Computer Science
Scientific paper
Jan 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995phdt........18l&link_type=abstract
Thesis (PH.D.)--THE UNIVERSITY OF TEXAS AT AUSTIN, 1995.Source: Dissertation Abstracts International, Volume: 56-10, Section: B,
Computer Science
L
Scientific paper
The purpose of this study has been to analyze the stability of Copenhagen Class (l) orbits. The Class (l) orbits are known to be stable when the Jacobian Constant (C) is greater than 3.552593. However, there have been no publications to explain the change from stable to unstable orbits after the first period doubling point of this class. The first period doubling of Class (l) orbits occurs at C = 3.67771921, where the orbit becomes critically stable (i.e. stability index k = 2). This value is denoted by C_sp {1}{1}, Class (l) orbits will then change to unstable orbits at C = 3.552593 (C _0). Within the range of C_0 < {rm C} < C_sp{1}{1 }, Class (l) orbits can bifurcate to unstable orbits with higher periods by a small energy change. Hence, these orbits are sensitive to the energy change in C _0 < {rm C} < C_sp{1 }{1}; this range will be analyzed in detail in this dissertation. An infinite number of bifurcations are expected in this region and bifurcations become highly condensed near C_0. It has been found here that the Feigenbaum sequence (or pitchfork sequence) does not exist for Class (l). However, a different sequence of bifurcations of unstable orbits that come from a stable family, namely Class (l), has been found. This type of bifurcation was also found by Contopoulos for a different dynamical system. Theoretically, at C _0, Class (l) bifurcates to an unstable orbit with infinite periodicity. The period doubling process of Class (l) is analyzed in the region of C_0 < {rm C } < C_sp{1}{1}. Bifurcation values of C, in a sequence of period doubling process, were also computed by using the numerical results obtained from the differential equations of motion. These values are theoretically related to the stability indices (or the eigenvalues of the monodromy matrix) of Class (l) orbits. It has been shown that the ratio { Delta C_sp{n}{1}over Delta C_sp{n{+}1}{1 }} converges to 4.0, as C approaches to C_0. Again, we should mention that this number is the result of the different period doubling sequence found in Class (l), rather than from the Feigenbaum sequence.
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