Regular & Chaotic Dynamics 

About: Regular & Chaotic Dynamics is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Hamiltonian system & Integrable system. It has an ISSN identifier of 1468-4845. Over the lifetime, 826 publications have been published receiving 9101 citations. The journal is also known as: RHD & Regulârnaâ i haotičeskaâ dinamika.

On the theory of motion of nonholonomic systems. The reducing-multiplier theorem

Abstract: This classical paper by S.A. Chaplygin presents a part of his research in non-holonomic mechanics. In this paper, Chaplygin suggests a general method for integration of the equations of motion for non-holonomic systems, which he himself called the “reducing-multiplier method”. The method is illustrated on two concrete problems from non-holonomic mechanics. This paper produced a considerable effect on the further development of the Russian non-holonomic community. With the help of Chaplygin’s reducing-multiplier theory the equations for quite a number of non-holonomic systems were solved (such systems are known as Chaplygin systems). First published about a hundred years ago, this work has not lost its scientific significance and is hoped to be estimated at its true worth by the English-speaking world.

A simple model for clock-actuated legged locomotion

Abstract: The spring-loaded inverted pendulum (SLIP) model describes well the steady-state center-of-mass motions of a diverse range of walking and running animals and robots. Here we ask whether the SLIP model can also explain the dynamic stability of these gaits, and we find that it cannot do so in many physically-relevant parameter ranges. We develop an actuated, lossy, clock-torqued SLIP, or CT-SLIP, with more realistic hip-motor torque inputs, that can capture the robust stability properties observed in most animals and some legged robots. Variations of CT-SLIP at a similar level of detail and complexity may also be appropriate for capturing the whole-system center-of-mass dynamics of locomotion of legged animals and robots varying widely in size and morphology. This paper contributes to a broader program to develop mathematical models, at varied levels of detail, that capture the dynamics of integrated organismal systems exhibiting integrated whole-body motion.

Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems

Abstract: This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.

The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere

Abstract: In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.

Richness of chaotic dynamics in nonholonomic models of a celtic stone

Abstract: We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.