Approximate general solution of 2-degrees of freedom dynamical systems using `les solutions precieuses' |
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| Authors: | CL Goudas KE Papadakis E Valaris |
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| Institution: | (1) Department of Mathematics, University of Patras, GR-26504 Patras, Greece;(2) Department of Engineering Sciences, University of Patras, GR-26504 Patras, Greece; E-mail |
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| Abstract: | This paper presents the procedure of a computational scheme leading to approximate general solution of the axi-symmetric,2-degrees
of freedom dynamical systems. Also the results of application of this scheme in two such systems of the non-linear double
oscillator with third and fifth order potentials in position variables. Their approximate general solution is constructed
by computing a dense set of families of periodic solutions and their presentation is made through plots of initial conditions.
The accuracy of the approximate general solution is defined by two error parameters, one giving a measure of the accuracy
of the integration and calculation of periodic solutions procedure, and the second the density in the initial conditions space
of the periodic solutions calculated. Due to the need to compute families of periodic solutions of large periods the numerical
integrations were carried out using the eighth order, variable step, R-K algorithm, which secured for almost all results presented
here conservation of the energy constant between 10-9 and 10-12 for single runs of any and all solutions. The accuracy of the approximate general solution is controlled by increasing the
number of family curves and also by `zooming' into parts of the space of initial conditions. All families of periodic solutions
were checked for their stability. The computation of such families within areas of `deterministic chaos' did not encounter
any difficulty other than poorer precision. Furthermore, on the basis of the stability study of the computed families, the
boundaries of areas of `order' and `chaos' were approximately defined. On the basis of these results it is concluded that
investigations in thePoincaré sections have to disclose 3 distinct types of areas of `order' and 2 distinct types of areas
of `chaos'. Verification of the `order'/`chaos' boundary calculation was made by working out several Poincaré surfaces of
sections.
This revised version was published online in July 2006 with corrections to the Cover Date. |
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| Keywords: | dynamical system axi-symmetric potential equilibrium points periodic solutions numerical integration periodic family curves approximate general solution stability stable/unstable periodic family arcs Poincaré surface of sections order chaos `third integrals' zooming into space of solution |
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