The Maximal Number of Limit Cycles in a Family of Polynomial Systems

The main objective of this paper is to study the number of limit cycles in a family of polynomial systems. Using bifurcation methods, we obtain the maximal number of limit cycles in global bifurcation.

Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation

We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as ‘tipping points’, is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods arementioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.

The Existence of Positive Solutions for a Nonlinear Sixth-Order Boundary Value Problem

By using the Krein-Rutman theorem and bifurcation methods, we discuss the existence of positive solutions for the boundary value problems of a sixth-order ordinary differential equation.