Order and chaos

Contopoulos, G.

doi:10.1007/3-540-57983-4_18

Cited by 62 publications

(66 citation statements)

References 71 publications

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“…Some other routes for the transition from order to chaos in conservative dynamical systems of 2-degrees of freedom have been found by Contopoulos and Zikides (1980), Contopoulos (1983aContopoulos ( ,b, 1991, Heggie (1983), Bier and Bountis (1984) and Pinotsis (1988). These scenarios of routes to chaos are mentioned by Contopoulos (1993Contopoulos ( , 2002. Also, some of the routes to chaos in dissipative and conservative systems we find in an article by Eckmann (1981).…”

Section: Introductionsupporting

confidence: 56%

Infinite Feigenbaum sequences and spirals in the vicinity of the Lagrangian periodic solutions

Pinotsis

2010

Celest Mech Dyn Astr

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We studied systematically cases of the families of non-symmetric periodic orbits in the planar restricted three-body problem. We took interesting information about the evolution, stability and termination of bifurcating families of various multiplicities. We found that the main families of simple non-symmetric periodic orbits present a similar dynamical structure and bifurcation pattern. As the Jacobi constant changes each branch of the characteristic of a main family spirals around a focal point-terminating point in x-at which the Jacobi constant is C ∞ =3 and their periodic orbits terminate at the corotation (at the Lagrangian point L 4 or L 5 ). As the family approaches asymptotically its termination point infinite changes of stability to instability and vice versa occur along its characteristic. Thus, infinite bifurcation points appear and each one of them produces infinite inverse Feigenbaum sequences. That is, every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate large chaotic regions near the corotation points L 4 , L 5 , which are unstable. As C varies along the spiral characteristic of every bifurcating family, which approaches its focal point, infinite loops, one inside the other, surrounding the unstable triangular points L 4 or L 5 are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals into the corotation points L 4 , L 5 with infinite period. This is a new mechanism that produces very large degree of stochasticity. These conclusions help us to comprehend better the motions around the points L 4 and L 5 of Lagrange.

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Section: Introductionsupporting

confidence: 56%

Infinite Feigenbaum sequences and spirals in the vicinity of the Lagrangian periodic solutions

Pinotsis

2010

Celest Mech Dyn Astr

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“…2(a), we show the different morphologies of three regular orbits around the periodic orbits in the centres of the three largest islands of regular motion present in the associated PSS. The model's basic barred shape in the (x, y)plane is mainly provided by trajectories around the main family of periodic orbits, the so-called x1 family (following the nomenclature of Contopoulos & Papayannopoulos 1980) which are orbits elongated along the bar, that is, along the x-axis in our case, and inside corotation. The stability island on the right-hand side gives rise to elliptical-like orbits, around the x2 family, elongated perpendicular to the bar, while the island on the left-hand side contains orbits which are elliptical like but retrograde, very slightly elongated perpendicular to the bar and around the family x4.…”

Section: The 2-dof M O D E L P Ot E N T I a Lmentioning

confidence: 99%

“…Based on the structure and crowding of the periodic orbits, Contopoulos (1980) showed that bars have to end before corotation, that is, r L > a, where r L is the Lagrangian, or corotation, radius. A Figure 8.…”

Section: Percentages Of Regular Orbits Versus Pattern Speedmentioning

confidence: 99%

Regular and chaotic orbits in barred galaxies - I. Applying the SALI/GALI method to explore their distribution in several models

Manos

Athanassoula

2011

Monthly Notices of the Royal Astronomical Society

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International audienceThe distinction between the chaotic and regular behaviour of orbits in galactic models is an important issue and can help our understanding of galactic dynamical evolution. In this paper, we deal with this issue by applying the Smaller ALingment Index (SALI) and Generalized ALingment Index (GALI) techniques to extensive samples of orbits obtained by integrating numerically the equations of motion in a barred galaxy potential. We estimate first the fraction of chaotic and regular orbits for the 2 degrees of freedom (d.o.f.) case [where the galaxy extends only in the (x, y) space] and show that it is a non-monotonic function of the energy. For the 3-d.o.f. extension of this model (in the z-direction), we give similar estimates, both by exploring different sets of initial conditions and by varying the model parameters, like the mass, size and pattern speed of the bar. We find that regular motion is more abundant at small radial distances from the centre of the galaxy, where the relative non-axisymmetric forcing is relatively weak, and at small distances from the equatorial plane, where the trapping around the stable periodic orbits is important. We also find that the variation in the bar pattern speed, within a realistic range of values, does not affect much the phase-space fraction of regular and chaotic motions. Using different sets of initial conditions, we show that chaotic motion is dominant in galaxy models whose bar component is more massive, while models with a fatter or thicker bar present generally more regular behaviour. Finally, we find that the fraction of orbits that are chaotic correlates strongly with the bar strength

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“…The form of the two integrals, besides the Hamiltonian, is given e.g. in Contopoulos (1994) where the main types of orbits are also analyzed. A case of particular interest for galactic dynamics is the perfect ellipsoid.…”

Section: Triaxial Systemsmentioning

confidence: 99%