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Cincotta, Pablo Miguel; Simó, Carles

doi:10.1023/a:1008355215603

Cited by 35 publications

(27 citation statements)

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“…An application to a 2D model of barred galaxy is given by Patsis et al (1997). The main advantage of this tool is its efficiency to separate regular and stochastic domains in rather short motion times, < ∼ 50 periods; f) In two recent papers, Cincotta & Simó (1999 proposed the conditional entropy of nearby orbits, I, defined through the arc length parameter along the orbit instead of the time as a random variable, as an efficient tool to investigate the phase space as well as to derive the LCN in realistic physical times (∼ 10 3 periods). In fact, the indicator J was defined here in such a way that it behaves in a similar manner than the logarithmic time-derivative of I, J = d log I/d log T .…”

Section: Discussionmentioning

confidence: 99%

Simple tools to study global dynamics in non-axisymmetric galactic potentials – I

Cincotta

Simó

2000

Astron. Astrophys. Suppl. Ser.

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350

319

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Abstract. In a first part we discuss the well-known problem of the motion of a star in a general non-axisymmetric 2D galactic potential by means of a very simple but almost universal system: the pendulum model. It is shown that both loop and box families of orbits arise as a natural consequence of the dynamics of the pendulum. An approximate invariant of motion is derived. A critical value of the latter sharply separates the domains of loops and boxes and a very simple computation allows to get a clear picture of the distribution of orbits on a given energy surface. Besides, a geometrical representation of the global phase space using the natural surface of section for the problem, the 2D sphere, is presented. This provides a better visualization of the dynamics.In a second part we introduce a new indicator of the basic dynamics, the Mean Exponential Growth factor of Nearby Orbits (MEGNO), that is suitable to investigate the phase space structure associated to a general Hamiltonian. When applied to the 2D logarithmic potential it is shown to be effective to obtain a picture of the global dynamics and, also, to derive good estimates of the largest Lyapunov characteristic number in realistic physical times. Comparisons with other techniques reveal that the MEGNO provides more information about the dynamics in the phase space than other wide used tools.Finally, we discuss the structure of the phase space associated to the 2D logarithmic potential for several values of the semiaxis ratio and energy. We focus our attention on the stability analysis of the principal periodic orbits and on the chaotic component. We obtain critical energy values for which connections between the main stochastic zones take place. In any case, the whole chaotic domain appears to be always confined to narrow filaments, with a Lyapunov time about three characteristic periods.Send offprint requests to: P.M. Cincotta

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Section: Discussionmentioning

confidence: 99%

Simple tools to study global dynamics in non-axisymmetric galactic potentials – I

Cincotta

Simó

2000

Astron. Astrophys. Suppl. Ser.

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350

319

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“…A detailed description including examples of the application of the MEGNO method may be found in series of papers (CSGN hereafter) (Cincotta & Simó 2000;Cincotta & Giordano 2001;Cincotta & Núnez 2000). Here we give a simple overview of the method in order to make the paper self-contained.…”

Section: Overview Of the Methodsmentioning

confidence: 99%

“…These calculations usually require quite a long time due to a slow convergence of λ 1 (t) to λ, which is often much longer than the lifetime of the studied model (Cincotta & Simó 2000). Consequently, different new approaches allowing to explore the phase space on relatively short time-scales have been proposed, e.g.…”

Section: Overview Of the Methodsmentioning

confidence: 99%

“…The MEGNO provides more informations than other indicators about the dynamics of the system (see CSGN), and its properties are justified by solid analytical investigations and numerical tests. It originates from the concept of the symmetric conditional entropy of nearby orbits (Cincotta & Simó 1999), and an heuristic observation that close regular trajectories are strongly correlated with each other, but in chaotic domains of the phase space the nearby orbits loose this correlation very quickly.…”

Section: Overview Of the Methodsmentioning

confidence: 99%

“…It is widely applied and well understood (e.g., Benettin et al 1976;Benettin et al 1980;Wisdom 1983;Froeschle 1984;Sussman & Wisdom 1988;Mikkola & Innanen 1995;Froeschlé & Lega 2000;Tancredi et al 2001). In this paper we consider an alternative technique invented by Cincotta et al (Cincotta & Simó 1999Cincotta & Giordano 2001;Cincotta & Núnez 2000). This method seems to have better computational properties than the LCN, and can be very useful for efficient discrimination of regions in parameter space corresponding to quasiperiodic and chaotic motions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global dynamics of planetary systems with the MEGNO criterion

Goździewski

Bois

Maciejewski

et al. 2001

A&A

124

114

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Abstract. In this paper we apply a new technique alternative to the numerically computed Lyapunov Characteristic Number (LCN) for studying the dynamical behaviour of planetary systems in the framework of the gravitational N -body problem. The method invented by P. Cincotta and C. Simó is called the Mean Exponential Growth of Nearby Orbits (MEGNO). It provides an efficient way for investigation of the fine structure of the phase space and its regular and chaotic components in any conservative Hamiltonian system. In this work we use it to study the dynamical behaviour of the multidimensional planetary systems. We investigate the recently discovered υ And planetary system, which consists of a star of 1.3 M and three Jupiter-size planets. The two outermost planets have eccentric orbits. This system appears to be one of the best candidates for dynamical studies. The mutual gravitational interaction between the two outermost planets is strong. Moreover the system can survive on a stellar evolutionary time scale as it is claimed by some authors (e.g., Rivera & Lissauer 2000b). As the main methodological result of this work, we confirm important properties of the MEGNO criterion such as its fast convergence, and short motion times (of the order of 10 4 times the longest orbital period) required to distinguish between regular and chaotic behaviors. Using the MEGNO technique we found that the presence of the innermost planet may cause the whole system to become chaotic with the Lyapunov time scale of the order of 10 3 -10 4 yr only. Chaos does not induce in this case visible irregular changes of the orbital elements, and therefore its presence can be overlooked by studying variations of the elements. We confirm explicitly the strong and sensitive dependence of the dynamical behaviour on the companion masses.

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Numerical Solutions

Musielak¹,

Quarles²

2017

SpringerBriefs in Astronomy

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Cited by 35 publications

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Simple tools to study global dynamics in non-axisymmetric galactic potentials – I

Simple tools to study global dynamics in non-axisymmetric galactic potentials – I

Global dynamics of planetary systems with the MEGNO criterion

Numerical Solutions

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