6 problems

Consider a nonlinear autonomous system … whose second-order form is … so that the functions $g_i$ act as oscillator frequencies and the functions $h_i$ are the remaining terms. Com…

Endpoint behavior conjecture. There exists $E_k>0$ such that if

Let $m,n$ be the mode indices, let $P$ be the parameter, and define … Let $\Theta_m$ and $\Theta_n$ be the corresponding modes with energy $E_{\Theta_m}$. Assume … Let $E_1$ and…

Let $m,n$ be the mode indices, let $P$ be the parameter, and let $\Theta_m$ and $\Theta_n$ denote the corresponding modes with energy $E_{\Theta_m}$. Assume … Let $E_2$ be the thre…

All-energy stability conjecture. Under these assumptions, $\Theta_m$ is linearly stable with respect to $\Theta_n$ for every $E_{\Theta_m}>0$.

Let $({\cal P}_0)$ be the discretized nonlinear algebraic problem associated with a nonlinear dynamical system, and let $({\cal P})$ be the corresponding optimization problem with…