Critical periodic orbit

For periodic orbits that undergo a bifurcation, some Lyapunov exponents may vanish so that the orbit becomes of neutral linear stability in the critical directions [32]. In such cases, the dynamics transverse to the periodic orbit... 

Due to the nonlinearities of the classical Hamiltonian, the periodic orbits undergo bifurcations at critical energies. At these bifurcations, the stability of the orbit changes and extra periodic orbits are created or existing ones annihilated [19]. These bifurcations have dramatic effects on the semiclassical amplitudes of the periodic orbits [49]. In particular, the comparison between the amplitudes of neutrally stable and unstable periodic orbits shows that the amplitude is expected to be globally lowered after a destabilization. 

Theorem 3.2. Suppose that the parameters ai, mi, and T[ are chosen so that (3.2) has a (linearly) asymptotically orbitally stable periodic solution (S(t), x(t)) with period T>0. Fix ai and ti > tj. Then there exist a critical value ml and a branch of periodic orbits of (3.1), with positive Xi component, bifurcating from the hypothesized orbit for m2 near ml-... 

The stable and unstable sets correspond to the stable and unstable manifolds introduced for rest points and periodic orbits in Chapter 1. Unfortunately, if the attractors are more complex than rest points or periodic orbits, the question of the existence of stable and unstable manifolds becomes a difficult topological problem. In the applications that follow, these more complicated attractors do not appear, so one can simply deal with the stable manifold theorem. The Butler-McGehee lemma (used in Chapter 1) played a critical role in the first uses of persistence. The following lemma is a generalization of this work. It can be found (with slightly different hypotheses) in [BW], [DRS], and [HaW]. (In particular, the local compactness is not needed if a stronger condition - asymptotic smoothness - is placed on the semidynamical system.)... 

Let us assume that a periodic orbit is stable, which implies that the eigenvalues A3, A4 are on the unit circle (Figure 4) and we assume that they are not equal to +1 or —1. If a parameter varies, then the eigenvalues A3, A4 are restricted to move on the unit circle, because they must be both inverse, A3 = I/A4 and complex conjugate. Consequently, the stability is conserved. However, if A3, A4 meet at the points +1 or — 1, then it is possible for them to go outside the unit circle, as shown in Figure 6, and thus generate instability. For this reason the orbits with A3 = A4 = 1 are called critical as far as stability is concerned. 

A new family of periodic orbits bifurcates from the T-periodic orbit x(t) of the original family, starting with the same period. Note that stability changes (critical points with respect to the stability). 

If for a periodic orbit of the planar family we have a = 0 (vertically critical orbit), then 3(t) is T-periodic. Then we have a bifurcation of a family of 3-dimensional periodic orbits from this point of the planar family, with the same period. 

A sub-critical Hopf bifurcation contains an interval of A in which the stable steady state is connected by an unstable periodic orbit. Therefore, in addition to the bifurcation point, a turning point exists that connects a stable with an unstable periodic solution. 

Normal Forms. In order to model essential nonlinear properties of a given flow nearby a critical point one can focus on a center manifold that locally contains all critical points, steady states, and periodic orbits. In the case of static bifurcations it turns out that a one dimensional parameter dependent ODE suffices to describe the dynamics inside the center manifold. For... 

For these spacecraft, the mission success is mainly binary, without reference to a period dtrring which service is provided. The situation is similar to the first phase (transport or travel) of exploration probes. The constrairrts set for the probe voyage (section 7.4.1.3 mass, consumption, ground-onboard liaisorr, trajectory, critical periods) are weaker for an orbital transport vehicle for which the reliability corrstraints namely come from the precision required for approach and docking and from appended safety objectives (see section 7.4.4). 

The study of critical cases gives rise to a number of questions why does an equilibrium state, or a periodic orbit, preserves its stability on the boundary in some cases and not in others What happens beyond the stability boundary ... 

Fig. 11.7.4. Sketch of the local structure of stability regions of periodic orbits near the boundary of a resonant zone far beyond the critical threshold.

We have already remarked that the problem concerning the loss of stability of periodic orbits in autonomous systems cannot always be reduced to a study of bifurcations of fixed points of the Poincare map. It may happen that the periodic orbit does not exist on the stability boundary and, therefore, the Poincare map is not defined at the critical parameter value. 

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.7.5. Period T of the periodic orbit born through a sub-critical Andronov-Hopf bifurcation versus the parameter a (6 = 1), as the cycle approaches the homoclinic loop. The origin is a saddle with 0.

The last comment on the Chua circuit concerns the bifurcations along the path 6 = 6 (see Fig. C.7.4). Notice that this sequence is very typical for many synunetric systems with saddle equilibrium states. We follow the stable periodic orbit starting from the super-critical Andronov-Hopf bifurcation of the non-trivial equilibrium states at a 3.908. As a increases, both separatrices tend to the stable periodic orbits. The last ones go through the pitch-fork bifurcations at a 4.496 and change into saddle type. Their size increases and at a 5.111, they merge with the homoclinic-8. This, as well as subsequent bifurcations, lead to the appearance of the strange attractor known as the double-scroll Chua s attractor in the Chua circuit. ... 

It is now known that the view of electrons in individual well-defined quantum states represents an approximation. The new quantum mechanics formulated in 1926 shows unambiguously that this model is strictly incorrect. The field of chemistry continues to adhere to the model, however. Pauli s scheme and the view that each electron is in a stationary state are the basis of the current approach to chemistry teaching and the electronic account of the periodic table. The fact that Pauli unwittingly contributed to the retention of the orbital model, albeit in modified form, is somewhat paradoxical in view of his frequent criticism of the older Bohr orbits model. For example Pauli writes,... 

This limit cycle represents a trajectory that starts at the static saddle point and ends after one period at the same saddle point. This trajectory is called the homoclinical orbit and will occur at some critical value jiuc- It has an infinite period and therefore this bifurcation point is called infinite period bifurcation . For p < hc the limit cycle disappears. This is the second most important type of dynamic bifurcation after Hopf bifurcation. 

In this chapter we have explored the structure of organic compounds. This is important since structure determines reactivity. We have seen that weak bonds are a source of reactivity. Strong bonds are made by good overlap of similar-sized orbitals (same row on periodic table). Bends or twists that decrease orbital overlap weaken bonds. Lewis structures and resonance forms along with electron flow arrows allow us to keep track of electrons and explain the changes that occur in reactions. VSEPR will help us predict the shape of molecules. Next we must review how bonds are made and broken, and what makes reactions favorable. Critical concepts and skills from this chapter are ... 

In the particular case p = 1, q = 2, we have a = m and the bifurcating orbits start with a period 2T. This type of bifurcation takes place when Ai = A2 = —1, which is also a critical point with respect to stability. 

In this way, we may picture orbital contraction as a purely quantum-mechanical effect, which arises from the existence of a short range well within the atom. The binding strength of this well increases with atomic number and, as a result, the critical condition for the appearance of the first bound state is satisfied around Z = 56. The condition for two bound states to occur inside the inner well is satisfied in a similar way at the onset of the 5/ period, giving rise to the actinide sequence. 

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