Sufficient conditions are obtained for the oscillation of a general form of a linear second-order differential equation with discontinuous solutions. The innovations are that the impulse effects are in mixed form and the results obtained are applicable even if the impulses are small. The novelty of the results is demonstrated by presenting an example of an oscillating equation to which previous oscillation theorems fail to apply.

Collision of equilibria with a splitting manifold has been locally studied, but might also be a contributing factor to global bifurcations. In particular, a boundary collision can be coincident with collision of a virtual equilibrium with a periodic orbit, giving an analogue to a homoclinic bifurcation. This type of bifurcation is demonstrated in a non-smooth climate application. Here, we describe the non-smooth bifurcation structure, as well as the smooth bifurcation structure for which the non-smooth homoclinic bifurcation is a limiting case.

The Lagrange-d'Alembert equations with constraints belonging to H1,∞ have been considered. A concept of weak solutions to these equations has been built. A global existence theorem for Cauchy problem has been obtained.

The main result ensures that the scalar problem $x''\,{=}\,f(x)$, $x(0)\,{=}\,x_0$, $x'(0)\,{=}\,x_1,$ has a nonconstant locally $W^{2,1}$ solution if and only if there exists a nontrivial interval $J$ such that $x_0 \in J$, $f \in L^1_{\rm loc}(J)$, $x_1^2+2\int_{x_0}^{y}{f(s)\,ds} > 0$ for almost all $y \in J$ and \[\frac{\max\{1,|f|\}}{ \sqrt{x_1^2+2\int_{x_0}^{\cdot}{f(s)\,ds}}} \in L^1_{\rm loc}(J).\] Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.