On a bifurcation scenario of a birth of attractor of Smale–Williams type
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 267-294
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We describe one possible scenario of destruction or of a birth of the hyperbolic attractors considering the Smale—Williams solenoid as an example. The content of the transition observed under variation of the control parameter is the pairwise merge of the orbits belonging to the attractor and to the unstable invariant set on the border of the basin of attraction, in the course of the set of bifurcations of the saddle-node type. The transition is not a single event, but occupies a finite interval on the control parameter axis. In an extended space of the state variables and the control parameter this scenario can be regarded as a mutual transformation of the stable and unstable solenoids one to each other. Several model systems are discussed manifesting this scenario e.g. the specially designed iterative maps and the physically realizable system of coupled alternately activated non-autonomous van der Pol oscillators. Detailed studies of inherent features and of the related statistical and scaling properties of the scenario are provided.
Isaeva O. B., Kuznetsov S. P., Sataev I. R., Pikovsky A., On a bifurcation scenario of a birth of attractor of Smale–Williams type, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 267-294
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