Transition to turbulence in a discrete Ginzburg-Landau model

We present a numerical study of the onset of turbulence in a discretized version of the complex Ginzburg-Landau equation. The transition point is determined by computing Lyapunov exponents, which show a first-order transition at a parameter value ${\mathrm{\alpha }}_1$ below the linear stability threshold for the uniform state. On further decreasing the parameter, the finite-time Lyapunov exponent remains positive only up to a characteristic transient time, after which the vortices get entangled and the asymptotic Lyapunov exponents become zero. The finite-time exponent goes to zero at ${\mathrm{\alpha }}_{\mathit{c}}$${\mathrm{\alpha }}_1$ as a power law.