Saddle-focus equilibrium state

It follows from the above theorem that a rough system on the plane may possess only rough equilibrium states (nodes, foci and saddles) and rough limit cycles. As for separatrices of saddles, they either tend asymptotically to a node, a focus, or a limit cycle in forward or backward time, or leave the region G after a finite interval of time. 

If the first non-zero Lyapunov value is positive and if all non-critical characteristic exponents (71,...,7n) lie to the left of the imaginary axis in the complex plane, then the equilibrium state is a complex saddle-focus, as shown in Fig. 9.3.2(b). Its stable manifold is and the unstable manifold coincides with the center manifold W, The trajectories lying neither in nor pass nearby the equilibrium state. 

Let us examine next the bifurcations of the system (11.5.1) in the multidimensional case. If Li < 0 (Fig. 11.5.4), then when // < 0, the equilibrium state O is stable (rough focus when p < 0, and a weak focus aX p = 0) and it attracts all trajectories in a small neighborhood of the origin. When > 0 the point O becomes a saddle-focus with a two-dimensional unstable manifold and an m-dimensional stable manifold. The edge of the unstable manifold is the stable periodic orbit which now attracts all trajectories, except those in the stable manifold of O. One multiplier of the periodic orbit was calculated in Theorem 11.1, this is po p) = 1 — 47r /a (0) -h o p). To find the others we... 

If Li > 0, the phase portraits are depicted in Fig. 11.5.5. Here, when // < 0, there exists a stable equilibrium state O (a focus) and a saddle periodic orbit whose m-dimensional stable manifold is the boundary of the attraction basin of O. As /i increases, the cycle shrinks towards to O and collapses into it at /i = 0. The equilibrium state O becomes a saddle-focus as soon as p increases through zero. 

We should, however, stress that such a reduction to the two-dimensional case is not always possible. In particular, it cannot be performed when the equilibrium state is a saddle-focus. Moreover, under certain conditions, we run into an important new phenomenon when infinitely many saddle periodic orbits coexist in a neighborhood of a homoclinic loop to a saddle-focus. Hence, the problem of finding the stability boundaries of periodic orbits in multidimensional systems requires a complete and incisive analysis of all cases of homoclinic loops of codimension one, both with simple and complex dynamics. This problem was solved by L. Shilnikov in the sixties. 

As we have seen above, the dynamics near the homoclinic loop to a saddle with real leading eigenvalues is essentially two-dimensional. New phenomena appear when we consider the case of a saddle-focus. Namely, we take a C -smooth (r > 2) system with an equilibrium state O of the saddle-focus saddle-focus (2,1) type (in the notation we introduced in Sec. 2.7). In other words, we assume that the equilibrium state has only one positive characteristic exponent 7 > 0, whereas the other characteristic exponents Ai, A2,..., are with negative real parts. Besides, we also assume that the leading (nearest to the imaginary axis) stable exponents consist of a complex conjugate pair Ai and A2 ... 

To distinguish between these two cases, we will call the equilibrium state a saddle in the first case, and a saddle-focus in the second case, for the sake of brevity. Note that this terminology differs from what we have used throughout the first part of this book. Namely, in this section we do not take into account whether the leading characteristic exponent A is real or complex. Thus, in this particular section, we call O a saddle if (13.5.1) and (13.5.2) are satisfied, even if Ai is complex. 

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 is similar to Case 1, but with Li(0) >0. As e —> —0, a saddle periodic trajectory shrinks into a stable point O. Upon moving through e = 0, the equilibrium state becomes a saddle-focus it spawns a two-dimensional unstable invariant manifold (i.e. the boundary Ss is dangerous). 

When the equilibrium state is topologically saddle, condition (C.2.8) distinguishes between the cases of a simple saddle and a saddle-focus. However, when the equilibrium is stable or completely unstable, the presence of complex characteristic roots does not necessarily imply that it is a focus. Indeed, if the nearest to the imaginary axis (i.e. the leading) characteristic root is real, the stable (or completely imstable) equilibrium state is a node independently of what other characteristic roots are. 

In the (a, 6)-parameter plane, find the transition boundary saddle-focus for the origin, and equations for its linear stable and unstable subspaces. Detect the curves in the parameter plane that correspond to the vanishing of the saddle value a of the equilibrium state at the origin. Find where the divergence of the vector field at the saddle-focus vanishes. Plot the curves found in the (a 6)-plane. ... 

Fig. C.6.12. Plot of the x-coordinate of the equilibrium state versus z at e = 0. The symbols Xmim a max and (x) denote, respectively, the maximal, minimal and averaged values of the x-coordinates of the stable limit cycle which bifurcates from a stable focus at AH and terminates in the separatrix loop to the saddle O (see the next figure) at the point H z cz 2.086.

Shilnikov, L. P. [1970] A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik 10, 91-102. 

Figure 11.3 State space for different initial conditions. The equilibrium point Pi ( o ) is a stable focus, P2 ( ) is an unstable saddle point, and P3 ( ) is stable.

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