Hyperbolic chaos in parametric oscillations of a string
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 3-10
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We outline a possibility of chaotic dynamics associated with a hyperbolic attractor of the Smale–Williams type in mechanical vibrations of a nonhomogeneous string with nonlinear dissipation arising due to parametric excitation of modes at the frequencies $\omega$ and $3\omega$, when the pump is supplied by means of the string tension variations alternately at frequencies of $2\omega$ and $6\omega$.
On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade
Regular and Chaotic Dynamics, 2000, vol. 5, no. 4, pp. 459-476
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We analyse dynamics generated by quadratic complex map at the accumulation point of the period-tripling cascade (see Golberg, Sinai, and Khanin, Usp. Mat. Nauk. V. 38, № 1, 1983, 159; Cvitanovic; and Myrheim, Phys. Lett. A94, № 8, 1983, 329). It is shown that in general this kind of the universal behavior does not survive the translation two-dimensional real maps violating the Cauchy–Riemann equations. In the extended parameter space of the two-dimensional maps the scaling properties are determined by two complex universal constants. One of them corresponds to perturbations retaining the map in the complex-analytic class and equals $\delta_1 \cong 4.6002-8.9812i$ in accordance with the mentioned works. The second constant $\delta_2 \cong 2.5872+1.8067i$ is responsible for violation of the analyticity. Graphical illustrations of scaling properties associated with both these constants are presented. We conclude that in the extended parameter space of the two-dimensional maps the period tripling universal behavior appears as a phenomenon of codimension $4$.
Isaeva O. B., Kuznetsov S. P., On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade, Regular and Chaotic Dynamics, 2000, vol. 5, no. 4, pp. 459-476