Bifurcation From S-Shaped Solution Curves in A Class of Sturm-Liouville Problems Related to Climate Modeling

摘要

We study a class of parameter-dependent Legendre-type ordinary differential equations of the form -d/dx (#kappa#p du/dx) = #mu#f(u) - g(u), which arise from simple energy-balance climate models. Here k is a positive function on the interval [-1, 1]; p denotes the polynomial x -> 1 - x~2; f and g are nonnegative functions on [0, infinity] satisfying certain monotonicity and convexity conditions; and #mu# is a nonnegative parameter. We are interested in the structure of the set sum of all pairs (#mu#, u), where #mu# implied by [0, infinity] and u implied by C~2 ((-1,1)) intersect C ([-1, 1]) is a nonnegative classical solution of the differential equation. Under suitable assumptions on f and g, the set sum contains an "S-shaped" branch sum_0 of trivial solutions, that is, pairs (#mu#, u) with u = const. We obtain sufficient conditions for the bifurcation of branches of nontrivial solutions from sum_0 and describe the global behavior of the bifurcating branches.