In a previous paper [Tonnelier, 2002] we conjectured that a Liénard system of the form ẋ = p(x) − y, ẏ = x where p is piecewise linear on n + 1 intervals has up to 2n limit cycles. We construct here a general class of functions p satisfying this conjecture. Limit cycles are obtained from the bifurcation of the linear center.

The paper investigates a stochastic model where two agents (persons, companies, institutions, states, software agents or other) learn interactive behavior in a series of alternating moves. Each agent is assumed to perform “stimulus–response–consequence” learning, as studied in psychology. In the presented model, the response of one agent to the other agent’s move is both the stimulus for the ot...

This paper is devoted to development of a new synthesis method of a multi-modal and 2-dimensional piecewise affine system that generates a desired stable limit cycle and investigation of some characteristics for the system. The new proposed method has some advantages on the convergence property of solution trajectories to a desired limit cycle in comparison with the previous one. First, the pro...

Liénard systems of the form ẍ+ ǫf(x)ẋ+x = 0, with f(x) an even function, are studied in the strongly nonlinear regime (ǫ → ∞). A method for obtaining the number, amplitude and loci of the limit cycles of these equations is derived. The accuracy of this method is checked in several examples. Lins-Melo-Pugh conjecture for the polynomial case is true in this regime.

Niels Lörch, Jiang Qian, Aashish Clerk, Florian Marquardt, and Klemens Hammerer Institut für Gravitationsphysik, Leibniz Universität Hannover and Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Callinstraße 38, 30167 Hannover, Germany Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany Arnold Sommerfeld Center for Theoret...

We study the attractor structure of standard block-sequential threshold dynamical systems. In a block-sequential update, the vertex set of the graph is partitioned into blocks, and the blocks are updated sequentially while the vertices within each block are updated in parallel. There are several notable previous results concerning the two extreme cases of block-sequential update: (i) sequential...

Creating and studying neurocognitive architectures is an active and increasing focus of research efforts. Based on our recent research that uses neural activity limit cycles in selforganizing maps (SOMs) to represent external stimuli, this study explores the use of such limit cycle attractors in a neurocognitive architecture for an open-loop arm reaching task. The goal is to learn to produce a ...

In this article, we study the planar cubic polynomial differential system ẋ = −yR(x, y) ẏ = xR(x, y) where R(x, y) = 0 is a conic and R(0, 0) 6= 0. We find a bound for the number of limit cycles which bifurcate from the period annulus of the center, under piecewise smooth cubic polynomial perturbations. Our results show that the piecewise smooth cubic system can have at least 1 more limit cycle...

This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows thatH(4) ≥ 20, whereH(n) is the Hilbert number for the second part of Hilbert’s 16th problem. It is well known that ...

I review recent advances in our understanding of accretion disks in transient systems− the dwarf novae and the soft X-ray transients. The primary theme will be the ongoing development of theory in response to the observations. The accretion disk limit cycle model appears to provide a unifying framework within which we may begin to understand what is seen in different types of interacting binary...