Hidden oscillation

From a computational point of view, in nonlinear dynamical systems periodic oscillations and chaotic attractors (a neighborhood of which is their attraction domain) can be regarded as self-exciting attractors or hidden attractors.

Here it is essential to consider numerical localization procedures in forward and backward time, since computation in backward time may localize unstable oscillation.

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Self-exciting attractor localization

Classical attractors in the well-known dynamical systems of Van der Pol, Beluosov–Zhabotinsky, Lorenz, Rössler, Chua and many others are self-exciting attractors and can be obtained numerically, with relative ease, by standard computational procedures.

Hidden attractor localization

The simplest examples of hidden oscillations are internal nested limit cycles in two-dimensional systems. Here hidden oscillations can be investigated using analytical methods (see, e.g., results on the second part of Hilbert's 16th problem ). Other examples of hidden oscillations are counterexamples to Aizerman's and Kalman's conjectures on absolute stability in automatic control theory (where unique stable equilibrium points and attracting periodic solutions coexist), which can be constructed for system dimensions not less than three and four respectively.

In 2010, for the first time, a chaotic hidden attractor was discovered [1] [2] in Chua's circuit, which is described by a three-dimensional dynamical system.

In the multi-dimensional case the integration of trajectories with random initial data is unlikely to provide localization of a hidden attractor, since a basin of attraction may be very small and the attractor dimension itself may be much less than the dimension of the considered system. Therefore for numerical localization of hidden attractors in multi-dimensional space it is necessary to develop special analytical-numerical computational procedures ,[3][4][5] which allow one to step away from equilibria (by an analytical method) and to choose initial data in an attraction domain of the hidden oscillation (which does not contain neighborhoods of equilibria) and then to perform trajectory computation there.

References

  1. ^ a b Kuznetsov N.V., Leonov G.A., Vagaitsev V.I. (2010). "Analytical-numerical method for attractor localization of generalized Chua's system". IFAC Proceedings Volumes (IFAC-PapersOnline) 4 (1). doi:10.3182/20100826-3-TR-4016.00009. 
  2. ^ a b Leonov G.A., Vagaitsev V.I., Kuznetsov N.V. (2011). "Localization of hidden Chua's attractors". Physics Letters, Section A 375 (23): 2230–2233. doi:10.1016/j.physleta.2011.04.037. 
  3. ^ Leonov G.A., Vagaitsev V.I., Kuznetsov N.V. (2010). "Algorithm for localizing Chua attractors based on the harmonic linearization method". Doklady Mathematics 82 (1): 663–666. doi:10.1134/S1064562410040411. http://www.math.spbu.ru/user/nk/PDF/2010_DAN_Hidden_Chua_Attractor_Localization_Harmonic_Balance.pdf. 
  4. ^ Leonov G.A., Kuznetsov N.V. (2011). "Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems". Doklady Mathematics 84 (1): 475–481. doi:10.1134/S1064562411040120. 
  5. ^ Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A. (2011). "Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits". Journal of Computer and Systems Sciences International 50 (4): 511–543. doi:10.1134/S106423071104006X. 

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