ON THE EXISTENCE OF INFINITE PERIOD-DOUBLING SEQUENCES IN A CLASS OF 4D-SEMI-SYMPLECTIC MAPPINGS

We investigate the existence of an infinite period-doubling sequence in the following class of semi-symplectic maps: a two-dimensional (2D) nonlinear constant Jacobian map with an infinite period-doubling sequence, linearly and weakly coupled with a 2D linear map. We introduce the property of semi-symplecticity as the generalization of symplecticity that incorporates uniform dissipation. We define Krein signatures for pairs of complex eigenvalues and show that they play the same role as in the symplectic case. The Krein signature alternates in the period-doubling sequence of a 2D constant Jacobian map, whereas the signatures of iterates of a 2D linear map form (almost always) an uncorrelated row. With these results we show that any finite coupling strength destroys the infinite period-doubling sequence of the conservative maps of our class in two ways: firstly, there are 'bubbles of instability'; secondly, and far more importantly, there are period-doubling bifurcations in which the newborn period-doubled orbits are unstable. So crucial parts of the period-doubling sequence are experimentally invisible. For the dissipative maps of our class the same conclusions hold, but only if the coupling strength is strong enough.