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We review the applications of order and chaos in various branches of astronomy. Order and chaos appear in generic dynamical
systems, including the sun and other stars, the solar system and galaxies, up to the whole Universe. We discuss in particular
the various types of orbits in galaxies, emphasizing the role of diffusion of chaotic orbits and the escapes to infinity.
Then we consider chaos in dissipative systems, like gas in a galaxy, chaos in relativity and cosmology, and chaos in stellar
pulsations and in the solar activity.

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... Henry Poincare introduces several geometric and topological notions into Celestial Mechanics, such as the notion of the Poincare Map and is one of the first to recognize the structure of the geometrical entity we would call in our epoch as the "strange attractor". The pioneering work of Henry Poincare opened a completely novel view of the kinematical and dynamical processes of gravitationally interacting physical bodies, in the description of our Solar System, in the formation and evolution of clusters of stars and galaxies, of the collision and interaction between galaxies, just to mention a few basic thematic issues which are extensively studied in our epoch (Contopoulos, 2003). ...

The dynamical interaction between binary systems is crucial in understanding the nature of orbital motion under the influence of gravitational potential. In our study, we focused on investigating the effects of dynamical forces on the regularity of binary pulsar orbits, which represent a pure two-body system. To incorporate the necessary time dependence and have a regular 3-D axisymmetric potential, we utilized the Rebound package as a numerical integrator. This package integrates the motion of particles under the influence of gravity, allowing for changing orbital parameters at a given instant, and providing a variety of integrators to be used. By analyzing the regularity properties of binaries and their sensitivity to initial conditions, we gained insight into the importance of considering even small perturbations to the system, as they can lead to significant changes in its dynamics.

In this paper, we firstly formulate a new hyperchaotic Hamiltonian system and demonstrate the existence of multi-equilibrium points in the system. The characteristics of equilibrium points, Lyapunov exponents and Poincaré sections are studied. Secondly, we investigate the complex dynamical behaviors of the system under holonomic constraint and nonholonomic constraint, respectively. The results show that the hyperchaotic system can generated by introducing constraint. Additionally, the hyperchaos control of the system is achieved by applying linear feedback control. The numerical simulations are carried out in order to analyze the complex phenomena of the systems.

The dynamics of a non-autonomous chaotic system with one cubic nonlinearity is studied through numerical and experimental investigations in this paper. A method for calculating Lyapunov exponents (LEs), Lyapunov dimension (LD) from time series is presented. Furthermore, some complex dynamic behaviors such as periodic, quasi-periodic motion and chaos which occurred in the system are analyzed, and a route to chaos, phase portraits, Poincare sections, bifurcation diagrams are observed. Finally, a first order differential controller for the non-autonomous system is designed. Also some dynamics such as Poincare sections, bifurcation diagrams for specific control parameter values of the controlled system are showed using numerical and experimental simulations.

We discuss the following fundamental concepts of galactic dynamics: (a) regular (smoothed) and irregular (random) forces, (b) truncation of the impact parameter, (c) the invariance of the Maxwellian velocity distribution, and (d) the Jeans theorem. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We discuss the effective stochastization time τe for gravitating systems in terms of the Krylov and GurzadyanSavvidi paradigm. The truncated Holtsmark distribution for a random force proposed by Rastorguev and Sementsov implies τe/τc α N0.20, where τc is the crossing time. We find in the case of the Petrovskaya distribution for a random force τe/τc α Nk, where k = 0.27–0.31, depending on the oblateness and rotation of the system, and τe/τc α N1/3/(ln N)1/2 when N » 1. The latter result agrees with those of Genkin (1969) and Gurzadyan & Kocharyan (2009) (k = 1/3). (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper illustrates the application of Lie transform normal-form theory to
the construction of the 1:2 resonant normal form corresponding to a wide class
of natural Hamiltonian systems. We show how to compute the bifurcations of the
main periodic orbits in a potential with double reflection symmetries. The
stability analysis of the normal modes and of the periodic orbits in general
position allows us to get overall informations on the phase-space structure of
systems in which this resonance is dominating. As an example we apply these
results to a class of models useful as galactic potentials.

