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Gubenko P. P., Koronovskii A. A., Moskalenko O. I. On peculiarities of application of the auxiliary system approach for the generalized chaotic synchronization regime detection. Izvestiya of Saratov University. Physics , 2025, vol. 25, iss. 4, pp. 408-413. DOI: 10.18500/1817-3020-2025-25-4-408-413, EDN: GQHDTE
On peculiarities of application of the auxiliary system approach for the generalized chaotic synchronization regime detection
Background and Objectives: In this paper we consider the peculiarity that arises during the generalized chaotic synchronization regime detection in systems with a complex attractor topology having internal symmetry. Materials and Methods: As the system under study we consider two modified Guckenheimer – Holmes discrete maps coupled unidirectionally. To detect the presence of generalized synchronization we calculate the spectrum of Lyapunov exponents and propose the modification of auxiliary system approach. Results: Considering that the symmetry in the dynamics of the autonomous response system can lead to multistability due to the drive system signal, the correct detection of generalized synchronization with the help of the auxiliary system approach may be complicated. Modification of the generalized synchronization criterion, which allows increasing the reliability of the method in such cases, has been proposed. Conclusion: The results obtained with the help of modified auxiliary system approach are in a good agreement with the calculation of Lyapunov exponents.
- Balanov A. G., Janson N. B., Postnov D. E., Sosnovtseva O. V. Synchronization: From Simple to Complex. Berlin, Springer, 2009. XIV, 426 p. https://doi.org/10.1007/978-3-540-72128-4
- Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Uiversal Concept in Nonlinear Sciences. Cambridge, Cambridge University Press, 2001. XIX, 411 p. https://doi.org/10.1119/1/1475332
- Rulkov N. F., Sushchik M. M., Tsimring L. S., Abarbanel H. D. I. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E, 1995, vol. 51, pp. 980–994. https://www.doi.org/10.1103/PhysRevE.51.980
- Koronovskii A. A., Moskalenko O. I., Selskii A. O. Intermittent generalized synchronization and modified system approach: Discrete maps. Phys. Rev. E, 2024, vol. 109, art. 064217. https://doi.org/10.1103/PhysRevE.109.064217
- Koronovskii A. A., Moskalenko O. I., Hramov A. E. On the use of chaotic synchronization for secure communication. Phys. Usp., 2009, vol. 52, no. 12, pp. 1213–1238. https://www.doi.org/10.3367/UFNe.0179.200912c.1281
- Kulagin N. D., Andreev A. V, Koronovskii A. A., Moskalenko O. I., Sergeev A. P., Badarin A. A., Hramov A. E. Intermittency in predicting the behavior of stochastic systems using reservoir computing. Phys. Rev. E, 2025, vol. 111, art. 024209. https://doi.org/10.1103/PhysRevE.111.024209
- Hramov A. E., Kulagin N. D., Pisarchik A. N., Andreev A. V. Strong and weak prediction of stochastic dynamics using reservoir computing. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2025, vol. 35, art. 033140. https://doi.org/10.1063/5.0252908
- Abarbanel H. D. I., Rulkov N. F., Sushchik M. M. Generalized synchronization of chaos: The auxiliary system approach. Phys. Rev. E, 1996, vol. 53, pp. 4528–4535. https://www.doi.org/10.1103/PhysRevE.53.4528
- Kocarev L., Parlitz U. Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett., 1996, vol. 76, pp. 1816–1819. https://doi.org/10.1103/PhysRevLett.76.1816
- Pyragas K. Properties of generalized synchronization of chaos. Nonlinear Analysis: Modelling and Control (Vilnius, IMI), 1998, no. 3, pp. 101–129. https://doi.org/10.15388/NA.1998.3.0.15261
- Zheng Z., Hu G. Generalized synchronization versus phase synchronization. Phys. Rev. E, 2000, vol. 62, pp. 7882–7885. https://doi.org/10.1103/PhysRevE.62.7882
- Moskalenko O. I., Koronovskii A. A., Hramov A. E. Inapplicability of an auxiliary-system approach to chaotic oscillators with mutual-type coupling and complex networks. Phys. Rev. E, 2013, vol. 87, art. 064901. https://doi.org/10.1103/PhysRevE.87.064901
- González-Miranda J. M. Synchronization of symmetric chaotic systems. Phys. Rev. E, 1996, vol. 53, pp. 5656–5669. https://doi.org/10.1103/PhysRevE.53.5656
- González-Miranda J. M. Bistable generalized synchronization of chaotic systems. Computer Physics Communications, 1999, vol. 121–122, pp. 429–431.
- Pecora L. M., Carroll T. L. Synchronization in chaotic systems. Phys. Rev. Lett., 1990, vol. 64, pp. 821–824. https://doi.org/10.1103/PhysRevLett.64.821
- Guan S., Lai C.-H., Wei G. W. Bistable chaos without symmetry in generalized synchronization. Phys. Rev. E, 2005, vol. 71, art. 036209. https://doi.org/10.1103/PhysRevE.71.036209
- Grebogi C., Ott E., Yorke J. A. Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation. Phys. Rev. Lett., 1983, vol. 50, pp. 935–938. https://doi.org/10.1103/PhysRevLett.50.935
- Proshin Yu. N., Shakirov M. A. Modelirovanie i vizualizatsiya neline’nykh dinamicheskikh system. Chast’ 1. Tochechnye otobrazheniya [Modeling and Visualization of Nonlinear Dynamic Systems. Part 1. Point Mappings]. Kazan, Kazan State University Publ., 2017. 36 p. Available at: https://kpfu.ru/portal/docs/F1367493855/Tochechnye.otobrazheniya.pdf (accessed September 20, 2025) (in Russian).
- Moskalenko O. I., Koronovskii A. A., Selskii A. O., Evstifeev E. V. A Method to detect the characteristics of intermittent generalized synchronization based on calculation of probability of the synchronous regime observation. Tech. Phys. Lett., 2024, vol. 50, pp. 209–212. https://doi.org/10.1134/S1063785023180116
- Hramov A. E., Koronovskii A. A. Intermittent generalized synchronization in unidirectionally coupled chaotic oscillators. Europhys. Lett., 2005, vol. 70, pp. 169–175. https://doi.org/10.1209/epl/i2004–10488–6
- Hramov A. E., Koronovskii A. A. Generalized synchronization: A modified system approach. Phys. Rev. E, 2005, vol. 71, art. 067201. https://doi.org/10.1103/PhysRevE.71.067201
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