Consider a polynomial Liénard system depending on three parameters a, b, c and with the following properties: (i) The origin is the unique equilibrium for all parameters. (ii). If a crosses zero, then the origin changes its stability, and a limit cycle bifurcates from the equilibrium. We investigate analytically this bifurcation in dependence on the parameters b and c and establish the existenc...

We prove that a group is hyperbolic relative to virtually nilpotent subgroups if and only if there exists a Gromov-hyperbolic metric space with bounded geometry on which it acts as a relatively hyperbolic group. As a consequence we obtain that any group hyperbolic relative to virtually nilpotent subgroups has finite asymptotic dimension. For these groups the Novikov conjecture holds. The class ...

We represent a filamentous actin molecule as a graph of finite-state machines (F-actin automaton). Each node in the graph takes three states — resting, excited, refractory. All nodes update their states simultaneously and by the same rule, in discrete time steps. Two rules are considered: threshold rule — a resting node is excited if it has at least one excited neighbour and narrow excitation i...

This paper studies the uid approximation, also known as the functional strong law-of-large-numbers, for a GI/G/1 queue under a processor-sharing service discipline. The uid (approximation) limit in general depends on the service time distribution, and the convergence is in general in the Skorohod J 1 topology. This is in contrast to the known result for the GI/G/1 queue under a FIFO service dis...

The limit cycle bifurcations of a Z2 equivariant planar Hamiltonian vector field of degree 7 under Z2 equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert’s 16th problem for degree 7, i.e., on the possible number of limit cycles that can bifurcate from a degree 7 planar...

We call Poincaré time the time associated to the Poincaré (or first return) map of a vector field. In this paper we prove the non-accumulation of isolated critical points of the Poincaré time T on hyperbolic polycycles of polynomial vector fields. The result is obtained by proving that the Poincaré time of a hyperbolic polycycle either has an unbounded principal part or is an almost regular fun...

In this paper, we consider the existence of limit cycles of coupled van der Pol equations by using S−degree theory due to Dylawerski et al., see [3].

This paper considers the limit cycles in the Liénard equation, described by €xþ f ðxÞ _ xþ gðxÞ 1⁄4 0, with Z2 symmetry (i.e., the vector filed is symmetric with the y-axis). Particular attention is given to the existence of small-amplitude (local) limit cycles around fine focus points when g(x) is a third-degree, odd polynomial function and f(x) is an even function. Such a system has three fix...