2009/04/17 by Norman R. Lebovitz, Norman Lebovitz
Earth and Planetary Sciences · Engineering · Environmental Science · Mathematics · Physics and Astronomy · #Attractor #Boundary (topology) #Classical mechanics #Dynamical systems theory #Fluid Dynamics and Turbulent Flows #Geology #Geometry #Geomorphology #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Mechanics #Paleontology #Periodic orbits #Phase space #Phase transition #Physics #Plant Water Relations and Carbon Dynamics #Shear (geology) #Shear flow #Statistical physics #Structural basin #Tree-ring climate responses #Turbulence #physics.flu-dyn
paper · pdf · doi:10.1088/0951-7715/22/11/004
11 pages; submitted for publication in Nonlinearity
arxiv created 2009/04/17 · openalex publication_date 2009/10/02 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
The basin of attraction of a stable equilibrium point is investigated for a dynamical system (W97) that has been used to model transition to turbulence in shear flows. The basin boundary contains a linearly unstable equilibrium point X lb which, in the self-sustaining scenario, plays a role in mediating the transition in that transition orbits cluster around its unstable manifold. However we find—for W97 with canonical parameter values—that this role is played not by X lb but rather by a periodic orbit also lying on the basin boundary. Moreover, it appears via numerical computations that all orbits beginning near X lb relaminarize. We offer numerical evidence that the parameter values of W97 are post-critical in the following sense: for some, subcritical parameter values, the basin boundary coincides with the stable manifold of X lb and only a subset of nearby orbits relaminarize, whereas for supercritical values the basin boundary is the union of two stable manifolds, one belonging to the periodic orbit and dominating the basin boundary, and the other belonging to X lb and detectable only as edge separating relaminarizing orbits of different characters. The periodic orbit appears at the critical parameter value via a homoclinic connection. This further leads to a proposal for the structure of the 'edge of chaos' somewhat different from that which has previously been proposed.