For infinitesimal, homologous perturbations, stability analysis has found the solar radiative interior thermally stable. It is considered for the first time here whether stability is preserved when finite amplitude nonhomologous perturbations are present. We argue that local heated regions may develop in the solar core due to magnetic instabilities. Simple numerical estimations are derived for the timescales of the decay of these events and, when heated bubbles are generated that rise towards the surface, of their rising motion. These estimations suggest that the solar core is in a metastable state. For more detailed analysis, we developed a numerical code to solve the differential equation system. Our calculations determined the conditions of metastability and the evolution of timescales. We obtained two principal results. One of them shows that small amplitude heating events (with energy surplus Q
o < 1026 ergs) contribute to subtle but long-lifetime heat waves and give the solar interior a persistently oscillating character. Interestingly, the slow decay of heat waves may make their accumulation possible and so their overlapping may contribute to the development of an intermittent, individual, local process of bubble generation, which may also be generated directly by stronger (Qo
> 1026 ergs) heating events. Our second principal result is that for heated regions with ΔT/T ≥ 10−4 and radius 105–106cm, the generated bubbles may travel distances larger than their linear size. We point out to some possible observable consequences of the obtained results.

A two-point model of an unisolated star cluster moving in a circular orbit in the Galactic plane is analyzed. The equations
of stellar motion are linearized in the neighborhood of the singular point at the zero-velocity surface (ZVS), and also in
the neighborhood of a point below the critical ZVS on a trajectory with less than the critical stellar ‘energy.’ We find the
eigenvalues and eigenvectors of these equations and point out the instability of the two singular points on the critical ZVS;
the separatrix connecting these points is determined numerically. For trajectories located below the critical ZVS, the absolute
values of the eigenvalues of the linearized equations of motion increase with decreasing energy of the star and decreasing
maximum distance between the trajectory and the cluster center of mass. This results in an increase of the numerical estimates
of the maximum characteristic Lyapunov exponents for trajectories located closer to the center of mass of the cluster. We
use Poincaré sections and the maximum characteristic Lyapunov exponents to analyze the properties of the stellar trajectories.
A number of periodic orbits for different stellar energies are found, and the properties of the trajectories in the vicinity
of these periodic orbits analyzed. Almost all the stellar trajectories considered are stochastic, with the degree of stochasticity
increasing with decreasing stellar energy. Domains with different degrees of stochasticity are identified in the Poincaré
maps.
PACS numbers98.10.+z–98.20.Di

This is a tutorial presentation of special features of galactic disc dynamics, which completes our introduction to galactic
dynamics initially presented in [30]. The emphasis is on topics where galactic dynamics and celestial mechanics share common
starting points and/or methods of approach. We start by giving some definitions and general notions on the link between observations
and dynamical modeling of discs. Then we focus on the application of resonant Hamiltonian perturbation theory in disc resonances.
By examining in detail the case of the Inner Lindblad resonance, we demonstrate how resonant perturbation theory leads to
an orbital theory of spiral structure in normal galaxies. Passing to barred galaxies, the phase space structure and the role
of chaos in the corotation region are analyzed. This is accomplished by a summary of the modern theory of invariant manifolds
of unstable periodic orbits in the vicinity of L1 or L2, which can interpret the generation of spiral patterns by chaotic orbits beyond corotation. Some additional topics, potentially
important for disc dynamics, are briefly commented.

This paper aims to illustrate the applications of resonant Hamiltonian normal
forms to some problems of galactic dynamics. We detail the construction of the
1:1 resonant normal form corresponding to a wide class of potentials with
self-similar elliptical equi-potentials and apply it to investigate relevant
features of the orbit structure of the system. We show how to compute the
bifurcation of the main periodic orbits in the symmetry planes of a triaxial
ellipsoid and in the meridional plane of an axi-symmetric spheroid and briefly
discuss how to refine these results with higher-order approaches.

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