On the zeros of the Abelian integrals for a class of Liénard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this paper, we investigated a class of Liénard systems of the form x'=y, y'=f(x)+y g(x) with deg f=5 and deg g=4. We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.

Australian waterfalls as tourism attractions

For a largely arid country with generally low relief, Australia has a remarkably large number and variety of waterfalls. Found mainly near the coast, close to where most of the population lives and near the major tourist resort areas, these landscape features have long been popular scenic attractions. As sights to see and places to enjoy a variety of recreational activities, waterfalls continue to play an important role in Australia’s tourism, even in seaside resort areas where the main attractions are sunshine, sandy beaches and surf. The aesthetic appeal of waterfalls and their value as recreational resources are recognized by the inclusion of many in national parks. Even here, demands of visitors and pressures from developers raise serious problems. This paper examines the way in which waterfalls have been developed and promoted as tourist attractions, demonstrating their importance to Australian tourism. It considers threats to the sustainable use of waterfall resources posed by power schemes and, particularly, by the tourist industry itself. Queensland’s Gold Coast is selected as a case study, and comparisons are made with other areas in which waterfalls have played important roles as tourist attractions, especially the Yorkshire coast of northeast England. The discussion draws largely on an examination of tourist literature from the nineteenth to the twenty-first century, including holiday brochures and guide books, as well as other published sources, together with field observation in various parts of the world