Competitive modes conditions for chaos in nonlinear dynamical systems

Consider a nonlinear autonomous system

$$\dot x_i=f_i(x_1,x_2,\ldots,x_n),\qquad i=1,2,\ldots,n,$$

whose second-order form is

$$\ddot x_i=-x_i g_i(x_1,x_2,\ldots,x_n)+h_i(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n),$$

so that the functions $g_i$ act as oscillator frequencies and the functions $h_i$ are the remaining terms. Competitive modes conjecture. The system is chaotic only if: (1) there exist at least two functions $g_i$; (2) at least two of them are competitive or nearly competitive, meaning that $g_i\backsimeq g_j>0$ at some time $t$; (3) at least one $g_i$ depends on evolution variables such as $t$; and (4) at least one $h_i$ depends on the system variables. The conjecture proposes necessary conditions for chaos based on competition between oscillator modes. It is presented as a way to identify chaotic parameter regimes, but no resolution or proof is supplied here.