Abstract. 

Using the numerical analysis of the bifurcations of elliptic cycles of the conservative generalized Mathieu equation and the Hamiltonian system of equations generated by it, it is shown that the transition to chaos in conservative systems does not occur as a result of the destruction of some mythical tori of the unperturbed system in accordance with the Kolmogorov-Arnold-Moser theory, but as a result of the birth of new complex multi-turn tori around elliptic cycles, which are born in accordance with the incomplete Feigenbaum-Sharkovsky-Magnitskii bifurcation scenario, which is characteristic for dissipative systems of differential equations.

Keywords: 

conservative, Hamiltonian and dissipative systems, bifurcations, dynamical chaos, FShM theory.

PP. 3-7.

DOI: 10.14357/20790279210201

 

References

1. Magnitskii N.A. Novyy podkhod k analizu gamil’tonovykh i konservativnykh sistem // Differentsial’nyye uravneniya. 20008. T.44. № 12. P. 1618-1627.

2. Magnitskii N.A. Teoriya dinamicheskogo khaosa. M.: Lenand, 2011. 320 p.

3. Magnitskii N.A. Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations. Chapter in Nonlinearity, Bifurcation and Chaos – Theory and Applications. Rijeca: InTech. 2012. P. 133-174.

4. Korol’kova M.A. Bifurkatsionnaya diagramma gamil’tonovoy sistemy Mat’ye-Magnitskogo // Trudy ISA RAN. 2012. T. 62. № 1. P. 69-71.1.