I have a system of $N$ coupled differential equations that are numerically integrated in time. Each equation $i$ is coupled to the other equations $(j,k,l,etc...)$ by

$$ \ddot{x}_i = -\sum_iK_ix_i - \sum_{jk}K_{ijk}x_jx_k-\sum_{jkl}K_{ijkl}x_jx_kx_l - \cdots$$

The only constraint is that $K_i > 0$. The constants $K_{ijk}$, $K_{ijkl}$, etc. can be anything, but I'm looking for values such that the system is numerically "stable" in the sense that values of $x_i$ oscillate about a small value given some small perturbation as an initial condition.

Given random intitial conditions of $x_i$, I numerically integrate this equation in time. I notice some peculiarities that lead to questions:

  1. The system is always "stable" in the sense that values of $x_i$ oscillate about $0$ if only the first term on the right hand side, $-\sum_iK_ix_i$, is retained. Why do higher order terms cause instability?
  2. Are there nonzero constants $K_{ijk}$, $K_{ijkl}$ that ensure stability?
  3. If so, how do you find these constants?
  4. Is there any field of math that studies the dynamical "stability" of systems of coupled equations like these?

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