Period adding and broken Farey tree sequence of bifurcations for mixed-mode oscillations and chaos in the simplest three-variable nonlinear system

A detailed study of the simplest three-variable model exhibiting mixed-mode oscillations and chaos is presented. We show that mixed-mode oscillations appear due to a sequence of bifurcations which is characterized by a combin ation of the Farey tree that is broken by chaotic windows and period adding . This scenario is supported by a family of one-dimensional return maps. Th e model also exhibits hysteresis between stable steady state and mixed mode s. (C) 2000 American Institute of Physics. [S0021- 9606(00)50439-X].