We outline a possibility of chaotic dynamics associated with a hyperbolic attractor of the Smale–Williams type in mechanical vibrations of a nonhomogeneous string with nonlinear dissipation arising due to parametric excitation of modes at the frequencies $\omega$ and $3\omega$, when the pump is supplied by means of the string tension variations alternately at frequencies of $2\omega$ and $6\omega$.

A phenomenon of interaction between three reactively coupled Van der Pol oscillators is considered in the paper. Phase equations are being obtained in second order approximation. Bifurcation analysis and Lyapunov’s exponent maps are used for illustrating system behavior. The paper consider significant features of reactive coupling. Complicated original system behavior with steering parameter growth is being considered too.

A motion of two identical pendulums connected by a linear elastic spring with an arbitrary stiffness is investigated. The system moves in an homogeneous gravitational field in a fixed vertical plane. The paper mainly studies the linear orbital stability of a periodic motion for which the pendulums accomplish identical oscillations with an arbitrary amplitude. This is one of two types of nonlinear normal oscillations. Perturbational equations depend on two parameters, the first one specifies the spring stiffness, and the second one defines the oscillation amplitude. Domains of stability and instability in a plane of these parameters are obtained.

Previously [1, 2] the problem of arbitrary linear and nonlinear oscillations of a small amplitude in a case of a small spring stiffness was investigated.

The question on possibility of writing the equations of motion of a nonholonomic system in the form of Lagrange’s equations of the 2nd kind for the minimal number of parameters is considered. The corresponding results of J. Hadamard and H. Beghin are discussed. It is proved that in the classic problem on rolling of a rigid body along a fixed plane without sliding the case when all three Chaplygin’s equations become Lagrange’s equations does not exist. For the same problem with two degrees of freedom the most general kind of nonholonomic constraints that provides the correct using Lagrange’s equations without multipliers, is established. Examples are given.

We consider a mechanical system inside a rolling ball and show that if the ideal constraints have spherical symmetry, the equations of motion have a Lagrangian form. Without symmetry, this is not true.

In our earlier paper [2] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.

We consider a nonholonomic model of movement of celtic stone on the plane. We show that, for certain values of parameters characterizing geometrical and physical properties of the stone, a strange Lorenz-like attractor is observed in the model. We have traced both scenarios of appearance and break-down of this attractor.

The paper deals with deviation based control algorithm for trajectory following of omni-wheeled mobile robot. The kinematic model and the dynamics of the robot actuators are described.

Pairs of bodies with regular rigid surfaces rolling onto each other in space form a nonholonomic system of a rather general type, posing several interesting control problems of which not much is known. The nonholonomy of such systems can be exploited in practical devices, which is very useful in robotic applications. In order to achieve all potential benefits, a deeper understanding of these types of systems and more practical algorithms for planning and controlling their motions are necessary. In this paper, we study the controllability aspect of this problem, giving a complete description of the reachable manifold for general pairs of bodies, and a constructive controllability algorithm for planning rolling motions for dexterous robot hands.