Search results for "Limit cycle"

More limit cycles than expected in Liénard equations

2007

The paper deals with classical polynomial Lienard equations, i.e. planar vector fields associated to scalar second order differential equations x"+ f(x)x' + x = 0 where f is a polynomial. We prove that for a well-chosen polynomial f of degree 6, the equation exhibits 4 limit cycles. It induces that for n ≥ 3 there exist polynomials f of degree 2n such that the related equations exhibit more than n limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Lienard equations as above, with f of degree 2n, the maximum number of limit cycles is n. The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classic…

Darboux integrable system with a triple point and pseudo-abelian integrals

2016

We study pseudo-abelian integrals associated with polynomial perturbations of Dar-boux integrable system with a triple point. Under some assumptions we prove the local boundedness of the number of their zeros. Assuming that this is the only non-genericity, we prove that the number of zeros of the corresponding pseudo-abelian integrals is bounded uniformly for nearby Darboux integrable foliations.

A generalization of Françoise's algorithm for calculating higher order Melnikov functions

2002

Abstract In [J. Differential Equations 146 (2) (1998) 320–335], Francoise gives an algorithm for calculating the first nonvanishing Melnikov function Ml of a small polynomial perturbation of a Hamiltonian vector field and shows that Ml is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form ω on the family of cycles implies that the form is algebraically relatively exact. We study here a simple example where Francoise's condition is not verified. We generalize Francoise's algorithm to this case and we show that Ml belongs to the C [ log t,t,1/t] module above the Abelian integrals. We also establish the linear differentia…

A novel digital control for DC/DC converters to improve steady-state performances

2006

This paper describes an innovative digital PWM control implementation for low voltage, high current DC-DC converters. The proposed technique, based on the use of a low resolution DAC, improves steady-state performances, minimizing limit cycle effects. The novel technique is tested on a FPGA-based single phase buck converter operating at 250 kHz. A detailed description of the proposed architecture is given and test results, simulation and experimental ones, are shown

Mathematical models and stability analysis of three-phase synchronous machines

2013

Abelian integrals and limit cycles

2006

Abstract The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.

Multiplicity of fixed points and growth of ε-neighborhoods of orbits

2012

We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on epsilon of the length of epsilon-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered before (Elezovic, Zubrinic, Zupanovic) in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity o…

Application of thermodynamics to driven systems

2007

Application of thermodynamics to driven systems is discussed. As particular examples, simple traffic flow models are considered. On a microscopic level, traffic flow is described by Bando's optimal velocity model in terms of accelerating and decelerating forces. It allows to introduce kinetic, potential, as well as total energy, which is the internal energy of the car system in view of thermodynamics. The latter is not conserved, although it has certain value in any of two possible stationary states corresponding either to fixed point or to limit cycle in the space of headways and velocities. On a mesoscopic level of description, the size n of car cluster is considered as a stochastic varia…

Design, Control, and Analysis of Nonlinear Circuits with Tunnel Diode with Piecewise Affine Dynamics

2019

Iterative Learning Applied to Hydraulic Pressure Control

2018

This paper addresses a performance limiting phenomenon that may occur in the pressure control of hydraulic actuators subjected to external velocity disturbances. It is demonstrated that under certain conditions a severe peaking of the control error may be observed that significantly degrades the performance of the system due to the presence of nonlinearities. The phenomenon is investigated numerically and experimentally using a system that requires pressure control of two hydraulic cylinders. It is demonstrated that the common solution of feed forwarding the velocity disturbance is not effective in reducing the peaking that occurs as a result of this phenomenon. To improve the system perfor…