The Separating Mechanisms in Poincaré Halfmaps

Abstract

The separating mechanisms occuring in a class of Poincar6 halfm aps that is induced by the flow of a saddle-focus are described quantitatively. The "critical spiral" is calculated explicitly inside the domain o f the map. A complete hierarchy, consisting of three different types of separating mechanisms, is demonstrated. It is characterized in terms o f the properties o f the critical spiral, as it depends on the canonical param eters o f the system.

Analytical Approximations to Type III Separation Thresholds in Poincaré Halfmaps

“…As was explained already in [11], the present dy namics is the time reverse of the one treated in [7] and [10]. Thus, whenever we talk about "trajectories", "entry" or "exit points", we adopt the opposite time direction, in order be consistent with the previous papers.…”

“…This Separation is called "type I" for values of the canonical parameters below the curve co2 = q + 1, and is denoted "type II" above the latter threshold [7], For a properly chosen second halfmap both types of separation give rise to a positive Lyapunov ex ponent [9] and hence chaotic solutions.…”

“…For the simplest cases, with only two regions in state space, the notion of Poincare halfmaps [6] turned out to be a fruitful concept. These maps (denoted here as P and P) are defined in the plane S, which separates the two regions, called f and T [7], When a saddle-focus type dynamics in three space (with an actual fixed point [8] and negative real eigen value) generates a halfmap, the mapping behaves discontinuously along a branch cut, the critical spiral y, being from the domain of the map [6,7], The latter property of halfmaps exhibits a separating mecha nism, which is due to trajectories that touch the plane S. They lie in between those orbits that exit the region close to the point of contact of the touching trajectory and those which perform almost one more turn around the real eigenvector of the saddle-focus.…”

“…For the simplest cases, with only two regions in state space, the notion of Poincare halfmaps [6] turned out to be a fruitful concept. These maps (denoted here as P and P) are defined in the plane S, which separates the two regions, called f and T [7], When a saddle-focus type dynamics in three space (with an actual fixed point [8] and negative real eigen value) generates a halfmap, the mapping behaves discontinuously along a branch cut, the critical spiral y, being from the domain of the map [6,7], The latter property of halfmaps exhibits a separating mecha nism, which is due to trajectories that touch the plane S. They lie in between those orbits that exit the region close to the point of contact of the touching trajectory and those which perform almost one more turn around the real eigenvector of the saddle-focus.Reprint requests to Dr. C. Kahlert, Institut für Physikalische und Theoretische Chemie der Universität Tübingen, Auf der Morgenstelle 8, D-7400 Tübingen, F.R.G. This Separation is called "type I" for values of the canonical parameters below the curve co2 = q + 1, and is denoted "type II" above the latter threshold [7], For a properly chosen second halfmap both types of separation give rise to a positive Lyapunov ex ponent [9] and hence chaotic solutions.…”

“…This yields a third (stronger) kind of separation, called "type III". Since no trajectory can touch the plane S more than twice, this is the most complicated behav ior possible for a single halfmap [7], The sets of parameter values, where doubly touching orbits first come into existence, form the curve in the 0-co-plane of the canonical parameter space [7], Beyond a countable infinity of curves Qk (k = 2, 3,...) exists. There the kth pair of break and restart points of y freshly appears; together with a doubly touching trajectory, which performes k turns around the real eigenvector of the saddle-focus in be tween its two points of contact (bk and ck) with S.…”