An error analysis of variational integrators is obtained, by blowing up the discrete variational principles, all of which have a singularity at zero time-step. Divisions by the time step lead to an order that is one less than observed in simulations, a deficit that is repaired with the help of a new past-future symmetry.
Skew critical problems occur in continuous and discrete nonholonomic Lagrangian systems. They are analogues of constrained optimization problems, where the objective is differentiated in directions given by an apriori distribution, instead of tangent directions to the constraint. We show semiglobal existence and uniqueness for nondegenerate skew critical problems, and show that the solutions of two skew critical problems have the same contact as the problems themselves. Also, we develop some infrastructure that is necessary to compute with contact order geometrically, directly on manifolds
Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vectors. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.
In the formalism of constrained mechanics, such as that which underlies the SHAKE and RATTLE methods of molecular dynamics, we present an algorithm to convert any one-step integration method to a variational integrator of the same order. The one-step method is arbitrary, and the conversion can be automated, resulting in a powerful and flexible approach to the generation of novel variational integrators with arbitrary order.
The physics of nonlinear, nonholonomically constrained mechanical systems are not well understood. In order to make some headway, experiments need to be devised to test the various proposed theories. As a small part toward the solution of this problem, this paper addresses the feasibility of building an electric circuit analog of a nonholonomic system and its accuracy in representing the dynamics of the mechanical system. Abstract-The performance of chaotic synchronization based on the extended Kalman filter (EKF) is investigated here.We first establish the relationship between the EKF-based synchronization method and two conventional synchronization method, drive-response and unidirectionally coupled methods. The performance of the EKF-based synchronization method in the presence of channel noise is then derived in terms of mean square error (MSE) between the drive and response systems for one-dimensional discrete-time systems. Compared with the optimal coupled synchronization method, the EKF-based synchronization method is shown to have the same MSE performance for chaotic systems with gradient square independent of the system states (Type-I systems). For chaotic systems with state-dependent gradient square (Type-II systems), the EKF-based method is found to have a smaller MSE. The averaged Cramér-Rao lower bound (CRLB) is introduced here as a performance measure. It is shown that the EKF-based method approaches the averaged CRLB for both Type-I and Type-II systems when noise level is low. Our theoretical results are verified by using Monte Carlo simulation on three popular one-dimensional chaotic systems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.