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: This work studies a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0, which has a linear saddle with its equilibrium point living in x > 0, and it is said that it is virtual.
Abstract: We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.
51 citations
TL;DR: The piecewise linear method of nonlinear control systems analysis is developed by invoking an engineering oriented approach, rather than the classical mathematical treatment, to facilitate the analytical investigations of a wide variety of control loops containing discontinuous nonlinearities.
Abstract: The piecewise linear method of nonlinear control systems analysis is developed by invoking an engineering oriented approach, rather than the classical mathematical treatment. A simple but formalistic procedure is suggested to facilitate the analytical investigations of a wide variety of control loops containing discontinuous nonlinearities.
51 citations
TL;DR: In this article, the Babuska-Aziziziz constant is explicitly determined for the piecewise constant interpolation function and the piece-wise linear one over triangles, which plays an essential role in the interpolation error estimation of the linear triangular finite element.
51 citations
TL;DR: Theoretical models of neural and genetic control networks consisting of N interacting elements are cast in the form of an N-dimensional system of piecewise linear (PL) ordinary differential equations as discussed by the authors.
Abstract: Theoretical models of neural and genetic control networks consisting of N interacting elements are cast in the form of an N-dimensional system of piecewise linear (PL) ordinary differential equations. The state transition diagram of this system, which represents transitions between distinct volumes in the N-dimensional phase space, is given as a directed graph on an N-dimensional hypercube in which each edge has only one orientation. Analytical results establish steady-states and limit cycles in these systems, and numerical results have identified chaotic dynamics for N ≥ 6.
51 citations
TL;DR: It is proved that under some conditions, such n-neuron neural networks can have at least 5n equilibrium points, 3n of which are locally stable and the others are unstable, based on the contraction mapping theorem and the theory of strict diagonal dominance matrix.
Abstract: In this paper, we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions. It is proved that under some conditions, such $n$ -neuron neural networks can have at least $5^{n}$ equilibrium points, $3^{n}$ of which are locally stable and the others are unstable, based on the contraction mapping theorem and the theory of strict diagonal dominance matrix. The investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous Mexican-hat-type activation function or discontinuous two-level activation functions. An illustrative example with computer simulations is presented to verify the theoretical analysis.
51 citations