Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order m

In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near-Hamiltonian system. For any positive integer $m(m\ge 1)$, we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for $m=2$, we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles.     

A dynamic free-entry oligopoly with sluggish entry and exit adjustments

We study a dynamic free-entry oligopoly with sluggish entry and exit of firms under general demand and cost functions. We show that the number of firms in a steady-state open-loop solution for a dynamic free-entry oligopoly is smaller than that at static equilibrium and that the number of firms in a steady-state memoryless closed-loop solution is larger than that in an open-loop solution.     

Ill-posedness of the Camassa–Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B_{p,r}^{1+1/p}$ for $p\in [1,\infty ],r\in (1,\infty ]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]).