Two-dimensional Poincare map

For three-dimensional systems (i,e. for two-dimensional Poincare maps) the characteristic equation is... 

To answer it, one must examine the two-dimensional Poincare map instead of the one-dimensional one, and evaluate the Jacobian of the former map. If its absolute value is larger than one, the map has no stable periodic points, and hence there are no stable orbits in a neighborhood of the homoclinic trar jectory because the product of the multipliers of the fixed point is equal to the determinant of the Jacobian matrix of the map. One can see from formula (13.4.2) that the value of the Jacobian is directly related to whether — 1 >0 or 2i/ — 1 < 0, or, equivalently, i/ > 1/2 or z/ < 1/2. Rephrasing in terms of the characteristic exponents of the saddle-focus, the above condition translates into whether the second saddle value o-q = Ai + 2ReA is positive or negative. It can be shown [100] that if <7 > 0 but ct2 < 0 (a < 6 in Fig. C.7.4), there may be stable periodic orbits near the loop, along with saddle ones. However, when (72 > 0 > O5 automatically), totally imstable periodic orbits emerge... 

Fig. C.7.11. Twisted (A < 0) and orientable A > 0) double homoclinic loops. The two-dimensional Poincare map has a distinctive hook-like shape after the separatrix value A becomes negative.

Another codimension-two homoclinic bifurcation in the Shimizu-Morioka model occurs at (a 0.605,6 0.549) on the curve H2 corresponding to the double homoclinic loops. At this point, the separatrix value A vanishes and the loops become twisted, i.e. we run into inclination-flip bifurcation [see Figs. 13.4.8 and C.7.11]. The geometry of the local two-dimensional Poincare map is shown in Fig. 13.4.5 and 13.4.6. To find out what our case corresponds to in terms of the classification in Sec. 13.6, we need also to determine the saddle index 1/ at this point. Again, as in the case of a homoclinic loop to the saddle-focus, it is very crucial to determine whether 1/ < 1/2 or 1/ > 1/2. Simple calculation shows that z/ > 1/2 for the given parameter values. Therefore,... 

A fragment of its (r, cr) bifurcation diagram is shown in Fig. C.7.14. Detect the points where the path cr = 10 intersect the curve HB of the homoclinic butterfly and the curve LA on which the one-dimensional separatrices of the saddle tend to the saddle periodic orbits. Find the point on the curve LA above which the Lorenz attractor does not arise upon crossing LA towards larger values of r. The dashed line passing through the T-point in Fig. C.7.14 corresponds to the moment of the creation of the hooks in the two-dimensional Poincare map when the separatrix value varishes A — 0 (see discussion on the Shimizu-Morioka model). ... 

The function P can be computed from either an analytical or a numerical representation of the flow field. In such a way, a 3-D convection problem is essentially reduced to a mapping between two-dimensional Poincare sections. In order to analyze the growth of interfacial area in a spatially periodic mixer, the initial distri-... 

In two-dimensional Hamiltonian systems, the trajectories can be visualized by means of the Poincare surface of section plot. It is also possible to study two-dimensional Hamiltonian systems using the two-dimensional symplectic mapping. A typical phase space portrait of generic nonhyperbolic phase space is... 

Phase plane plots are not the best means of investigating these complex dynamics, for in such cases (which are at least 3-dimensional) the three (or more) dimensional phase planes can be quite complex as shown in Figures 16 and 17 (A-2) for two of the best known attractors, the Lorenz strange attractor [80] and the Rossler strange attractor [82, 83]. Instead stroboscopic maps for forced systems (nonautonomous) and Poincare maps for autonomous systems are better suited for investigating these types of complex dynamic behavior. 

The classical dynamics of a system can also be analyzed on the so-caUed Poincare surface of section (PSS). Hamiltonian flow in the entire phase space then reduces to a Poincare map on a surface of section. One important property of the Poincare map is that it is area-preserving for time-independent systems with two DOFs. In such systems Poincare showed that all dynamical information can be inferred from the properties of trajectories when they cross a PSS. For example, if a classical trajectory is restricted to a simple two-dimensional toms, then the associated Poincare map will generate closed KAM curves, an evident result considering the intersection between the toms and the surface of section. If a Poincare map generates highly erratic points on a surface of section, the trajectory under study should be chaotic. The Poincare map has been a powerful tool for understanding chemical reaction dynamics in few-dimensional systems. 

Further analysis of the system (3.2) requires understanding the dynamics generated by the Poincare map in the interior of Q. For general periodic systems, such an understanding is beyond current knowledge. Fortunately, system (3.2) - in addition to being two-dimensional - has the property that it is competitive. A beautiful theory for such systems has recently been constructed. The next section is devoted to the principal result of this theory. 

