Universal criterion for the breakup of invariant tori in dissipative systems

The transition from quasiperiodicity to chaos is studied in a two-dimensional dissipative map with the inverse-golden-mean rotation number. On the basis of a decimation scheme, it is argued that the (minimal) slope of the critical iterated circle map is proportional to the effective Jacobian determinant. Approaching the zero-Jacobian-determinant limit, the factor of proportion becomes a universal constant. Numerical investigation on the dissipative standard map suggests that this universal number could become observable in experiments.