Saddle value

If f(x) and g(x) are concave functions, then a solution to the above maximum problem is equivalent to the solution of the saddle value problem that requires finding x° and A°, which satisfy... 

Exercise Prove that if/( i, 2) Is differentiable in x and x2, and has a saddle point36 at x°,x ), i.e., satisfies a condition similar to the saddle value problem, then... 

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. 

In the case of a non-zero saddle value, we present the classical result by Andronov and Leontovich on the birth of a unique limit cycle at the bifurcation of the separatrix loop. Our proof differs from the original proof in [9] where Andronov and Leontovich essentially used the topology of the plane. However, following Andronov and Leontovich we present our proof under a minimal smoothness requirement (C ). 

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. 

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. 

The bifurcations of periodic orbits from a homoclinic loop of a multidimensional saddle equilibrium state are considered in Sec. 13.4. First, the conditions for the birth of a stable periodic orbit are found. These conditions stipulate that the unstable manifold of the equilibrium state must be one-dimensional and the saddle value must be negative. In fact, the precise theorem (Theorem 13.6) is a direct generalization of the Andronov-Leontovich theorem to the multi-dimensional case. We emphasize again that in comparison with the original proof due to Shilnikov [130], our proof here requires only the -smoothness of the vector field. 

We end this section with a consideration of the homoclinic loop to a saddle-focus whose unstable manifold is one-dimensional. It is shown that when the saddle value is positive, infinitely many saddle periodic orbits coexist near such a homoclinic loop of the saddle-focus (Theorem 13.8). 

An analogous situation occurs when the system has a separatrix loop to a non-resonant saddle (i.e. its saddle value cr = Ai + A2 0) which is the a -limit of a separatrix of another saddle Oi (see condition (E) and Fig. 8.1.5). In this case, the bifurcation surface is also unattainable from one side, where close nonrough systems may have a heteroclinic connection, as shown in Fig. 8.1.6(b). 

A more complicated example is given by codimension-two problems as those shown in Fig. 8.4.1, which include one or two saddle-foci. The saddle values are assumed to be negative at all equilibrium states. The parameters which govern the bifurcations are introduced in the following way let //i(2) be a deviation of ri2(r2i) from Wq (W02) saddle-foci, or let it... 

We prove the main theorem on the birth of a stable periodic orbit from a homoclinic loop to a saddle with the negative saddle value in Sec. 13.4. 

The problem of stability of a separatrix loop on a plane is easily solved when the so-called saddle value... 

The above coefficients tJi,..., are called saddle values. Although they are not all invariant with respect to smooth coordinate transformations, however the number of the first non-zero saddle value is defined by the system uniquely (because it is the coefficient of the first non-zero resonant term), and is therefore invariant. 

It also follows that when all saddle values vanish (i.e. in the infinitely degenerate case) the system is locally reduced to the linear form. 

As before, we assume that the homoclinic loop F adjoins the saddle from the side of positive x and y. Note that this assumption fixes the direction of the X and y axes and, therefore, it holds the sign of the nth saddle value an fixed. Let us choose a small d > 0 and define the cross-sections So and S by x = d and y = d, respectively. 

Bifurcation of a limit cycle from a scparatrix loop of a saddle with non-zero saddle value... 

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. 

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 ... 

In her proof of the above theorem, Leontovich had assumed C -smoothness for the system, where r > 4n + 6. First of all, she proved that when the first saddle value is close to zero, a system near the saddle can be transformed into... 

On the next step of the proof Leontovich had evaluated the local map. She considered, in fact, the map from the cross-section Si y = d to 5o x = d, i.e. the inverse of the local map To in our notations. Note that by assumption of the theorem only the last saddle value an is bounded away from zero, whereas ai,.., an-i are small. Therefore, by rescaling time variable the system may... 

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. 

Fig. 13.3.1. Bifurcation diagram for the homoclinic loop to a saddle with zero saddle value (A + 7 = 0) in the orientable case (0 < A < 1). Parameter p governs the splitting of the loop the sign of e is opposite to the sign of the saddle value.

Since < 0, the saddle value is positive, and hence the periodic orbit bifurcating from the separatrix loop is unstable here. 

L2 corresponds to a double separatrix loop on the Mobius band. The saddle value (Tq is positive on this curve ... 

Ls (the negative -semi-axis) corresponds to a simple separatrix loop with a positive saddle value ao] and... 

Let us consider first the case of a negative saddle value a... 

Theorem 13.6. (Shilnikov [130]) When the saddle value a is negative at the saddle a single stable periodic orbit L is bom from the homoclinic loop for p> 0. The separatrix Fi tends to L as t +oo. For there are no peri-... 

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