A system, which is a discontinuous and noninvertible concatenation of two area\|preserving maps, may display a dissipation showing linear time dipendence and is addressed as “quasi-dissipative”. Three quasi-dissipative systems are studied. They are: a model of an electronic relaxation oscillator with over-voltage protection, a kicked rotor subjected to a piecewise-continuous force field, and a similar rotor but with an oscillating discontinuity border. I the systems two unique features, named a period-doubling bifurcation cascade interrupted by a border-collision bifurcation, and some elliptic islands, which attract iterations outside, have been observed.
