Periodic orbit, elliptic

This study [14] has shown that a period-doubling bifurcation associated with the Fermi resonance occurs in this subsystem at the energy E = 3061.3 cm-1 (with Egp = 0). Below the Fermi bifurcation, there exist edge periodic orbits of normal type, which are labeled by ( i, 22 -)normai- At the Fermi bifurcation, a new periodic orbit of type (2,1°, ->Fermi appears by period doubling around a period of 2T = 100 fs. This orbit is surrounded by an elliptic island that forms a region of local modes in phase space. Therefore, another family of edge periodic orbits of local type are bom after the Fermi bifurcation that may be labeled by the integers (n n , -)iocai- They are distinct... 

At still higher energies, the elliptic island undergoes a typical cascade of bifurcations in which subsidiary elliptic islands of periods 6, 5, 4, 3 are successively created, which leads to the global destruction of the main elliptic island to the benefit of the surrounding chaotic zone. The cascade ends with a period-doubling bifurcation at Ed, above which the periodic orbit 0 is hyperbolic with reflection, and the main elliptic island has disappeared... 

As the energy increases in the interval E < E < Ea, the orbits Y and 2 progressively shift toward the symmetric-stretch orbit 0 and merge at the subcritical antipitchfork bifurcation. Just below this bifurcation, 1 and 2 are elliptic while 0 is still hyperbolic (without reflection). Between and Ea, the periodic orbits 1 and 2 may either remain of elliptic type or become hyperbolic in the energy interval [ , "1 such that E < Edd> < Edd < Ea. 

The resonant invariant curves correspond to the resonant elliptic periodic orbits, in the rotating frame. 

A resonant n/n = p/q elliptic periodic orbit is a multiple fixed point on the resonant invariant curve. The angle i) changes during one iteration by... 

If the system has symmetries (as is the case with the restricted problem), usually the symmetric periodic orbits survive (but not always ). The resonant fixed points that survive correspond to monoparametric families of elliptic periodic orbits, in the rotating frame. These families bifurcate from the circular family, at the corresponding circular resonant orbits. From the above analysis we come to the conclusion that out of the infinite set of resonant elliptic periodic orbits of the two-body problem, with the same semimajor axes and the same eccentricities, but different orientations, as shown in Figure 15, only a finite number survive as periodic orbits in the rotating frame, and in most cases only two, usually, but not always, are symmetric. 

The paper is organized as follows in Section 2 and 3 we define the Fast Lyapunov Indicator and give some examples on the 2 dimensional standard map and on a Hamiltonian model. The special case of periodic orbits will be detailed in 4 and thanks to a model of linear elliptic rotation we will be able to recover the structure of the phase space in the vicinity of a noble torus. The use of the FLI for detecting the transition between the stable Nekhoroshev regime to the diffusive Chirikov s one will be recalled in Section 5. In 6 and 7 we will make use of the FLI results for the detection of the Arnold s diffusion. 

Let us consider the elliptic point (1.239,0) of the standard map (equation 5) for e = 0.7, which corresponds to a periodic orbit of order 4. We have computed the evolution of the FLI with time for ten orbits regularly spaced on the x-axis in the interval [1.239,1.339] starting from the periodic point. 

Again, we have a qualitative good agreement with the other two methods, but in the case of the LMA we expect also to have a quantitative agreement. Actually this method is based on the linear approximation as well as the model of elliptic rotation introduced for explaining the FLI for periodic orbits. 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

Thus some of the fluid elements move on aperiodic chaotic trajectories and others on quasiperiodic orbits. The quasiperiodic orbits are invariant surfaces in the phase space that form the boundaries of the chaotic layers and limit the motion of the chaotic trajectories. There is a similar structure around each elliptic periodic orbit resulting from broken resonant tori that are also surrounded by invariant tori forming isolated islands inside the chaotic region. 

