Dense orbit

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

The simplicity of the shift map allows us to prove another astonishing fact the existence of a dense orbit. An orbit of a dynamical system is said to be dense if it comes arbitrarily close to any point in the domain of the mapping. Following Devaney (1992), we choose the seed... 

Closely related to the notion of dense orbits is the concept of transitivity of a dynamical system. A dynamical system is transitive if for any two points x and X2 in the domain of the system there exists an orbit that visits both x and X2 to any prescribed accuracy. Obviously, the existence of a dense orbit ensures transitivity of the corresponding dynamical system. Thus, the shift map is transitive. 

In general, the Lyapunov exponent A depends on the starting value xq. For seeds of dense orbits of the mapping /, however, the hmit operation in (2.2.38) assures that the Lyapunov exponent is independent of the starting value, since the dense orbit will explore all the points of the domain of /. 

Irrational flow yields dense orbits) Consider the flow on the torus given... 

Dense orbit for the decimal shift map) Consider a map of the unit interval into itself. An orbit x, is said to be dense if it eventually gets arbitrarily close to every point in the interval. Such an orbit has to hop around rather crazily More precisely, given any e > 0 and any point p e [0,1], the orbit x is dense if there is some finite n such that x - p < . ... 

Binary shift map) Show that the binary shift map x , = 2x (modi) has sensitive dependence on initial conditions, infinitely many periodic and aperiodic orbits, and a dense orbit. (Hint Redo Exercises 10.3.7 and 10.3.8, but write x as a binary number, not a decimal.)... 

The compound Lajln has Tc = 10.4 K. Because La is hypoelectronic and In is hyperelectronic, I expect electron transfer to take place to the extent allowed by the approximate electroneutrality principle.13 The unit cube would then consist of 2 La, La, and In+, with In+ having no need for a metallic orbital and thus having valence 6 with the bonds showing mainly pivoting resonance among the twelve positions. The increase in valence of In and also of La (to 3 f ) and the assumption of the densely packed A15 structure account for the decrease in volume by 14.3%. Because the holes are fixed on the In + atoms, only the electrons move with the phonon, explaining the increase in Tc. 

The density estimates in Table 7.1 show a distinction between the structures of the planets, with Mercury, Venus, Earth and Mars all having mean densities consistent with a rocky internal structure. The Earth-like nature of their composition, orbital periods and distance from the Sun enable these to be classified as the terrestrial planets. Jupiter, Saturn and Uranus have very low densities and are simple gas giants, perhaps with a very small rocky core. Neptune and Pluto clearly contain more dense materials, perhaps a mixture of gas, rock and ice. 

The orbits are dense in a state space region i.e. the orbits fills the phase space zone of the strange attractor fl. 

Problem 9-15. Could the same predictions be made from a simple electron repulsion argument If n pairs of electrons must be accommodated in the valence-shell molecular orbitals, then assume simply that they will be as far apart as possible. Up to four electrons will push each other as far apart as possible, to create linear geometry more than four must be distributed more densely, so that the angle between the substituents will be less that 180°. Does this simple hypothesis explain everything that Walsh s rules do Is there any advantage to using Walsh s correlation diagram analysis ... 

In the Bohr atom, as it is commonly now depicted, electrons -which have a mass just 0.00055 times that of the proton, but an equal and opposite electric charge - orbit around a nucleus of protons and neutrons, packed together with an awesome density. If matter were uniformly as dense as the nucleus rather than containing so much empty space, a thimbleful would weigh about a billion tonnes. ... 

Which is all very well, but the Bohr atom is wrong. The picture of a dense nucleus surrounded by electrons is accurate enough, but they do not follow nice elliptical orbits like those of the planets. Venus and Mars follow Newton s laws, but electrons are governed by the... 

Warm air rises because it is less dense than the surrounding air. Within an orbiting space station, however, there is no up or down, hence nowhere for the warm air to rise. It instead remains hovering around the flame. But this warm air contains the products of combustion—primarily water vapor and carbon dioxide—which eventually smother the flame. 

Now, the eigenenergies of the Hamiltonian can be detected directly if the time dependence of the above average exhibits quantum beats. This will be the case if the spectrum is not too dense and the linewidths are smaller than the level spacings. From a Fourier transform of the autocorrelation function, we then obtain an expression of the form (2.26)-(2.27), which can be evaluated semiclassically in terms of periodic orbits and their quantum phases. 

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