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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Journal of statistical physics 48 (1987), S. 231-254 
    ISSN: 1572-9613
    Keywords: Discrete dynamics ; circle maps ; noise ; mean first passage times ; metastability ; escape rates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The problem of escape from a domain of attraction is applied to the case of discrete dynamical systems possessing stable and unstable fixed points. In the presence of noise, the otherwise stable fixed point of a nonlinear map becomes metastable, due to noise-induced hopping events, which eventually pass the unstable fixed point. Exact integral equations for the moments of the first passage time variable are derived, as well as an upper bound for the first moment. In the limit of weak noise, the integral equation for the first moment, i.e., the mean first passage time (MFPT), is treated, both numerically and analytically. The exponential leading part of the MFPT is given by the ratio of the noise-induced invariant probability at the stable fixed point and unstable fixed point, respectively. The evaluation of the prefactor is more subtle: It is characterized by a jump at the exit boundaries, which is the result of a discontinuous boundary layer function obeying an inhomogeneous integral equation. The jump at the boundary is shown to be always less than one-half of the maximum value of the MFPT. On the basis of a clear-cut separation of time scales, the MFPT is related to the escape rate to leave the domain of attraction and other transport coefficients, such as the diffusion coefficient. Alternatively, the rate can also be obtained if one evaluates the current-carrying flux that results if particles are continuously injected into the domain of attraction and captured beyond the exit boundaries. The two methods are shown to yield identical results for the escape rate of the weak noise result for the MFPT, respectively. As a byproduct of this study, we obtain general analytic expressions for the invariant probability of noisy maps with a small amount of nonlinearity.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Celestial mechanics and dynamical astronomy 66 (1996), S. 203-228 
    ISSN: 1572-9478
    Keywords: Restricted three-body problem ; chaotic phenomena ; scattering
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider the scattering motion of the planar restricted three-body problem with two equal masses on a circular orbit. Using the methods of chaotic scattering we present results on the structure of scattering functions. Their connection with primitive periodic orbits and the underlying chaotic saddle are studied. Numerical evidence is presented which suggests that in some intervals of the Jacobi integral the system is hyperbolic. The Smale horseshoe found there is built from a countable infinite number of primitive periodic orbits, where the parabolic orbits play a fundamental role.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Celestial mechanics and dynamical astronomy 71 (1998), S. 167-189 
    ISSN: 1572-9478
    Keywords: restricted three‐body problem ; chaotic phenomena ; scattering
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the scattering motion of the planar restricted three‐body problem for small mass parameters μ. We consider the symmetric periodic orbits of this system with μ = 0 that collide with the singularity together with the circular and parabolic solutions of the Kepler problem. These divide the parameter space in a natural way and characterize the main features of the scattering problem for small non‐vanishing μ. Indeed, continuation of these orbits yields the primitive periodic orbits of the system for small μ. For different regions of the parameter space, we present scattering functions and discuss the structure of the chaotic saddle. We show that for μ 〈 μc and any Jacobi integral there exist departures from hyperbolicity due to regions of stable motion in phase space. Numerical bounds for μc are given.
    Type of Medium: Electronic Resource
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