investigated by the Lyapunov exponent computation. In our approximation, a hundred Si atoms (top of the AFM tip) interact with C atoms of the nanostructure. Тhis interaction is described by Lennard-Jones potential, and the distance between the top and the center of the cantilever mass is a constant. Complex influence of the initial tip-nanostructure distance and nanostructure size on stability has been examined. We discuss a possible new mode of the AFM operation based on the Lyapunov exponent computation. Maxima and minima of the Lyapunov exponent show where certain parts of the elementary cells are placed.

We transform stock price time series to the time dependent force parameters using Newton’s second law (a kind of the information filtering). The force acts on a point in the space of data. One of ten force parameters, if its value in a short time interval is extremely large, announces increasing of the time series uncertainty. This delayed effect of the nonlinear force we interpret as sensitivity to initial conditions, for chaos combined with stochasticity, considering the time series generated by nonlinear differential equations with fluctuating function included. Our aim is to improve understanding and rough prediction of processes in complex systems.

The study of the magnetic properties of materials have attracted the interest of scientists for ages. Knowing the magnetic properties of materials do their best possible use in industry and technology, as well as better understanding of some yet unexplained phenomena in nature related to the Earth's magnetic field. Several models that describe the given phenomenon are known, such as the domino model, two-dimensional Ising model and model of inversion magnetic Earth's field. In computer experiments whose results are presented in this paper, it is used a simplified two-dimensional Ising model, one version of the model related spins. Computer experiments (simulations) are made in the mathematical software Maple. It has been observed reversing the direction of magnetization in a small change of inhomogeneity energy interactions. Consideration of this issue could be of importance for the study of ferromagnetic material.