  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 ODE's 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 $d\sim 50$. 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.