We have discussed the typical manifestation of periodic orbits on a Poincare map as fixed points that are either elliptic or hyperbolic. Let us now consider the properties of motion nearby these fixed points in terms of their stability properties. This is accomplished by a straightforward linear stability analysis about the fixed point. We can carry out such an analysis on any fixed point, whether or not the surrounding phase space is chaotic (as long as we can find the fixed point). 

Moreover, short time decay rate constants can be estimated very accurately by linearization of the dynamics(8) around the HC periodic orbit(14). The eigenvalues of the Jacobian matrix propagator of the return map are = exp(ia), =exp(--ia) for elliptic (stable) fixed... 

Now consider the hypothetical problem of trying to teach the physics of space flight during the period in time between the formulation of Kepler s laws and the publication of Newton s laws. Such a course would introduce Kepler s laws to explain why all spacecraft proceed on elliptical orbits around a nearby heavenly body with the center of mass of that heavenly body in one of the focal points. It would further introduce a second principle to describe course corrections, and define the orbital jump to go from one ellipse to another. It would present a table for each type of known spacecraft with the bum time for its rockets to go from one tabulated course to another reachable tabulated course. Students completing this course could run mission control, but they would be confused about what is going on during the orbital jump and how it follows from Kepler s laws. 

In general, if a particle is bound (E < 0) it will oscillate (classically) between some limits r = a, and r = b. For example, in an elliptic orbit of a hydrogen atom, the radius oscillates periodically between inner and outer limits. Only for a circular orbit is there no oscillation. Among the eigenvalues which have the same n, the one with lowest l has the largest amplitude in the vicinity of the nucleus. 

Following Sommerfeld s proposal of elliptical electron orbits in 1915, Bohr amended his original theory, which had included only circular orbits. 14 A 1922 paper in Zeitschriftfur Physik outlined the "Aufbauprinzip" by which electrons are fed into atomic subshells. There was a neat correlation between periodic groups containing 2, 8, 8, 18,... 

Eccentricity of the Earth s orbit, which varies from almost circular to strongly elliptical with a periodicity of about 95,800 years (these changes modulate precession). 

M 8] [P 7] The effect of switching between the flow fields (a) and (d) given in Figure 1.18 at various periods T was analyzed (see Figure 1.19) [28], For very high alterations, a simple superposition of the flow fields is achieved. Elliptic fixed points surrounded by closed orbits (tori) of various periods are found. 

An obvious possible improvement of the Bohr model was to bring it better into line with Kepler s model of the solar sxstem, which placed the planets in elliptical, rather than circular, orbits. Sommerfeld managed to solve this problem by the introduction of two extra quantum numbers in addition to the principal quantum number (n) of the Bohr model, and the formulation of general quantization rules for periodic systems, which contained the Bohr conjecture as a special case. 

But no fine structure - yet - until in 1915 Bohr considered the effect of relativistic variation of mass with velocity in elliptical orbits under the inverse square law of binding, and pointed out that the consequential precessional motion of the ellipses would introduce new periodicities into the motion of the electron, whose consequences would be satellite lines in the spectra. The details of the dynamics were worked out independently by SOMMERFELD [38] and WILSON [39] in 1915/16 based on a generalisation of Bohr s quantization, namely, the quantization of action the values of the phase integrals Jf = fpj.d, - of classical mechanics should be constrained to assume only integral multiples of h. 

Many other complex interactions are involved in climatic effects, among them lateral and vertical perturbations of ocean currents, changes in prevailing winds, periodicity in the earth s tilt (1 1/4°, 21,000-year cycle), position of elliptical orbit (97,000-year cycle), and changes in the solar flux (correlated with sunspots, ca. 13-year-cycle) [75]. As the periods of these factors differ, the warming effect of some will be augmented by in-phase peaks at times, and will be decreased or eliminated by out-of-phase peaks at others. So, the net effect is at best difficult to predict. Again, like atmospheric moisture, these variables are beyond our control. 

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