First-return map

The Poincare map is also called the first-return map, because... 

Fig. 10. Stroboscopic plot of two variables of the network, recorded at frequency Q. of the target waves in compartment I. (a) The closed curve reveals the presence of a quasi-periodic dynamics, (b) First return map associated with the stroboscopic plot. The phase 6 of the curve is defined as the angle that forms the vector joining the centre of the cycle (the cross) to the state point with a reference direction. Parameters of Equation (1) are identical to those of Figure 9. The window length is / = 8.

Fig. 10 (a), (b) and (c) are two-dimensional projections of the phase portraits of the chaotic states Co, Ct and cf respectively. The corresponding first return maps obtained with the... 

Fig. 9.5 Schematic representation of Type-I intermittency. The plot on the LHS shows a return map for the system as it appears just below and precisely at the critical parameter value Tc. The plot on the RHS shows the return map for r > r. Note how, for r > Vc, X = Xc appears to first attract" then repel trajectories.

Now consider the general case Given a system x = f(x) with a closed orbit, how can we tell whether the orbit is stable is not Equivalently, we ask whether the corresponding fixed point x of the Poincare map is stable. Let Vq be an infinitesimal perturbation such that x +V(, is in S. Then after the first return to S,... 

Second, the Lorenz map may remind you of a Poincare map (Section 8.7). In both cases we re trying to simplify the analysis of a differential equation by reducing it to an iterated map of some kind. But there s an important distinction To construct a Poincare map for a three-dimensional flow, we compute a trajectory s successive intersections with a two-dimensional surface. The Poincare map takes a point on that surface, specified by two coordinates, and then tells us how those two coordinates change after the first return to the surface. The Lorenz map is different because it characterizes the trajectory by only one number, not two. This simpler approach works only if the attractor is very flat, i.e., close to two-dimensional, as the Lorenz attractor is. 

Strength Responders were thoroughly debriefed when they returned to the CP from operations in the hot zone. Logistical support was timely in processing requests once they were established. Additional maps were available at the CP within the first hour. Railroad consists received at the CP within the first hour. Written preplans were used for searches of mill facilities GVW Fire Department walks down all Avondale facilities annually. 

Fig. 2.7 shows the graph of T in A. The shape of the graph explains the mapping s name. All points with x < 0 are mapped monotonically to —oo. Points with x > 1 are first mapped to 3(1 — x) < 0 and then also to —oo. Thus, none of the points outside A will ever be mapped into A. This is an important property. It implies that whenever a point of A is mapped outside A this point will never return to A. Thus, this property is called the never-come-hackpiopeTty. It facilitates appreciably the analysis of the tent map. 

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