Stability of periodic orbits in no-slip billiards

Rigid bodies collision maps in dimension-two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads to the study of no-slip billiards—planar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change after collisions with the boundary of the billiard domain. This paper, which continues the investigation initiated in Cox and Feres (2017 Dynamical Systems, Ergodic Theory, and Probability: in Memory of Chernov (Providence, RI: American Mathematical Society), is mainly focused on the issue of stability of periodic orbits in no-slip planar billiards. We prove Lyapunov stability of periodic orbits in polygonal billiards of this kind and, for general billiards domains, we obtain curvature thresholds for linear stability at commonly occurring period-2 orbits. More specifically, we prove that: (i) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than π, no-slip billiard maps admit elliptic period-2 orbits; (ii) polygonal no-slip billiards under this same corner angle condition always contain small invariant neighborhoods of the periodic point on which, up to smooth conjugacy, orbits of the return map lie on concentric circles; in particular the system cannot be ergodic with respect to the canonical invariant billiard measure; (iii) the no-slip version of the Sinai billiard must contain linearly stable periodic orbits of period 2 and, more generally, we obtain a curvature threshold at which the period-2 orbits go from being hyperbolic to being elliptic; (iv) finally, we make a number of observations concerning periodic orbits in wedge and triangular billiards. Our linear stability results extend those of Wojtkowski for the no-slip Sinai billiard

No-slip Billiards

We investigate the dynamics of no-slip billiards, a model in which small rotating disks may exchange linear and angular momentum at collisions with the boundary. A general theory of rigid body collisions in is developed, which returns the known dimension two model as a special case but generalizes to higher dimensions. We give new results on periodicity and boundedness of orbits which suggest that a class of billiards (including all polygons) is not ergodic. Computer generated phase portraits demonstrate non-ergodic features, suggesting chaotic no-slip billiards cannot easily be constructed using the common techniques for generating chaos in standard billiards. However, Sinai type dispersing billiards, which are always ergodic in the case of standard billiards, appear to be ergodic above a certain curvature threshold

Geometry and Dynamics of Rolling Systems

Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive collision forces. When it is desired or necessary to account for linear/angular momentum exchange in collisions involving a spherical body, a type of billiard system often referred to as no-slip has been used. Previous work indicated that no-slip billiards resemble non-holonomic systems, specifically, systems consisting of a ball rolling on surface. In prior research, such connections were only observed numerically and were restricted to very special surfaces. In this thesis, it is shown that no-slip billiard and rolling systems are directly related to each other under very general conditions. Our main result shows that no-slip billiards are truly the non-holonomic counterpart to standard billiard systems. In addition, to the best of our knowledge, we use a novel from of the rolling equations, showing that these systems are a one-parameter perturbation of the geodesic equation on a Riemannian manifold. This opens up a new area of investigation in the theory of geometric dynamical systems, concerning what we call rolling flows. We introduced the main concepts related to the rolling flow but we leave further development for future research

Classical and quantum mechanics with chaos

This thesis is concerned with the study, classically and quantum mechanically, of the square billiard with particular attention to chaos in both cases. Classically, we show that the rotating square billiard has two regular limits with a mixture of order and chaos between, depending on an energy parameter, E. This parameter ranges from -2w(^2) to oo, where w is the angular rotation, corresponding to the two integrable limits. The rotating square billiard has simple enough geometry to permit us to elucidate that the mechanism for chaos with rotation or curved trajectories is not flyaway, as previously suggested, but rather the accumulation of angular dispersion from a rotating line. Furthermore, we find periodic cycles which have asymmetric trajectories, below the value of E at which phase space becomes disjointed. These trajectories exhibit both left and right hand curvatures due to the fine balance between Centrifugal and Coriolis forces. Quantum mechanically, we compare the spectral analysis results for the square billiard with three different theoretical distribution functions. A new feature in the study is the correspondence we find, by utilising the Berry-Robnik parameter q, between classical E and a quantum rotation parameter w. The parameter q gives the ratio of chaotic quantum phase volume which we can link to the ratio of chaotic phase volume found classically for varying values of E. We find good correspondence, in particular, the different values of q as w is varied reflect the births and subsequent destructions of the different periodic cycles. We also study wave packet dynamics, necessitating the adaptation of a one dimensional unitary integration method to the two dimensional square billiard. In concluding we suggest how this work may be used, with the aid of the chaotic phase volumes calculated, in future directions for research work

Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u

Laboratory directed research and development. FY 1995 progress report

This document presents an overview of Laboratory Directed Research and Development Programs at Los Alamos. The nine technical disciplines in which research is described include materials, engineering and base technologies, plasma, fluids, and particle beams, chemistry, mathematics and computational science, atmic and molecular physics, geoscience, space science, and astrophysics, nuclear and particle physics, and biosciences. Brief descriptions are provided in the above programs