Narrow Results

This paper gives a short overview of various applications of stabilization by vibration, along with the exposition of the geometrical mechanism of this phenomenon. More specifically, the following observation is described: a rapidly vibrated holonomic system can be approximated by a certain associated nonholonomic system. It turns out that effective forces in some rapidly vibrated (holonomic) systems are the constraint forces of an associated auxiliary nonholonomic constraint. In particular, we review a simple but remarkable connection between the curvature of the pursuit curve (the tractrix) on the one hand and the effective force on the pendulum with vibrating support. The latter observation is a part of a recently discovered close relationship between two standard classical problems in mechanics: (1) the pendulum whose suspension point executes fast periodic motion along a given curve, and (2) the Chaplygin skate (known also as the Prytz planimeter, or the "bicycle"). The former is holonomic, the latter is nonholonomic. The holonomy of the skate shows up in the effective motion of the pendulum. This relationship between the pendulum with a twirled pivot and the Chaplygin skate has somewhat unexpected physical manifestations, such as the drift of suspended particles in acoustic waves. Finally, a higher-dimensional example of "geodesic motion" on a vibrating surface is described.

Dynamical behavior of a nonsmooth master system which is coupled to a nonsmooth Nonlinear Energy Sink (NES) during free and forced oscillations is studied analytically and numerically. Invariant manifolds of the system and their stable zones at different time scales are revealed and finally application of coupled nonsmooth NES to the passive control process of the main nonsmooth system is highlighted.

We provide the nine topological global phase portraits in the Poincaré disk of the family of the centers of Kukles polynomial differential systems of the form $ẋ=-y,$$ẏ=x+a{x}^{5}y+b{x}^3{y}^3+cx{y}^5,$ where $x,y\in ℝ$ and $a,b,c$ are real parameters satisfying $a^2+b^2+c^2\ne 0.$ Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree $6,$ and second inside the class of all polynomial differential systems of degree $6.$

The effect of multiplicative stochastic perturbations on planar Hamiltonian systems is investigated. It is assumed that perturbations fade with time and preserve a stable equilibrium of the limiting system. The paper investigates bifurcations associated with changes in the stability of the equilibrium and with the appearance of new stochastically stable states in the perturbed system. It is shown that depending on the structure and the parameters of the decaying perturbations, the equilibrium can remain stable or become unstable. In some intermediate cases, a practical stability of the equilibrium with estimates for the length of the stability interval is justified. The performed stability analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions.