Unstable separatrices

Theorem 10.6. Let ReC i < 1. Then for any small e 7 0 the point 0 w = 0) is stable. Moreover the map T possesses a saddle periodic trajectory 01,02,03) of period three which lies apart from the point O over a distance 0 e). One of the two unstable separatrices of each point Oi tends to O, the other unstable separatrix leaves a neighborhood of the origin. The stable separatrices of the periodic trajectory form a boundary of the basin of attraction of the point O see Fig. 10.6.1). 

Point O is stable the function (2i — SR sin 3(f) is a Lyapunov function for small R. Clearly, the stable separatrices of the saddle points tend to infinity as t —> — 00. Otherwise, they had to tend to a completely unstable periodic trajectory or a completely unstable equilibrium state but there is none. Another possibility is that a stable separatrices of one saddle might coincide with the unstable one of the other saddle thereby forming a separatrix cycle as that shown in Fig. 10.6.2, but with four saddles however this hypothesis contradicts to the negative divergence condition. The unstable separatrices cannot tend to infinity as t -> +cx). To prove this, check that when R is large, V <0 for the function V in (10.6.12). Therefore, all trajectories of the system, as t increases, must get inside some closed curve V == C with C sufficiently large, where they remain forever. The same behavior applies to the separatrices of the saddle. Thus, the only option for the unstable separatrices of the point Oi is that one tends to O and the other to 0 as shown in Fig. 10.6.6. 

As in the previous case, the stable separatrices of the saddle points diverge to infinity as t —00. To show that the unstable separatrices of the points 0 behave as they are depicted in Fig. 10.6.7 one tends to infinity as t 4-00 while the second tends to the origin, we must show that all together they do not go to infinity simultaneously. To do this let us consider system (10.6.11) for large R, Introduce a new variable z — R and make the transformation of time dt —> zdt. We obtain ... 

Let us introduce coordinates (x y) near the saddle O. Without loss of generality we may assume that the saddle resides at the origin for all p. We may also assume that the unstable separatrices are tangent at O to the y-axis and the stable separatrices are tangent to the a -axis. Thus, the system can be written in the following form... 

The unstable manifold of O is one-dimensional, and the stable manifold is n-dimensional. Suppose that one of the unstable separatrices (Fi) tends to O as i - -oo, thereby forming a homoclinic loop, as shown in Fig. 13.6.1. 

When (7 = 10, 6 = 8/3 and r 13.926, both of the one-dimensional unstable separatrices Fi and F2 of the saddle 0(0,0,0) return to the saddle, along the same direction (the positive z semi-axis). They form a geometrical configuration called a homoclinic butterfly (Fig. 13,6.3). Note that a homoclinic butterfly may only occur in with n > 3. 

The homoclinic-8 consists of a pair of homoclinic loops to a saddle such that the unstable separatrices T i and F2 come out from the saddle in the opposite directions at t = -00 (Fig. 13.7.1). We will consider the case where the unstable manifold of the saddle O is one-dimensional i.e. the saddle has only one positive characteristic exponent 7 > 0 all other exponents are assumed to have negative real parts Re < 0, j = 1,..., n. Moreover, we assume that the saddle value a is negative ... 

The Cherry flow is a flow on a two-dimensional torus with two equilibrium states a saddle and an unstable node both unstable separatrices are stable one stable separatrix is a-limit to a node and the other lies in the closure of the unstable separatrices and it is P -stable [see Fig. 13.7.4(a)]. The closure of the unstable separatrices is a quasiminimal set which contains the saddle O and a continuum of unclosed P-stable trajectories. The rotation number for such flows is defined in the same way as for flows on a torus without equilibrium states. Since there is no periodic orbits in a Cherry flow. 

The first example illustrates one of the most typical bifurcations which occur in dissipative systems namely a stable periodic orbit L adheres to the homoclinic loop of a saddle. Denote the unstable separatrices of the saddle by Fi and F2. Let Fi form a homoclinic loop at the bifurcation point. Denote the limit set of the second separatrix by D(F2). In the general case fI(F2) is an attractor for instance, a stable equilibrium state, a stable periodic trajectory, or a stable torus, etc. Since inunediately after bifurcation a representative point will follow closely along F2, it seems likely that fl(F2) will become its new attractor. 

