Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories

Abstract

Сформулированы уравнения и проведено численное исследование хаотических автоколебаний в системах, построенных на основе тройного шарнирного механизма Тёрстона-Уикса-Ханта-Маккея. Рассмотрены варианты систем с голономной механической связью трех ротаторов и систем, где три ротатора взаимодействуют посредством потенциальных сил. Представлены и обсуждаются характеристики хаотических режимов (показатели Ляпунова, спектры мощности). Хаотическая динамика исследованных моделей ассоциируется с гиперболическим аттрактором, по крайней мере, при условии относительно небольшой надкритичности автоколебательного режима, что следует из проведенного численного анализа распределений углов пересечения устойчивых и неустойчивых многообразий принадлежащих аттрактору фазовых траекторий. В системах на базе ротаторов с потенциальным взаимодействием, начиная с некоторого уровня надкритичности, гиперболичность нарушается.

Numerical test for hyperbolicity in chaotic systems with multiple time delays

“…It means that we could also detect this situation taking the unit matrix as a metric and using the standard dot product instead of Eq. (35). However the angles computed in this way would depend on the discretization step h. The inner product (35) provides a correct asymptotic behavior of the angles as h → 0 when the numerical scheme converges to the original differential equations.…”

“…(35). However the angles computed in this way would depend on the discretization step h. The inner product (35) provides a correct asymptotic behavior of the angles as h → 0 when the numerical scheme converges to the original differential equations. This is the case because Eq.…”

“…[29] its fast and economical reformulation was suggested. This approach may be regarded as an extension of Lyapunov analyses, well-established and applied successively not only for low-dimensional systems but for spatiotemporal systems too [30][31][32][33][34][35][36][37].…”

Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study

“…Hunt and MacKay showed that the free motion of this mechanism with proper selection of parameters provides a feasible example of the Anosov flow on compact two-dimensional manifold of negative curvature [9]. The special case, when the constraint equation is simplified (in certain asymptotics in parameters) and reduces to a condition of vanishing sum of cosines of the three angles of revolution of the rotators about their fixed axes, was discussed in detail in [10,11]. The configuration space in this case is a two-dimensional curved surface, known as the Schwarz P-surface [12].…”

Numerical test for hyperbolicity of chaotic dynamics in time-delay systems

“…This method was suggested initially in Ref. [11] and then developed and used with modifications in [12][13][14][15][16][17][18][19][20][21][22]. It may be regarded as an extended version of Lyapunov analyses, well-established and applied successively not only for low-dimensional systems but for spatiotemporal systems too.…”