For this simple two-dimensional system, the linearized Poincare map degenerates to a 1 X1 matrix, i.e., a number. Exercise 8.7.1 asks you to show explicitly that... 

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. 

Many of the studies mentioned above made use of a method conceived of by Poincare in the late 1800s to project the motion of thousands of isomerizing molecules onto a special map of the dynamics. We will describe how this method is implemented in a practical sense. However, before beginning that discussion, we will try to guide the reader through a thought experiment in which we try to visualize the three-dimensional phase space of a two-dimensional prototypical model for isomerization. 

To visualize some of the effects described in the previous section, Poincare showed that the behavior of two degree-of-freedom nonlinear systems can be profitably studied by mapping the dynamics onto a well-chosen plane. This is because the conservation of energy requires all trajectories to wander on a three-dimensional hypersurface. In his honor, these maps are often referred to as Poincare maps. The plane chosen to map the dynamics onto is referred to as a surface of section. 

The stroboscopic and Poincare maps are different from the phase plane in that they plot the variables on the trajectory at specific chosen and repeated time intervals. For example, for the forced two-dimensional system, these points are taken at every forcing period. For the Poincare map, the interval of strobing is not as obvious as in the case of the forced system and many techniques can be applied. Different planes can be used in order to get a deeper insight into the nature of strange attractors in these cases. A periodic solution (limit cycle) on the phase plane will appear as one point on the stroboscopic (or Poincare) map. When period doubling takes place, period 2... 

In the same section we give the bifurcation diagrams for the codimension two case with a first zero saddle value and a non-zero first separatrix value (the second term of the Dulac sequence) at the bifurcation point. Leontovich s method is based on the construction of a Poincare map, which allows one to consider homoclinic loops on non-orientable two-dimensional surfaces as well, where a small-neighborhood of the separatrix loop may be a Mobius band. Here, we discuss the bifurcation diagrams for both cases. 

We can now describe the behavior of trajectories in a small neighborhood of the periodic trajectory L to which the fixed point O of the Poincare map corresponds. In the two-dimensional case the behavior of trajectories is shown in Fig. 10.2.4, and a higher-dimensional case in Fig. 10.2.5. The invariant strongly stable manifold Wff (the imion of the trajectories which start from the points of Wq on the cross-section) partitions a neighborhood of L into a node and a saddle region. In the node region all trajectories wind towards L... 

As for the original map (10.3.1) the fixed point O is asymptotically stable when Ik < 0 and is a saddle when Ik > 0. In the latter case the stable and unstable manifolds of O are the manifolds and, respectively. In terms of the Poincare map of the system of differential equations, the corresponding periodic trajectory L is stable when Ik < 0, or a saddle when Ik > 0. Note that in the saddle case the two-dimensional unstable manifold W L) is, in a neighborhood of the periodic trajectory, a Mobius band. 

If the mapping (11.6.2) is the Poincare map of an autonomous system of differential equations, then the invariant curve corresponds to a two-dimensional smooth invariant torus (see Fig. 11.6.3). It is stable if L < 0, or it is saddle with a three-dimensional unstable manifold and an (m -h 2)-dimensional stable manifold if L > 0. Recall from Sec. 3.4, that the motion on the torus is determined by the Poincare rotation number if the rotation number v is irrational, then trajectories on the torus are quasiperiodic with a frequency rate u] otherwise, if the rotation number is rational, then there are resonant periodic orbits on a torus. 

We have already established in the last section that when the first Lyapunov value does not vanish, the passage over the stability boundary 9Jl p e) = 0 is accompanied by the appearance of an invariant two-dimensional torus (in the associated Poincare map this corresponds to the appearance of an invariant closed curve). If we are not interested in the behavior of the trajectories on the torus, we can restrict our consideration to the study of one-parameter families transverse to 9Jl, In this case Theorem 11.4 in Sec. 11.6, gives a complete description of the bifurcation structure. In order to examine the... 

The closed invariant curve Wq for the Poincare map on the cross-section is the loci of intersection of an invariant two-dimensional torus W with the cross-section. The torus is smooth if the invariant curve is smooth, and it is non-smooth otherwise. If the original non-autonomous system does not have a global cross-section, then other configurations of W are also possible, as... 

We will define below the quantity A in terms of the Poincare map. It is an analogue of the separatrix value A introduced in Secs. 13.1 and 13.2 for the two-dimensional case. Recall that A is always non-zero in dimension two. However, in the multi-dimensional case the non-vanishing of. 4 is an essential assumption. 

For q>3 the two Poincare splitting obstructions along Ycx for a map f M----->X from an n-dimensional geometric Poincare comple... 

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