This paper describes three human-inspired approaches to balancing in highly dynamic environments. In this particular work, we focus on balancing on a bongo board a common device used for human balance and coordination training as an example of a highly dynamic environment. The three approaches were developed to overcome limitations in robot hardware. Starting with an approach based around a sim...

Limit cycle stability analysis and adaptive control of a multi-compartment model for a pressure-limited respirator and lung mechanics system VijaySekhar Chellaboina a; Wassim M. Haddad b; Hancao Li b;James M. Bailey c a Advanced Technology Center, Tata Consultancy Services, Hyderabad, India 500081 b School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA c ...

The period annuli of the planar vector field x′ = −yF (x, y), y′ = xF (x, y), where the set {F (x, y) = 0} consists of k different isolated points, is defined by k + 1 concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n. Additionally, ...

Limit cycles are common in hybrid systems. However the nonsmooth dynamics of such systems makes stability analysis difficult. This paper uses recent extensions of trajectory sensitivity analysis to obtain the characteristic multipliers of nonsmooth limit cycles. The stability of a limit cycle is determined by its characteristic multipliers. The concepts are illustrated using a coupled tank syst...

The Existential Hilbert Problem is a weak version of the part b of the Hilbert 16-th problem which also asks not only about the number, but also about position of limit cycles of (1). The problem about finiteness of number of limit cycles for an individual polynomial line field (1) is called Dulac problem, since the pioneering work of Dulac [Du], who claimed in 1923 to solve this problem, but a...

This paper presents some new results which we obtained recently for the study of limit cycles of nonlinear dynamical systems. Particular attention is given to small limit cycles of generalized Liénard systems in the vicinity of the origin. New results for a number of cases of the Liénard systems are presented with the Hilbert number, b H ði; jÞ 1⁄4 b H ðj; iÞ, for j = 4, i = 10,11,12,13; j = 5,...

In this work, Liénard equations are considered. The limit cycles of these systems are studied by applying the homotopy analysis method. The amplitude and frequency obtained with this methodology are in good agreement with those calculated by computational methods. This puts in evidence that the homotopy analysis method is an useful tool to solve nonlinear differential equations.

We discuss a previous attempt at a microscopic counting of the entropy of asymptot-ically flat non-extremal black-holes. This method used string dualities to relate 4 and 5 dimensional black holes to the BTZ black hole. We show how the dualities can be justified in a certain limit, equivalent to a near horizon limit, but the resulting spacetime is no longer asymptotically flat.

Consider the planar ordinary differential equation ẋ = −y(1 − y)m, ẏ = x(1 − y)m, where m is a positive integer number. We study the maximum number of zeroes of the Abelian integral M that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field. One of the key points of our approach is that we obtain a simple ex...

Using Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center ẋ = −y((x + y)/2) and ẏ = x((x + y)/2) with m ≥ 1, when we perturb it inside the whole class of polynomial vector fields of degree n. The positive integers m and n are arbitrary. As far as we know there is only one paper that provide ...