Topic
Piecewise linear function
About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.
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TL;DR: A novel manifold piecewise-linear (MPL) system as a control object that can generate both chaotic attractors and periodic attractors is proposed and a novel occasional proportional feedback (OPF) method is proposed to apply to the MPL.
Abstract: This paper considers a basic approach to generalize techniques of controlling chaos. First, we propose a novel manifold piecewise-linear (MPL) system as a control object. The novel MPL can generate both chaotic attractors and periodic attractors. Second, we propose a novel occasional proportional feedback (OPF) method and apply it to the MPL. The OPF can change the form of the return map of the MPL-it can change chaos into a periodic attractor and change a periodic attractor into chaos. The OPF functions can be guaranteed theoretically. Third, we propose an implementation example of the OPF and confirm its function in the laboratory.
40 citations
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TL;DR: Numerical experiments confirm the theoretical predictions on the improved spatial convergence and indicate that the Crank--Nicolson scheme is not always superior over the implicit Euler scheme in practice.
Abstract: The quasi-static elastoplastic evolution problem with combined isotropic and kinematic hardening is considered with emphasis on optimal convergence of the lowest order scheme. In each time-step of a generalized midpoint scheme such as the implicit Euler or the Crank--Nicolson scheme, the spatial discretization consists of minimizing a convex but nonsmooth function on a subspace of continuous piecewise linear, resp., piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which, for smooth data and a smooth exact solution, proves linear convergence as the mesh-size tends to zero. Strong convergence of the time-derivatives is established under mild conditions on the mesh- and time-step sizes. Numerical experiments confirm our theoretical predictions on the improved spatial convergence and indicate that the Crank--Nicolson scheme is not always superior over the implicit Euler scheme in practice.
40 citations
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TL;DR: In this article, the expression of the Green's function related with a first order periodic differential equation with piecewise constant argument was obtained and comparison results for the treated linear operator were derived by studying the sign of the obtained Green's functions.
40 citations
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TL;DR: In this article, the authors studied geodesies in metric complexes and showed that a simplicial complex can be metrized by assigning to each simplex a £ M a linear simplex in some Euclidean space Rfc so that face relations correspond to isometries.
Abstract: A simplicial complex M is metrized by assigning to each simplex a £ M a linear simplex a* in some Euclidean space Rfc so that face relations correspond to isometries. An equivalence class of metrized complexes under the relation generated by subdivisions and isometries is called a metric complex; it consists primarily of a polyhedron M with an intrinsic metric pmThis paper studies geodesies in metric complexes. Let P e M; then the tangent space 7p(M) is canonically isometric to an orthogonal product of cones from P, Rk x i>p(M); once k is as large as possible. vpQA) is called the normal geometry at P in M. Let PX be a tangent direction at P in vp(M). I define numbers k+(PX) and kJ¡PX), called the maximum and minimum curvatures at P in the direction PX. THEOREM. Let M be a complete, simply-connected metric complex which is a p.l n-manifold without boundary. Assume k+(PX) < 0 for all P e M and all PX Ç vp(M). Then M is p.l. isomorphic to R". This is analogous to a well-known theorem for smooth manifolds by E. Cartan and J. Hadamard. THEOREM (ROUGHLY). Let M be a complete metric complex which is a p.L n-manifold without boundary. Assume (1) there is a number k ^ 0 such that k_(PX) > k whenever P is in the (n — 2)-skeleton of M and whenever PX Ç pp(M); (2) the Simplexes of M are bounded in size and shape. Then M Is compact. This is analogous to a weak form of a well-known theorem of S. B. Myers for smooth manifolds.
40 citations
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09 Dec 2003
TL;DR: This work proposes a method for identifying the parameters of the local models when choosing an adapted weighting function, this function allowing to select the data for which each local model is active and is able to solve simultaneously the data allocation and the parameter estimation.
Abstract: During the last years, a number of methodological papers on models with discrete parameter shifts have revived interest in the so-called regime switching models. Piecewise linear models are attractive when modelling a wide range of nonlinear system and determining simultaneously i) the data partition ii) the time instant of change iii) the parameter values of the different local models. This is a difficult problem for which no solution exists in the general case and we show here some aspects and particular results concerning the problem of off line learning of switching time series. We propose a method for identifying the parameters of the local models when choosing an adapted weighting function, this function allowing to select the data for which each local model is active. Indeed the proposed method is able to solve simultaneously the data allocation and the parameter estimation. The feasibility and the performance of the procedure is demonstrated using several academic examples.
40 citations