Saddle periodic orbit

A more vivid characteristics of systems with complex behaviors is the presence of a Poincare homoclinic trajectory, i.e. a trajectory which is biasymptotic to a saddle periodic orbit as t —> oo. The existence of a homoclinic orbit which lies at the transverse intersection of the stable and unstable... 

The spiral-like shape of this attractor follows from the shape of homoclinic loops to a saddle-focus (2, 1) which appear to form its skeleton. Its wildness is due to the simultaneous existence of saddle periodic orbits of different topological type and both rough and non-rough Poincare homoclinic orbits. 

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

A similar effect occurs when a saddle-saddle periodic orbit (with one multiplier equal to 1 and the rest of the multipliers both inside and outside of the unit circle) disappears. If the stable and unstable manifolds of the saddle-saddle periodic orbits intersect across two (at least) smooth tori, then the disappearance of such a periodic orbit is followed by the birth of a limit set in which an infinite set of smooth saddle invariant tori is dense [6]. 

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). 

Fig 7.5.1. Saddle periodic orbit in R are distinguished by the topology of the stable and unstable invariant manifolds which may be homeomorphic to a cylinder (left) or a Mobius band (right). 

For example, the following theorem shows that a Morse-Smale system cannot have a homoclinic trajectory to a saddle periodic orbit. 

Theorem 7.11. Let L be a saddle periodic orbits and let P be its homoclinic trajectory along which Wf and intersect transversely. Then, any small neighborhood of L JT contains infinitely many saddle periodic orbits. 

An analogous picture takes place in the case of three-dimensional flows possessing the chain Qi < Q < Q2 where Q denotes a saddle periodic orbit, and Qi and Q2 stand for either saddle equilibrium states or periodic orbits. 

The value 9 is also a modulus of topological equivalence in the case of a three-dimensional fiow which has two saddle periodic orbits such that an unstable manifold of one periodic orbit has a quadratic tangency with a stable manifold of another orbit along a heteroclinic trajectory. 

There are some other occurrences of moduli in structurally unstable three-dimensional systems of codimension-one with simple dynamics. For example, consider a three-dimensional system with a saddle-focus O and a saddle periodic orbit L. Let i 2 = p iu), and A3 be the characteristic roots at O such that /o < 0, cj > 0, A3 > 0, i.e. assume the saddle-focus has type (2,1) let i/ < 1 and I7I > 1 be the multipliers of the orbit L. Let one of the two sepa-ratrices P of O tend to L as t -> +00, i.e. T W[, as shown in Fig. 8.3.2. This condition gives the simplest structural instability. All nearby systems with similar trajectory behavior form a surface B of codimension-one. Belogui [28] had found that the value... 

Fig. 8.3.2. A structurally unstable heteroclinic trajectory connecting a saddle-focus and a saddle periodic orbit with positive multipliers, i.e. both manifolds of the saddle cycle are homeomorphic to a cylinder.

Fig. 11.3.7. Scenario of a saddle-node bifurcation of periodic orbits in The stable periodic orbit and the saddle periodic orbit in (a) coalesce at /i = 0 in (b) into a saddle-node periodic orbit, and then vanishes in (c). The unstable manifold of the saddle-node orbit in (c) is homeomorphic to a semi-cylinder. A trajectory following the path in (b) slows down along a transverse direction near the virtual saddle-node periodic orbit (i.e., the ghost of the saddle-node orbit in (b)) so that its local segment is similar to a compressed spring.

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. 

Fig 11.5.5. A subcritical Andronov-Hopf bifurcation, (a) An attraction basin of a stable focus is bounded by a stable manifold of a saddle periodic orbit, (b) The periodic orbit narrows to the stable focus at /x = 0, and the latter becomes a saddle-focus (1,2). 

Let us consider a one-parameter family of n-dimensional C -smooth (r > 2) systems having a saddle-node periodic orbit L at /i = 0. We assume that /jL is the governing parameter for local bifurcations. Thus (recall Fig. 11.3.7), for /i < 0, there exist stable and saddle periodic orbits which collapse into one orbit L at /X = 0. The local imstable set is homeomorphic to a half-cylinder... 

The invariant manifold depends continuously on p. At p = 0, it coincides with W, When /x < 0, it is the imion of the mist able manifold of the saddle periodic orbit L p) with the stable periodic orbit L p) (where L p) are the periodic orbits into which the saddle-node bifurcates ). In the case of torus, for p> 0, the Poincare rotation number on Tfj, tends to zero as /x -> +0. Thus, on the /x-axis there are infinitely many (practically indistinguishable as p -hO) resonant zones which correspond to periodic orbits on 7 with rational rotation numbers, as well as an infinite set (typically, a Cantor set) of irrational values of p for which the motion on is quasiperiodic. 

We will study the case m = 0 in Sec. 12.4 in connection with the problem of the blue sky catastrophe . In the case m > 2, infinitely many saddle periodic orbits are born (see Theorem 12.5) when the saddle-node disappears moreover, even hyperbolic attractors may arise here (see [139]). We do not discuss such kind of bifurcations in this book. 

