In a one-parameter study of a noninvertible family of maps of the plane arising in the context of a numerical integration scheme, Lorenz studied a sequence of transitions from an attracting fixed point to "computational chaos". As part of the transition sequence, he proposed the following as a possible scenario for the breakup of an invariant circle (IC): the IC develops regions of increasingly sharper curvature until at a critical parameter value it develops cusps; beyond this parameter value, the IC fails to persist, and the system exhibits chaotic behavior on an invariant set with loops [Computational chaos - a prelude to computational instability, Physica D 35 (1989) 299]. We investigate this problem in more detail and show that the IC is actually destroyed in a global bifurcation before it has a chance to develop cusps. Instead, the global unstable manifolds of saddle-type periodic points are the objects which develop cusps and subsequently "loops" or "antennae". The one-parameter study is better understood when embedded in the full two-parameter space and viewed in the context of the two-parameter Arnold horn structure. Certain elements of the interplay of noninvertibility with this structure, the associated ICs, periodic points and global bifurcations are examined.
Bibliographical noteFunding Information:
Besides being motivated by the original work of Professor E. Lorenz, we acknowledge extensive discussions with him when this paper was first being developed in the early and mid 1990s. He declined, however, being a co-author of the paper, something that we believe he deserved, and we note this here. We also acknowledge discussions with Prof. R. Rico-Martinez of the Instituto Tecnologico de Celaya (Mexico) and Prof. C. Mira of the Complexe Scientifique de Rangueil, Toulouse, France. We also acknowledge the support of the National Science Foundation (IGK and BBP: grant no. DMS-9973926), and the Swiss Office of Energy (CEF). Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the funding agencies.
- Invariant circles
- Noninvertible maps