Complex saddle-focus

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. 

Formula (10.4.20) is similar to the formula (10.4.14) for the non-resonant case and the only difference is that in. the case of a weak resonance only a finite number of the Lyapunov values Li,..., Lp is defined (for example, only L is defined when N = b). If at least one of these Lyapunov values is non-zero, then Theorem 10.3 holds i.e. depending on the sign of the first non-zero Lyapunov value the fixed point is either a stable complex focus or an unstable complex focus (a complex saddle-focus in the multi-dimensional case). 

The existence of complex dynamics near a homoclinic loop to a saddle-focus was discovered by L. Shilnikov for the three-dimensional case in [131]. Subsequently, the four-dimensional case was considered in [132] and the general case in [136]. 

In Sec. 13.5 we consider the bifurcation of the homoclinic loop of a saddle without any restrictions on the dimensions of its stable and imstable manifolds. We prove a theorem which gives the conditions for the birth of a single periodic orbit from the loop [134], and also formulate (without proof) a theorem on complex dynamics in a neighborhood of a homoclinic loop to a saddle-focus. Here, we show how the non-local center manifold theorem (Chap. 6 of Part I) can be used for simple saddles to reduce our analysis to known results (Theorem 13.6). 

Here, the saddle-focus has two pairs of complex-conjugate characteristic exponents and the divergence of the vector field is non-vanishing at the saddle-focus. 

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. 

We also show in Secs. 13.4 and 13.5 (the latter deals with the case where the dimension of the unstable manifold of the saddle is greater than one) that in other cases either a saddle periodic orbit is born from the loop, or a system exhibits complex dynamics (the case of a saddle-focus). 

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 ... 

This theorem is a part of a more general assertion [including also the case of a multi-dimensional unstable manifold as well as saddle-foci of types (2,1) and (2, 2)] on complex dynamics near the homoclinic loop of a saddle-focus [136]. Condition p < 1 also known as the Shilnikov condition is very important here, because the structure of the phase space near the homoclinic loop is drastically changed in comparison to the case p > 1 covered by Theorem 13.6. The main bifurcations in the boundary case p = 1, when a small perturbation trigging the system into a homoclinic explosion from simple dynamics (p > 1) to complex dynamics (p < 1) were first considered in [29]. 

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. 

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 case of a saddle-focus of a three-dimensional system the condition a = 0 reads as Ai + ReA2 = 0 where Ai is a real root and A2,a are the pair of complex-conjugate roots. This can be written as... 

C.7. 86.1 Assume there is a homoclinic loop to a saddle-focus in the Shimizu-Morioka model (like a T-point). Without computing the characteristic exponent of the saddle-focus, what can we say about the local structure is it trivial (one periodic orbit), or complex (infinitely many periodic orbits) ... 

Instead of nodal lines in closed systems we are interested in the statistics of NPs for open chaotic billiards since they form vortex centers and thereby shape the entire flow pattern (K.-F. Berggren et.al., 1999). Thus we will focus on nodal points and their spatial distributions and try to characterize chaos in terms of such distributions. The question we wish to ask is simply if one can find a distinct difference between the distributions for nominally regular and irregular billiards. The answer to this question is clearly positive as it is seen from fig. 3. As shown qualitatively NPs and saddles are both spaced less regularly in chaotic billiard in comparison to the integrable billiard. The mean density of NPs for a complex RGF (9) equals to k2/A-k (M.V. Berry et.al., 1986). This formula is satisfied with good accuracy in both chaotic and integrable billiards. 

Fig. 9. Linear stability diagram illustrating (21). The bifurcation parameter B is plotted against the wave number n. (a) and (b) are regions of complex eigenvalues < > (b) shows region corresponding to an unstable focus (c) region corresponding to a saddle point. The vertical lines indicate the allowed discrete values of n. A = 2, D, = 8 103 D2 = 1.6 10-3.

In conventional transition-state theory, we place the dividing surface between reactants and products at the saddle point, perpendicular to the minimum-energy path, and focus our attention on the activated complex. That is, we write the reaction... 

The third assumption is that the energy states of the reactants as well as the (shortlived) activated complex are populated according to the Boltzmann distribution. Now, we focus on activated complexes where the reaction coordinate q is fixed at the saddle-point value and where the associated momentum ispj, i.e., with a position and momentum in the range (q, q - -dq ) and (p, p +dp ). The probability of finding such... 

In fact, the unstable UPS branch represents the saddle point (or separatrix) that separates the regions of attraction of the NUP state (or, below p2, the UPl state) from that of the largely-reoriented UP2 state. At this point, it might also be interesting to note that the UPl state represents a stable node at p pi (the relevant stabihty exponents are real and negative). Then, between pi and p2, it changes to a focus (the stability exponents become complex). At p2, the real part of the complex pair of stability exponents passes through zero and then becomes positive. 

The stochastic aspect of a complex bifurcation arising in a two variables chemical system is studied. The dynamics reduces, in a suitable region of the phase space, to a normal form for which both roots of the characteristic equation vanish simultaneously. In conditions close to this degenerate situation, the normal form can be viewed as a perturbation of an exactly soluble hamiltonian system, of hamiltonian h, which exhibits a homoclinic trajectory, h = 0. BAESENS and IMICOLIS [l ] have shown that the phase portrait of the dissipative sytem displays two steady states that coalesce, a focus F and a saddle S. [ Moreover, as one moves in the parameter space, a limit cycle surrounding F, bifurcates from a homoclinic trajectory and then disappears by Hopf bifurcation. ... 

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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