It follows immediately from this theorem that if the essential map (12.2.15) has a rough stable (unstable) periodic orbit at some uj, then there is a sequence of intervals Sk of values of p which acciunulate at +0, such that the difference u) ix) — k) remains close to a at /i <5, and the system has, respectively, a rough stable or saddle periodic orbit at // (Jfc for all sufficiently large k. 

Let us now consider the question concerning what happens when is non-smooth. For the first time, this question was studied in [3] where it was discovered that the possibility of the breakdown of the invariant manifold causes an onset of chaos at such bifurcations. In particular, sufficient conditions (the so-called big lobe and small lobe conditions) were given in [3] for the creation of infinitely many saddle periodic orbits upon the disappearance of a saddle-node in the non-smooth case. Subsequent studies have shown that these conditions may be further refined so we may reformulate them as follows. 

Theorem 12.5. If the big lobe condition is satisfied then the system has infinitely many saddle periodic orbits for all small fi > 0. 

This is a one-dimensional map which may have stable and unstable fixed points (at the zeros of They correspond to stable and saddle periodic orbits... 

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). 

This result gives us the last known principal (codimension one) stability boundary for periodic orbits. We will see below (Theorems 13.9 and 13.10) that the other cases of bifurcations of a homoclinic loop lead either to complex dynamics (infinitely many periodic orbits), or to the birth of a single saddle periodic orbit. 

The situation which we consider here is a particular case of Theorem 13.9 of the next section. It follows from this theorem (applied to the system in the reversed time) that a single saddle periodic orbit L is born from a homoclinic loop it has an m-dimensional stable manifold and a two-dimensional unstable manifold. This result is similar to Theorem 13.6. Note, however, that in the case of a negative saddle value the main result (the birth of a unique stable limit cycle) holds without any additional non-degeneracy requirements (the leading stable eigenvalue Ai is nowhere required to be simple or real). On the contrary, when the saddle value is positive, a violation of the non-degeneracy assumptions (1) and (2) leads to more bifurcations. We will study this problem in Sec. 13.6. 

Fig. 13.4.7. The fixed point M corresponds to a saddle periodic orbit that emerges from the homoclinic loop. Its unstable multiplier is positive when A > 0, and it is negative in the case i4 < 0.

Theorem 13.7. If a homoclinic loop T to a saddle with a positive saddle value satisfies both conditions (1) and (2), then a single saddle periodic orbit L p) is born from the loop for Ap < 0. The unstable manifold of L fi) is two-dimensional and orientable when A > 0 (then there is only one positive... 

Fig. 13.4.9. The birth of a saddle periodic orbit from a homoclinic loop to a saddle with the positive saddle value.

Theorem 13.8. If p < 1, then there exists infinitely many saddle periodic orbits in any neighborhood of the loop F. 

We have shown the existence of infinitely many saddle fixed points of the map T which correspond to saddle periodic orbits (with two-dimensional unstable and m-dimensional stable manifolds) of the system. Those with even fc s have a negative unstable multiplier, and their unstable manifolds are non-orientable. The periodic orbits corresponding to odd fc s have a positive unstable multiplier, and hence, orientable unstable manifolds. 

Theorem 13.10. (Shilnikov [136]) Let a saddle-focus O have a homoclinic loop r which satisfies the non-degeneracy conditions (1) and (2). Then in an arbitrarily small neighborhood of P, there exist infinitely many saddle periodic orbits. 

Case A corresponds to the boundary between positive and negative saddle values. Cases B and C correspond to a violation of the non-degeneracy conditions (1) and (2) of Theorem 13.4.2, respectively (the birth of a saddle periodic orbit from a homoclinic loop with positive saddle value). Condition (3) in the last two cases is necessary to exclude the transition to complex dynamics via these bifurcations (some of the cases with complex dynamics were studied in [44, 70, 78, 96, 79, 71, 72]). 

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. 

Let us describe the essential bifurcations in this system on the path 6 = 2 as fjL increases. On the left of the curve AH, the equilibrium state 0 is stable. It undergoes the super-critical Andronov-Hopf bifurcation on the curve AH. The stable periodic orbit becomes a saddle through the period-doubling bifurcation that occurs on the curve PD. Figure C.6.7 shows the unstable manifold of the saddle periodic orbit homeomorphic to a Mobius band. As a increases further, the saddle periodic orbit becomes the homoclinic loop to the saddle point 0(0,0,0,) at a 5.545. What can one say about the multipliers of the periodic orbit as it gets closer do the loop Can the saddle periodic orbit shown in this figure get pulled apart from the double stable orbit after the fiip bifurcation In other words, in what ways are such paired orbits linked in in R ... 

Fig. C.6.7. Shown is a piece of the stable manifold of the saddle periodic orbit (dark circle) at a 3.2 courtesy of H. Osinga and B. Krauskopf [181].

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