Along stable separatrix surfaces, integral trajectories tend to the critical point (with increasing time), while along unstable separatrix surfaces they move away from the critical point (with increasing time). Condition 2 consists in the fact that A-i- = A., that is, the stable separatrix surface A+ of the point (x+, +) coincides with the unstable separatrix surface A- of the point (x, ). FVom this it follows, in particular, that ffo( +> +) = o( - -) ... 

The inequality K, e) < K )( i ) independence of the integrab /i, /2, fz on the common level surface M123 imply that > 0. Stable and unstable separatrix surfaces of the two indicated periodic (in t) solutions can be set as intersections of a three-dimensional manifold M123 by hyperplanes of the form... 

D) if there is a semi-stable (double) limit cycle, the system may not have simultaneously an unstable separatrix of a saddle which tends to the cycle as t -> -hoo and a stable separatrix of a saddle which tends to the cycle as t —00, as shown in Fig. 8.1.4 and... 

Fig. 12.1.5. Behavior of the unstable separatrix before (e < 0) and after (e > 0) on-edge bifurcation.

Let us now consider the case of a heteroclinic cycle with two saddles 0 and O2, Let the unstable manifolds of both saddles be one-dimensional and let an unstable separatrix Fi of Oi tend to O2 as t —> H-oo and an unstable... 

C12 and C21 (fc = 1,. ) such that at /i the unstable separatrix Fi of Oi i = 12 ) makes k complete rotations along U and enters the saddle Qj i) thereby forming a heteroclinic connection. The curves are defined by the equations pj = hkij p>i) where hkij is some smooth function defined on an open subset of the positive /ii-axis such that the first derivative of hkij tends uniformly (with respect to k) to zero as /ii 0. The exact structure of the bifurcation set corresponding to heteroclinic connections is quite different depending on whether the equilibria Oi are saddles or saddle-foci. 

The heteroclinic cycles including the saddles whose unstable manifolds have different dimensions were first studied in [34, 35]. This study mostly focused on systems with complex dynamics. Let us, however, discuss here a case where the dynamics is simple. Let a three-dimensional infinitely smooth system have two equilibrium states 0 and O2 with real characteristic exponents, respectively, 7 > 0 > Ai > A2 and 772 > 1 > 0 > (i.e. the unstable manifold of 0 is onedimensional and the unstable manifold of O2 is two-dimensional). Suppose that the two-dimensional manifolds (Oi) and W 02) have a transverse intersection along a heteroclinic trajectory To (which lies neither in the corresponding strongly stable manifold, nor in the strongly unstable manifold). Suppose also that the one-dimensional unstable separatrix of Oi coincides with the one-dimensional stable separatrix of ( 2j so that a structurally unstable heteroclinic orbit F exists (Fig. 13.7.24). The additional non-degeneracy assumptions here are that the saddle values are non-zero and that the extended unstable manifold of Oi is transverse to the extended stable manifold of O2 at the points of the structurally unstable heteroclinic orbit F. 

This bifurcation has codimension two the governing parameters (/ii,/X2) are chosen here to be the coordinates of the point of intersection of the onedimensional unstable separatrix of 0 with some cross-section transverse to the one-dimensional stable separatrix of the other saddle O2. Since the... 

It may also seem sensible, if there are multiple solutions, to ask which of the states is the most stable In fact, however, this is not a valid question, partly because we have only been asking about very small disturbances. Each of the two stable states has a domain of attraction . If we start with a particular initial concentration of A the system will move to one or other. Some initial conditions go to the low extent of reaction state (generally those for which 1 — a is low initially), the remainder go to the upper stationary state. The shading in Fig. 6.9 shows which initial states go to which final stationary state. It is clear from the figure that the middle branch of (unstable) solutions plays the role of a boundary between the two stable states, and so is sometimes known as a separatrix (in one-dimensional systems only, though). 

Nitromethane shows the simplest residue curve map with one unstable curved separatrix dividing the triangle in two basic distillation regions. Methanol and acetonitrile give rise two binary azeotropic mixtures and three distillation regions that are bounded by two unstable curved separatrices. Water shows the most complicated residue curve maps, due to the presence of a ternary azeotrope and a miscibility gap with both the n-hexane and the ethyl acetate component. In all four cases, the heteroazeotrope (binary or ternary) has the lowest boiling temperature of the system. As it can be seen in Table 3, all entrainers except water provide the n-hexane-rich phase Zw as distillate product with a purity better than 0.91. Water is not a desirable entrainer because of the existence of ternary azeotrope whose n-hexane-rich phase has a water purity much lower (0.70). Considering in Table 3 the split... 

