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My questions concerns the stability analysis of the following dynamical system :

$\dfrac{d}{dt} a_{i}(t) = D_{i} + \displaystyle{\sum_{j=1}^{n}L_{ij}a_{j}(t) + \sum_{j=1}^{n}\sum_{k=1}^{n} C_{ijk} a_{j}(t)a_{k}(t)}$

defined for the state variables $[a_1,a_2,...,a_n]^{T}$ with the real numbers $D_i, L_{ij}, C_{ijk}$ for $i,j,k=1,...,n.$ The coefficients $C_{ijk}$ are assumed to be symmetric in $j$ and $k$ : $C_{ijk}=C_{ikj}$

What are the conditions for the asymptotically stability of this system ?

Thank you.

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  • $\begingroup$ Is C matrix positive(negative) definite? $\endgroup$
    – percusse
    Commented Oct 7, 2013 at 8:01
  • $\begingroup$ Let me know when you've solved this one - your equation incorporates the Lorenz system and many others, even just for $n=3$. $\endgroup$
    – user25199
    Commented Oct 7, 2013 at 10:52

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Assuming you have an equilibrium solution $a^*$, the first thing to look at is the linear stability. Generically, the eigenvalues of the Jacobian matrix of the right side at $a^*$ will tell you this (the exceptions being when there are eigenvalues with real part $0$ and any others are all in the left half plane).

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