First separatrix value

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. 

Let us now consider the case of codimension two in more detail. Recall that this case is distinguished by two conditions the first is the existence of a separatrix loop, and the second condition is the vanishing of the first saddle value t7o while the first separatrix value s is non-zero. The latter is equivalent to i4 1. We will assume that A <1 because the case A> 1 follows directly by a reversion of time. 

Two-dimensional systems having a separatrix loop to a saddle with non-zero first saddle value ao form a bifurcation set of codimension one. Therefore, we can study such homoclinic bifurcations using one-parameter families. 

In the case of a saddle (the leading characteristic exponent Ai is real), the bifurcation diagram depends on the signs of the separatrix values Ai and A2, as well as on the way the homoclinic loops F1 and F2 enter the saddle at t = -f 00. Let us consider first the case where F1 and F2 enter the saddle tangentially to each other, i.e. bifurcations of the stable homoclinic butterfly. 

Numerical integration of equations (2) and (3) with initial values for X,Y on the limit cycle and with one of the rate constants oscillating as in equation (4) or (5) may result in a transition of the X,Y trajectory across the separatrix towards the stationary state. The occurrence of a transition is dependent on the parameters g, u) and 0. For extremely small amplitude perturbations (g - -0), the trajectory continues to oscillate close to the limit cycle. As g is increased, however, transitions may occur. The time taken for a transition is then primarily a function of the frequency of the perturbation. The time from the onset of the oscillating perturbation to the time at which the trajectory attains the lower steady state (At) is plotted in Figure 3 as a function of with all other parameters held constant. The arrow marks the minimum value for At which occurs when the frequency of the external perturbation exactly equals that of the unperturbed limit cycle itself. The second minimum occurs at the first harmonic of the limit cycle. Qualitatively similar results are obtained when numerical integration is carried out with differing values for g and 0. 

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. 

If the bottom product point lies on the edge 2-4 (Fig. 5.28b), the point 5, as the value of V/L parameter is increased, first goes along the reversible distillation trajectory within the face 2-3-4 until it meets the an-line (x = ), then along the reversible distillation trajectory within the an-surface up to azeotrope 13 (at V/L = 1). Simultaneously, in the face 2-3-4, in the trajectory of reversible distillation, after point x / a stable node arises, and the point engenders a separatrix 5 - that divides the whole separatrix bundle Reg into two separate trajectory bundles. j... 

Nonlinearity of separatrix trajectory bundles is taken into consideration only at the third stage, if it is necessary to determine precisely the value of (L/ y) "". Usually to solve practical tasks, it is sufficient to confine oneself to the first two stages of the algorithm. 

In Chap. 13 we will consider the bifurcations of a homoclinic loop to a saddle equilibrium state. We start with the two-dimensional case. First of all, we investigate the question of the stability of the separatrix loop in the generic case (non-zero saddle value), as well as in the case of a zero saddle value. Next, we elaborate on the cases of arbitrarily finite codimensions where the so-called Dulac sequence is constructed, which allows one to determine the stability of the loop via the sign of the first non-zero term in this sequence. 

The case of zero saddle value was considered by E. A. Leontovich in 1951. Her main result is presented in Sec. 13.3, rephrased in somewhat different terms in the case of codimension n (i.e. when exactly the first (n — 1) terms in the Dulac sequence are zero) not more than n limit cycles can bifurcate from a separatrix loop on the plane moreover, this estimate is sharp. 

The question of the bifurcations of a separatrix loop to a saddle with zero saddle value (Tq was first considered by E. Leontovich. She had proven the following theorem ... 

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. 

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