We have already mentioned the sharp difference in the relaxation times outside the region between the null dines ( 1 s) and inside it (as high as hundreds of seconds). The motion outside this region depends on the "fastest reaction. Inside this region the relaxation rate is dependent on the complicated complex of rate constants, and in the general case we cannot suggest that the reaction rate is limited by some reaction. The common trajectory near which the relaxation is retarded is no more than a specific trajectory that is a separatrix going from the unstable into the stable steady... 

The phase plane plot of Figure 2 illustrates the behavior of the concentrations of X and Y within this region. Initial concentrations of X and Y corresponding to a point above the broken line will evolve in time to the limit cycle. The broken line represents the separatrix of the middle unstable steady state which has the stability characteristics of a saddle point. Initial values for X and Y corresponding to a point below the separatrix will evolve to the stable state to the right of the diagram. 

On the basis of the information obtained within the framework of such an elementary analysis, one can build a phase portrait of the studied system. As it may be seen in Fig. 10, the tetrahedron has three stable SPs located at the apexes of its base. The basins of SPs are separated by the proper separatrix surfaces. The first of these passes through SPs of simplexes (14), (4), (23), (123), and the second one through SPs of simplexes (14), (13), (123), (12). Each basin consists of a single cell with the unstable SP which is common for all basins and located on the edge (14). 

The basic picture discussed above is quite general in Hamiltonian systems. Of particular importance is the concept of stable and unstable manifolds associated with unstable periodic orbits. Trajectories along the stable manifold will be mapped toward the periodic orbit, whereas trajectories along the unstable manifold will be mapped away from the periodic orbit. It turns out that the union of segments of the stable and unstable manifolds is very useful in defining the reaction separatrix and calculating the flux crossing the separatrix in few-dimensional systems. 

Figure 14. An Hel2 surface of section for an unstable trajectory which forms a collision complex. The total energy is —2661.6 cm . Also shown are the reaction separatrix and the intramolecular bottleneck, (a) Graph showing the full dynamics of the trajectory. (b)-(f) Graphs illustrating the trajectory over five consecutive time ranges. These graphs are arranged to demonstrate the manner in which the trajectory moves with respect to the bottleneck and the separatrix. [From M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986).]...

Figure 21. Construction of the exact separatrix on the surface of section for a symmetric double-well model potential, (a) The unstable manifold (b) Superposition of the stable (dashed) and unstable (solid) manifolds, (c) The exact separatrix, which is a union of portions of the above manifolds, (d) Turnstiles superimposed on the separatrix. [From S. K. Gray and S. A. Rice, J. Chem. Phys. 86, 2020 (1987).]...

Since the manifold Mq is a NHIM, it changes continuously, under a small perturbation, into a new NHIM M - Moreover, the separatrix Wq changes, continuously and locally near M, into the stable manifold and the unstable one W" of the NHIM M. Note, however, that, in general, and W no longer coincide with each other to form a single manifold globally. Then, the Lie transformation method brings the total Hamiltonian H x.I, 0) into the Fenichel normal form locally near the manifold M. ... 

Equation (64) shows that the distance d x,a) exhibits an oscillatory dependence as a function of x. In other words, d x, a) changes between plus and minus values as initial conditions shift on the separatrix. This means that the stable and unstable manifolds have transverse intersections. See Fig. 14 showing how the oscillatory change of the integral implies the occurrence of transverse intersections. The existence of transverse intersections between stable and unstable manifolds leads to horseshoe dynamics—that is, chaos. Thus, the Melnikov integral given by Eq. (64) indicates that this system exhibits chaotic behavior. 

The degree of freedom (q,p) has a resonant term Vcosq. There exist unstable fixed points ( = 7i,p = 0), and the separatrix orbits connecting them. The separatrix orbits of the nonlinear resonance are given by the following ... 

First, we consider those orbits starting from the initial conditions with (/ = 0, 9 = 0o) and (q,p) on the separatrix Eq. (76) near the unstable fixed point ( = —7i,p = 0). In other words, we will see how a piece of the unstable manifold with a different initial value of 0 intersects with the stable manifold of the NHIM with q = n,p = 0). Here, the dimension of the pieces of the unstable manifold is 1, since we fix the initial conditions of (/, 0), and the dimension of the stable manifold is 3. 

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