High-Dimensional Chaos in Dissipative and Driven Dynamical Systems

Cuntz

Monthly Notices of the Royal Astronomical Society

2012

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We provide a detailed theoretical study aimed at the observational finding about the ν Octantis binary system that indicates the possible existence of a Jupiter‐type planet in this system. If a prograde planetary orbit is assumed, it has earlier been argued that the planet, if existing, should be located outside the zone of orbital stability. However, a previous study by Eberle & Cuntz concludes that the planet is most likely stable if assumed to be in a retrograde orbit with respect to the secondary system component. In the present work, we significantly augment this study by taking into account the observationally deduced uncertainty ranges of the orbital parameters for the stellar components and the suggested planet. Furthermore, our study employs additional mathematical methods, which include monitoring the Jacobi constant, the zero velocity function and the maximum Lyapunov exponent. We again find that the suggested planet is indeed possible if assumed to be in a retrograde orbit, but it is virtually impossible if assumed in a prograde orbit. Its existence is found to be consistent with the deduced system parameters of the binary components and of the suggested planet, including the associated uncertainty bars given by observations.

Section: Theoretical Approachmentioning

confidence: 99%

The stability of the suggested planet in the ν Octantis system: a numerical and statistical study

Cuntz

Monthly Notices of the Royal Astronomical Society

2012

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“…A dynamical system with n degrees of freedom is represented in 2n phase space; thus, to fully determine the stability of the system all 2n Lyapunov exponents must be calculated. The Lyapunov exponents are the most commonly used tools to determine the onset of chaos and chaotic behaviour of both dissipative (e.g., Musielak & Musielak 2009, and references therein) and non-dissipative systems of orbital mechanics (e.g., Lissauer 1999,and references therein). The positive Lyapunov exponents measure the rate of divergence of neighbouring orbits, whereas negative exponents measure the convergence rates between stable manifolds.…”

Section: Lyapunov Exponentsmentioning

confidence: 99%

“…The positive Lyapunov exponents measure the rate of divergence of neighbouring orbits, whereas negative exponents measure the convergence rates between stable manifolds. For dissipative dynamical systems the sum of all Lyapunov exponents is less than 0 (e.g., Musielak & Musielak 2009); however, for Hamiltonian (non-dissipative) systems the sum is equal to 0 (e.g., Hilborn 1994).…”

Section: Lyapunov Exponentsmentioning

confidence: 99%

The instability transition for the restricted 3-body problem

Eberle

et al. 2011

A&A

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Aims. We establish a criterion for the stability of planetary orbits in stellar binary systems by using Lyapunov exponents and power spectra for the special case of the circular restricted 3-body problem (CR3BP). The criterion augments our earlier results given in the two previous papers of this series where stability criteria have been developed based on the Jacobi constant and the hodograph method.Methods. The centerpiece of our method is the concept of Lyapunov exponents, which are incorporated into the analysis of orbital stability by integrating the Jacobian of the CR3BP and orthogonalizing the tangent vectors via a well-established algorithm originally developed by Wolf et al. The criterion for orbital stability based on the Lyapunov exponents is independently verified by using power spectra. The obtained results are compared to results presented in the two previous papers of this series. Results. It is shown that the maximum Lyapunov exponent can be used as an indicator for chaotic behaviour of planetary orbits, which is consistent with previous applications of this method, particularly studies for the Solar System. The chaotic behaviour corresponds to either orbital stability or instability, and it depends solely on the mass ratio μ of the binary components and the initial distance ratio ρ 0 of the planet relative to the stellar separation distance. Detailed case studies are presented for μ = 0.3 and 0.5. The stability limits are characterized based on the value of the maximum Lyapunov exponent. However, chaos theory as well as the concept of Lyapunov time prevents us from predicting exactly when the planet is ejected. Our method is also able to indicate evidence of quasi-periodicity. Conclusions. For different mass ratios of the stellar components, we are able to characterize stability limits for the CR3BP based on the value of the maximum Lyapunov exponent. This theoretical result allows us to link the study of planetary orbital stability to chaos theory noting that there is a large array of literature on the properties and significance of Lyapunov exponents. Although our results are given for the special case of the CR3BP, we expect that it may be possible to augment the proposed Lyapunov exponent criterion to studies of planets in generalized stellar binary systems, which is strongly motivated by existing observational results as well as results expected from ongoing and future planet search missions.

“…He also showed that solutions could leave two equilibria and then return to them asymptotically, thus forming a heteroclinic orbit. The above concepts originally introduced by Poincaré are now commonly used in modern theories of chaos [Thompson and Stewart, 1986, Hilborn, 1994, Musielak and Musielak, 2009.…”

Section: Poincaré's Qualitative Methods and Chaosmentioning

confidence: 99%

“…The third case considers a Lyapunov exponent that is exactly equal to zero, which means that the two trajectories are parallel to each other and by induction will remain so unless another force arises to disrupt the system. The mathematical definition of the Lyapunov exponent [Hilborn, 1994, Musielak andMusielak, 2009] is…”

Section: Maximum Lyapunov Exponentmentioning

confidence: 99%

The three-body problem

2014

Rep. Prog. Phys.

Abstract. The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis, search for periodic orbits and resonances, and application of the results to some interesting astronomical and space dynamical settings. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications.