For general dissipative dynamical systems we study what fraction of solutions exhibit chaotic behavior depending on the dimensionality $d$ of the phase space. We find that a system of $d$ globally coupled ODEs with quadratic and cubic non-linearities with random coefficients and initial conditions, the probability of a trajectory to be chaotic increases universally from $sim 10^{-5} - 10^{-4}$ for $d=3$ to essentially one for $dsim 50$. In the limit of large $d$, the invariant measure of the dynamical systems exhibits universal scaling that depends on the degree of non-linearity but does not depend on the choice of coefficients, and the largest Lyapunov exponent converges to a universal scaling limit. Using statistical arguments, we provide analytical explanations for the observed scaling and for the probability of chaos.
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