The classical Morse theory proceeds by considering sublevel sets f^{−1}(−∞,a] of a Morse function f:M→R, where M is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets f^{−1}(a) and give conditions under which the topology of f^{−1}(a) changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse functions, the topology of a regular level f^{−1}(a) always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold M. When f is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the base space. (Counter-)examples and applications to celestial mechanics are also discussed.

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We consider a damped/driven nonlinear Schr\"odinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n\in\mathbb{N}$, under Dirichlet boundary conditions \[ u_t-\nu\Delta u+i|u|^2u=\sqrt{\nu}\eta(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad \nu>0, \] where $\eta(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ \| u(t)\|_m^2 \le C\nu^{-m}, $ uniformy in $t\ge0$ and $\nu>0$. In this work we prove that for small $\nu>0$ and any initial data, with large probability the Sobolev norms $\|u(t,\cdot)\|_m$ of the solutions with $m>2$ become large at least to the order of $\nu^{-\kappa_{n,m}}$ with $\kappa_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}{\nu})$.

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Consider a system of periodic pendulum lattice with analytic weak coupling: i x ̈i +sinxi =−ε 􏰀 ∂xiβα(xj,xj+1,xj+2), xi =xi+N, i∈Z, j =i−2 where N 􏰃 3 is an integer, ε > 0 is a small parameter and the function βα is an analytic function of a certain form. It is shown in this paper that for small enough ε, the system admits motions such that the energy transfers between the pendulums in any predetermined order.