Birth of periodic orbit

Birth of periodic orbits from a homoclinic loop (the case dim W =l)... 

At the birth or death of periodic orbits, the amplitude of oscillations is zero but the time period is finite... 

A straightforward generalization of two-dimensional bifurcations was developed soon after. So were some natural modifications such as, for instance, the bifurcation of a two-dimensional invariant torus from a periodic orbit. Also it became evident that the bifurcation of a homoclinic loop in high-dimensional space does not always lead to the birth of only a periodic orbit. A question which remained open for a long time was could there be other codimension-one bifurcations of periodic orbits Only one new bifurcation has so far been discovered recently in connection with the so-called blue-sky catastrophe as found in [152]. All these high-dimensional bifurcations are presented in detail in Part II of this book. 

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. 

The idea of revolutionary progress in certain periods as developed by Kuhn [9] and very recently by McAllister [10], has some bearing on what I will discuss. It will be argued that theoretical organic chemistry has known three periods of dramatic change. The first of these periods (1850-1875) witnessed the birth of the structural formula and its development from formal representation to a reflection of physical reality. The second (1910-1935) saw the advent of quantum mechanics and the concepts of the electron pair, resonance and mesomerism, and hybridisation. In the third one (1955-1980), already mentioned, it is perhaps the succesful application of molecular orbital theory to chemical reactions, made possible by a very fruitful interplay of calculations and concepts, which is most significant. 

In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

A totally diflFerent situation becomes possible in the case where the system does not have a global cross-section, and when is not a manifold. In this case (Sec. 12.4), the disappearance of the saddle-node periodic orbit may, under some additional conditions, give birth to another (unique and stable) periodic orbit. When this periodic orbit approaches the stability boundary, both its length and period increases to infinity. This phenomenon is called a hlue-sky catastrophe. Since no physical model is presently known for which this bifurcation occurs, we illustrate it by a number of natural examples. 

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]. 

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). 

An analogous situation also appears in a classical problem on the birth of an invariant torus from a periodic orbit minor details of the structure of the... 

In this section we discuss what happens when a pair of complex-conjugate characteristic exponents of an equilibrium state crosses over the imaginary axis. The loss of stability here is directly connected to the birth, or vice versa, the disappearance of a periodic orbit. This bifurcation is the simplest mechanism for transition from a stationary regime to oscillations, and it allows one to give a proper interpretation of numerous physical phenomena. For this reason this bifurcation has traditionally played a special role in the theory of bifurcations. 

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. 

Fig. 13.4.1. Birth of a stable periodic orbit from a separatrix loop to the saddle with (Tq < 0.

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. 

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]). 

Of special consideration are systems with symmetry where both separatrix loops approach together the saddle point. Such a situation is rather trivial namely when the loops split inwards, each gives the birth to a single stable limit cycle, in view of Theorem 13.4.1. When the loops split outwards, the stability migrates to a large-amplitude symmetric stable periodic orbit that bifurcates from the homoclinic-8 as shown in Fig. 13.7.2. And that is it. This is the reason why the theory below focuses primarily on non-symmetric systems. 

Fig. 13.7.2. The bifurcations of the homoclinic-8 in the symmetric case. An outward breakdown of both homoclinic loops gives birth to a large symmetric periodic orbit. When the loops split inwards, a periodic orbit bifurcates from each of the loops.

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