The dynamics of a Chaplygin sleigh

The problem of the motion of a Chaplygin sleigh on horizontal and inclined surfaces is considered. The possibility of representing the equations of motion in Hamiltonian form and of integration using Liouville's theorem (with a redundant algebra of integrals) is investigated. The asymptotics for the rectilinear uniformly accelerated sliding of a sleigh along the line of steepest descent are determined in the case of an inclined plane. The zones in the plane of the initial conditions, corresponding to a different behaviour of the sleigh, are constructed using numerical calculations. The boundaries of these domains are of a complex fractal nature, which enables a conclusion to be drawn concerning the probable character from of the dynamic behaviour.     

Dynamics of the Chaplygin sleigh on a cylinder

This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found.In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.     

On final motions of a Chaplygin ball on a rough plane

A heavy balanced nonhomogeneous ball moving on a rough horizontal plane is considered. The classical model of a “marble” body means a single point of contact, where sliding is impossible. We suggest that the contact forces be described by Coulomb’s law and show that in the final motion there is no sliding. Another, relatively new, contact model is the “rubber” ball: there is no sliding and no spinning. We treat this situation by applying a local Coulomb law within a small contact area. It is proved that the final motion of a ball with such friction is the motion of the “rubber” ball.     

Limit periodic motions in some systems with aftereffect under resonance condition

Volterra-type integrodifferential equations and their solutions are considered which, when the time increases without limit, exponentially tend to periodic modes. In the critical case of stability, when the characteristic equation has a pair of pure imaginary roots and the remaining roots have negative real parts, the problem of the existence of limit periodic solutions with resonance, caused by coincidence between the periodic part of the limit external periodic perturbation and the natural frequency of the linearized system, is solved. It is shown that, if the right-hand side of the equation is an analytic function and the existence of limit periodic solutions is determined by terms of the (2m + 1)-th order, these solutions are represented by power series in the arbitrary initial values of the non-critical variables and the parameter μ^1/(2m+1), where μ is a small parameter, characterizing the magnitude of the maximum external periodic perturbation. The amplitude equations are presented.     

Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint

The main goal of the article is to suggest a two-dimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energy-conserving nonholonomic dynamics. In this connection, we consider a Chaplygin sleigh on a plane, supposing that the nonholonomic constraint switches periodically in such a way that it is located alternately at each of three legs supporting the sleigh. We assume that at the initiation of the constraint the respective element (“knife edge”) is directed along the local velocity vector and becomes instantly fixed relative to the sleigh till the next switch. Differential equations of the mathematical model are formulated and an analytical derivation of mapping for the state evolution on the switching period is provided. The dynamics take place with conservation of the mechanical energy, which plays the role of one of the parameters responsible for the type of the dynamic behavior. At the same time, the Liouville theorem does not hold, and the phase volume can undergo compression or expansion in certain state space domains. Numerical simulations reveal phenomena characteristic of nonholonomic systems with complex dynamics (like the rattleback or the Chaplygin top). In particular, on the energy surface attractors associated with regular sustained motions can occur, settling in domains of prevalent phase volume compression together with repellers in domains of the phase volume expansion. In addition, chaotic and quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics.     

Regular and chaotic recurrence in FPU cell-chains

In contrast to the classical Fermi–Pasta–Ulam (FPU) chain, the inhomogeneous FPU chain shows nearly all the principal resonances. Using this fact, we can construct a periodic FPU chain of low dimension, for instance a FPU cell of four degrees-of-freedom, that can be used as a building block for a FPU cell-chain. This is a new type of chain. Differences between chains in nearest-neighbor interaction and those in overall interaction are caused by symmetry. We will show some striking results on the dynamics of FPU cell-chains where near-integrable behavior or chaos plays a part near stable equilibrium.     

On the dynamics of a periodic Colpitts oscillator forced by periodic and chaotic signals

In this paper, we have examined effects of forcing a periodic Colpitts oscillator with periodic and chaotic signals for different values of coupling factors. The forcing signal is generated in a master bias-tuned Colpitts oscillator having identical structure as that of the slave periodic oscillator. Numerically solving the system equations, it is observed that the slave oscillator goes to chaotic state through a period-doubling route for increasing strengths of the forcing periodic signal. For forcing with chaotic signal, the transition to chaos is observed but the route to chaos is not clearly detectable due to random variations of the forcing signal strength. The chaos produced in the slave Colpitts oscillator for a chaotic forcing is found to be in a phase-synchronized state with the forced chaos for some values of the coupling factor. We also perform a hardware experiment in the radio frequency range with prototype Colpitts oscillator circuits and the experimental observations are in agreement with the numerical simulation results.     

Stabilizing periodic orbits in a chaotic semiconductor laser

We consider the single mode dynamics of a self-pulsating semiconductor laser model under modulated injection current. The observation of a chaotic regime for a wide range of parameters suggests the usefulness of this model for high-speed secure optical communications. A possible way to achieve symbol encoding by using such a laser as a light source is to control its chaotic dynamics. We study the stabilization of periodic orbits embedded in the chaotic invariant set of the system by applying perturbations on the amplitude of the injection current.     

Weak periodic shift operator and generalized-periodic motions

We introduce the notions of a weak periodic system and its generalized-periodic motions. Such systems include systems generated by ordinary differential equations, equations with retarded argument, Volterra type equations, etc. We present a criterion for the existence of generalized-periodic motions and establish their main properties.     

Single rub-impacting periodic motions of a rigid constrained rotor system

This paper discusses the existence of single rub-impacting period-n motions for a kind of rotor systems with rigid constraints. The ranges of parameters for period-2 motions are also given. An example of this method is given.