DOI: https://doi.org/10.30970/jps.01.513
NUMERICAL ANALYSIS OF NON-LINEAR CONSERVATIVE OSCILLATORS
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Journal of Physical Studies 1(4), 513–520 (1997)
DOI: https://doi.org/10.30970/jps.01.513 NUMERICAL ANALYSIS OF NON-LINEAR CONSERVATIVE OSCILLATORS |
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T. Vasylenko, L. Sinitsky, Ja. Shmigelsky
Lviv State University, Chair of Radioelectronics
19 Drahomanov Str., Lviv, UA–290005, Ukraine
The paper provides an analysis of bifurcation phenomena in the dynamic of conservative oscillators when their equations are discretised by various numerical methods. The Euler–Cromer method, trapezoidal, rectangular rules and conservative methods are considered. For each of them the dependence of the energy and the frequency of oscillation are evaluated as a function of the step of integration. Nevertheless the conservative methods preserve the energy conservation law the change of the frequency of oscillation has the same order just as for all other methods mentioned above. Conditions for existence of quasy-periodic motions and evaluation of their frequencies are established. Chaotic behavour caused by discretisation was investigated in some papers. More detailed analysis of such phenomena permitted to establish the values of parameters when such effects are impossible. Special attention was devoted to a problem of distinguishing chaotic behavour from quasy-periodic motion. For this purpose detailed analysis of spectrum of both motions was carried out. In particular it was shown that the change of energy due to the discretisation cannot be considered as a main sign of an appearance of chaotic behavour: there exists a wide interval steps of integration where energy changes to a marked degree during a period. Nevertheless chaotic motions are not observed. The existence of chaos was assured in the case when spectrum changes discontiniously under smoothly increasing of the energy.