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Conference publications

Abstracts

XIX conference

On the period-doubling bifurcation

Sidorov S.V.

Peoples' Friendship University Educational and Research Institute of Gravitation and Cosmology Russia, 117198 Moscow, ul. Maclay, 6.

1 pp. (accepted)

Investigation of the mechanisms of chaotic behavior in dissipative dynamical systems of differential equations is still an urgent task, despite the obvious progress made in solving this problem.

In a study the mechanism of period-doubling bifurcations in nonlinear dissipative systems of ordinary differential equations and partial differential equations of evolution type.

It is shown that in real ODE systems based on the mechanism of period-doubling bifurcation of the limit cycle lies in the appearance of the monodromy matrix of periodic solutions of complex pairs, but not complex-conjugate Floquet exponents. After this bifurcation the real part of one of the Floquet exponents of the pair tends to zero, the other - to minus infinity, while the imaginary part of the pair remain equal  i  / T. In passing one of the Floquet exponents in the positive real half-plane limit cycle loses its stability and double period cycle is born.

Solutions to systems of differential equations with the dimension of more than three may take the form of two-dimensional invariant tori, provided the topological product of two limit cycles: primary and secondary source formed as a result of Andronov-Hopf bifurcation in the buckling of the original cycle. As an example of such a system is considered the second boundary value problem for the nonstationary Ginzburg-Landau equation.

It is shown that in this problem is a bifurcation of type fork, resulting in the loss of stability of the homogeneous periodic solution is born a couple of stable inhomogeneous periodic solutions. Later both of these solutions, losing stability, generate two-dimensional invariant tori, which in the course of the parameter, increasing in size, closer to each other. Solutions are allowed to merge disaster period-doubling bifurcations of the tori in the original primary cycle. Similarly, two-dimensional invariant tori are born with a 4 -, 8 -, etc. multiple of the period, relative to the period of the initial cycle.



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